NOTE: The text below basically consists of Chapter 12 in the monograph Quantum Geometry, but with all the formulae left out. It can be understood without reference to the rest of that monograph The numbered superscripts indicate notes that appear at the end of the main text, but ahead of the references.

It is with sorrow that we report that Dr. Eduard Prugovecki passed away at his home in Lake Chapala Mexico on October 13th 2003

Historical and Epistemological Perspectives on
Developments in Relativity and Quantum Theory

What is number? What are space and time? What is mind, and what is matter? I do not say that we can here and now give definitive answers to all those ancient questions, but I do say that a method has been discovered by which . . . we can make successive approximations to the truth, in which each new stage results from an improvement, not a rejection, of what has gone before.
        In the welter of conflicting fanaticisms, one of the unifying forces is scientific truthfulness, by which I mean the habit of basing our beliefs upon observations and inferences as impersonal, and as much divested of local and temperamental bias, as is possible for human beings.

Bertrand Russell (1945)

The founders of relativity theory and of quantum mechanics were as concerned with the epistemological aspects and mathematical consistency of these theories, as they were with their empirical accuracy as reflected by experimental tests. In fact, some of them gave to epistemological scope and soundness preference over immediately apparent agreement with experiment, since they were acutely aware that all raw empirical data are submitted to a considerable amount of theoretical analysis and interpretation, before they are eventually released for publication. Of necessity, all such interpretations reflect the experimentalists' conscious or subconscious biases. Hence, the outcome is prone to various kinds of errors, ranging from systematic ones, due to the faulty design of apparatus or erroneous analysis of the raw data, to the subtle ones, due to misinterpretation or unwarranted extrapolation.
    Nowhere is the setting of priority on sound epistemology ahead of the immediate agreement with experiment better illustrated than in Einstein's (1907) response to Kaufmann's (1905, 1906) negative experimental verdict on Einstein's (1905) at-that-time-just formulated special theory of relativity, and to the claim that the just-acquired experimental evidence provided indubitable verification of Abraham's (1902, 1903) theory of the electron. G. Holton describes that situation as follows: “We know what Einstein did when he heard about Kaufmann's results – one of the foremost experimentalists in Europe disproving this unknown person's work. Einstein did not respond for nearly two years. Finally, ... [in 1907] Einstein wrote that he had not found any obvious errors in Kaufmann's article, but that the theory that was being proved by Kaufmann's data was a theory of so much smaller generality than his own, and therefore so much less probable, that he would prefer for the time being to stay with it. Actually, it took until 1916 for a fault in Kaufmann's experimental equipment to be discovered.” (Holton, 1980, p. 92).
    Einstein himself made clear1 the reasons for his primary concern with epistemological questions when he wrote: “[A physical] theory must not contradict empirical facts. However evident this demand may in the first place appear, its application turns out to be quite delicate. For it is often, perhaps even always, possible to adhere to a general theoretical foundation by securing the adaptation of the theory to the facts by means of additional artificial assumptions.” (Einstein, 1949, p. 23).   
    This fundamental concern with sound epistemology, as reflected by the internal consistency and “elegance” of the advocated theoretical ideas, was exhibited in equal measure by the main founders of quantum theory – as amply witnessed in the writings of Bohr (1934, 1955, 1961), Born, Dirac and Heisenberg. For example, in a paper entitled “Why We Believe in Einstein's Theory?”, Dirac (1980) asserts that the real basis for that belief does not lie merely in the experimental evidence itself; rather: “It is the essential beauty of the theory which, I feel, is the real reason for believing in it.” And, in a similar vein, Heisenberg (1971) comments: “If predictive power were the only criterion of truth, Ptolomy's astronomy would be no worse than Newton's.”    
    Unfortunately, after the Second World War this attitude towards epistemology and foundational issues in quantum physics became reversed2, as leading physicists of the post-war generation obviously decided that, contrary to the opinions of their great predecessors, it was legitimate to secure “the adaptation of the theory to the facts by means of addi-tional artificial assumptions”. Thus, soon after the “triumph” of renormalization theory, Dirac (1951) felt compelled to point out in print that: “Recent work by Lamb, Schwinger and Feynman and others has been very successful . . . but the resulting theory is an ugly and incomplete one.” He reiterated and expanded on this theme on many occasions. For example, in a 1968 lecture entitled “Methods in Theoretical Physics”, in which he explained the methodology and epistemology of his approach to developing new physical theories, he stated3: “The difficulty with divergencies proved to be a very bad one. No progress was made for twenty years. Then a development came, initiated by Lamb's discovery and explanation of the Lamb shift, which fundamentally changed the character of theoretical physics. It involved setting up rules for discarding the infinities, rules which are precise, so as to leave well-defined residues that can be compared with experiment. But still one is using working rules and not regular mathematics. Most theoretical physicists nowadays appear to be satisfied with this situation, but I am not. I believe that theoretical physics has gone on the wrong track with such developments and one should not be complacent about it.” In the end, true to his initial verdict4, in his very last paper he concluded: “I want to emphasize that many of these modern quantum field theories are not reliable at all, even though many people are working on them and their work sometimes gets detailed results.”  (Dirac, 1987, p. 196).

    Although, unfortunately, the many admonitions that were publicly pronounced by Dirac from 1951 until his death in 1984 have remained largely unheeded, the past decade has witnessed a gradual revival of interest in foundational questions. It is hoped that the present monograph will contribute to that revival in a constructive manner, which would reestablish the high standards for mathematical truth and epistemological soundness in science, to which the founders of twentieth century physics devoted their professional lives. Consequently, it is fitting, now that all the basic technicalities implicit in the formulation of quantum geometries have been presented in the preceding ten chapters, to devote this concluding chapter5 to a clearly stated analysis of the epistemological meaning and significance of the physical ideas underlying the present mathematical framework for quantum geometries, framed against the historical background that has shaped those ideas.
    We shall start, therefore, by reviewing the clash of philosophies that marked the rather turbulent early development of quantum theory in the pre-World War II years. We shall then describe the radical shift in values that characterized the post-World War II developments in quantum physics. These historical and sociological factors will help set into the proper perspective the multitude of glaring inconsistencies in conventional relativistic quantum theories, that  have been simply glossed over, or even totally ignored, during the past four decades. After that, we shall analyze the most essential epistemological aspects that underlie the mathematical framework described in Chapters 3-11 of this monograph, keeping those historical perspectives in mind.

12.1. Positivism vs. Realism in Relativity Theory and Quantum Mechanics

The advent of the orthodox interpretation of quantum mechanics in the mid-1920s gave rise to what one of the leading contemporary philosophers of science, K.R. Popper, has called a “schism” in twen-tieth century physics: “The two greatest physicists, Einstein and Bohr, perhaps the two greatest thinkers of the twentieth century, disagreed with one another. And their disagree-ment was as complete at the time of Einstein's death in 1955 as it had been at the Solvay meeting in 1927.” (Popper, 1976, p. 91).
    During the 1920s and 1930s this schism was manifested as a sharp division of the leading physicists of the first half of this century into two camps: the  cohesive Copenhagen school, led by Bohr, which included Heisenberg and Pauli as its other two leading propo-nents, with Born and Dirac as sympathizers, and a disunited opposition to that school, whose most outspoken representative was Einstein, but which also included such distin-guished physicists as Planck and Schrödinger, and which was eventually also joined by de Broglie and Landé. To this day, there are many myths and misconceptions about the posi-tions held by the main protagonists of the various public debates to which this schism has given rise – of which the Bohr-Einstein debate is the best known. This is closely connected to the still prevailing misconceptions about the degree of success which Bohr had in solv-ing the basic epistemological issues confronting quantum theory. Through no fault of Bohr, the myths and misconceptions are in this regard so widespread that M. Gell-Mann once felt compelled to remark that: “Niels Bohr brainwashed a whole generation of physi-cists into thinking that the job [of an adequate interpretation of quantum mechanics] was done 50 years ago” (Gell-Mann, 1979, p. 29).
    A critical examination of the main textbooks on quantum mechanics, which have shaped the beliefs held by most physicists since the thirties, seems to support this bluntly stated charge. Fortunately, in recent years, such publications as those by Popper (1982), MacKinnon (1982), Folse (1985), Murdoch (1989), Selleri (1990), and others, are beginning to set the record straight, by depicting and analyzing, amongst other things, the reasons behind the misconception that Bohr was the “winner” in the Bohr-Einstein debate. On the other hand, these and other similar studies are primarily written by scientific realists, and therefore sometimes tend to give the over-simplified impression of a clash between realism and positivism, with Einstein being cast as the “realist”, and Bohr as the “positivist”. For the more detached observer, who sees merit in both these most important streams in twentieth century philosophy, the situation appears to be considerably more complex.
    First of all, from a broader historical perspective (Mehra and Rechenberg, 1982; Pais, 1982), the above classification of the philosophical beliefs held by Einstein is very much a function of the time period in his life which one chooses to examine. Indeed, in their heyday logical positivists were proud to point out that both special and general relativity were the outgrowth of a positivistic epistemology (cf., e.g., Ayer, 1946), which can be traced to Mach. Even a cursory reading of Einstein's main papers on these subjects confirms their judgment. In fact, if the operationalist attitude is expurgated from Einstein's 1905 paper, which launched special relativity, much of its basic motivation disappears. Indeed, Einstein was not the one to discover the Lorentz transformations; rather, he was the one to give to Lorentz transformations a straightforward operational interpretation, which did not rely on preconceived ideas about the nature of physical reality, in general, and about the intrinsic properties of the electron, in particular – i.e., the type of ideas which Lorentz was advocating at that time. Eventually, that simple and elegant operationalistic approach gave rise to far-reaching consequences, that would have been inconceivable without it. Similarly, Einstein's 1916 paper, in which clas-sical general relativity was formulated in its final form, is operationally motivated and founded, even to the extent that it contains such extreme anti-realist statements as that the “requirement of general co-variance takes away from space and time the last remnant of physical objectivity” (Einstein, 1916, p. 117).
    Thus, in some of his writings Popper had to admit that: “It is an interesting fact that Einstein himself was for years a dogmatic positivist and operationalist.”  (Popper, 1976, p. 96). But then he hastened to add that Einstein “later rejected this interpretation: he told me in 1950 that he regretted no mistake he ever made as much as this mistake.” (ibid., p. 97).
    Regardless of whether Einstein's recantation was as extreme as all that, it remains a historical fact that, on one hand, by 1920 Einstein started to embrace the cause of realism; but, on the other hand, after that time he never came even close to matching any of the great achievements of his 1905–1916 period, during which his entire mode of thinking was heavily influenced by operational considerations. This perhaps contributed6 to the fate en-dured for a long time by Einstein's crown achievement, namely his classical theory of general relativity (CGR). One of the most prominent historians of the subject, J. Stachel, has recently described that fate as follows:
    “From the late 1920s until the late 1950s, general relativity was considered by most physicists a detour well off the main highway of physics, which ran through quantum theory. ... The low estimate of general relativity was not unconnected with the prevalence of a pragmatic attitude toward physics among its practitioners. Only the calculation of a testable number counted as valid theoretical physics. This attitude often was associated with an un-critical acceptance of a positivistic and operationalistic outlook on science. ... In recent years the situation has changed ... Difficulties encountered by the quantum field theory program made theorists more sympathetic to such explorations [as the relationship between general relativity and quantum theory]. Suggestions that the foundations of quantum mechanics might be subject to critical scrutiny and alteration were no longer taken as signs of mental incompetence.” (Stachel, 1989, pp. 1-2).
    Indeed, Bohr's attitude is often depicted as being staunchly positivistic, so that, historically speaking, it is fair to identify the uncritical acceptance of his ideas with an “uncritical acceptance of positivistic and operationalistic outlook on science”. However, the mode of thinking which led Bohr to his complementarity principle was influenced by philosophical ideas which fundamentally transcended the tenets of any form of positivism. In fact, Jammer (1966) seems to have been the first historian of twentieth-century physics to point out the influence of Kierkegaard's existentialistic and irrationalistic philosophy on Bohr. More recently, Folse (1985), Murdoch (1989) and Selleri (1990) have documented this influence via Bohr's father and via his mentor, the Danish philosopher Harald Høffding. However, the fact that Bohr went well beyond a merely “positivistic and operationalistic out-look on science” in his writings should be evident to anyone familiar with logical positivism. Thus, Bohr's insistence that either only sharp position or only sharp momentum can be measured, which has influenced the thinking of entire generations of physicists, has not been so much an outgrowth of operationalism, as much as a reflection of “the impossibility of overcoming the conflict between thesis and antithesis – with a consequent existential pessimism – [which] was one of the cardinal features of existentialist philoso-phy” (Selleri, 1990, p. 348) that subconsciously influenced Bohr's thinking (Folse, 1985). Indeed, as pointed out in Chapter 1, and as further discussed later in this chapter, this “either-or” stance towards measurement outcomes runs counter to what is actually operationally feasible in practice, where, on one hand, truly sharp values of position or momen-tum can never be measured, and, on the other hand, information about unsharp simultane-ous values of both position and momentum is always available to those willing to look for it.
    Thus, the influence of positivism on Bohr, and, in turn, Bohr's direct or indirect in-fluence on entire generations of physicists, is no doubt responsible for the following ver-dict, which still reflects an opinion widely held amongst quantum physicists, and especially those elementary particle physicists who have wholeheartedly embraced the instrumentalist doctrine discussed in the next two sections: “As every physicist knows, or is supposed to have been taught, physics does not deal with physical reality. Physics deals with mathematically describable patterns in our observations. It is only these patterns in our observations that can be tested empirically.” (Stapp, 1991, p. 1). Indeed, a very close collaborator of Bohr confirms the following: “When asked whether the algorithm of quantum mechanics could be considered as somehow mirroring an underlying quantum world, Bohr would answer, ‘There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature’.” (Peterson, 1985, p. 305). On the other hand, Bohr made many statements to the opposite effect, so that a recent analyst of his philosophy has arrived at the following over-all appraisal: “Just as to the religious apologist it is never God's existence which is really at issue, but His nature that needs defense and elaboration, so to Bohr it was never the existence of the objects of quantum mechanical description which was in question, but only how to understand that description.” (Folse, 1985, p. 224). And another recent analyst of his philosophy describes it by the (intentionally) contradictory terms of “instrumentalistic realism”, claiming that "the realist component and the instrumentalist  component are, so to speak, complementary sides to the phenomenon that is Bohr" (Murdoch, 1989, p. 222), but emphasizing that “it would be quite wrong to de-scribe Bohr as a weak instrumentalist” – least of all of the kind that bears any relationship to the brand of strong conventionalistic instrumentalism discussed in the next two sections.
    However, as we shall discuss in Sec. 12.3, Bohr's insistence that “the results of observations must be expressed in unambiguous language with suitable application of the terminology of classical physics” (Bohr, 1961, p. 39), rather than with a suitable application of the terminology of some new language, specifically designed for quantum theory, has nothing to do with positivism, op-erationalism, or any form of empiricism. Perhaps, the dictum that Kant's “inability to conceive of another geometry convinced him that there could be no other” (Kline, 1980, p. 76) could be also applied to Bohr vis-à-vis the possibility of introducing new quantum geometries – or purely quantum languages, in general. Indeed, although Bohr never referred to Kant in his writings, and never even acknowledged any influence of Kant's philosophy on his own, there is a certain parallelism between their epistemic stances: “Bohr's claim that the classical concepts are necessary for an objective description of experience may seem similar to Kant's view that the concepts of space, time, and causality can be known to apply to experienced phenomena a priori. Furthermore his view that these concepts apply only to phenomenal objects and cannot be used to characterize an independent physical reality seems to parallel Kant's ban on the application of these concepts to a transphenomenal reality.” (Folse, 1985, p. 217).
    In any event, one can speculate that the hidden influence of Kierkegaard's existentialist philosophy might have removed from Bohr any incentive to look into new nonclassical possibilities with regard to the geometries adopted in quantum mechanics. For the success of such an enterprise would have threatened to resolve the “conflict between thesis and an-tithesis”, to which Bohr was exposed during his formative years. In Bohr's mind, such a “conflict” might have very well taken the form of the con-flict between sharp simultaneous measurements of position and momentum, of various spin components, etc. And the exis-tentialist side of Bohr might have been predisposed to see this “conflict” as a manifestation of a “complementarity prin-ciple”, rather than allow for the possibility of realistic, and there-fore necessarily unsharp values for those quantities to be in-corporated into new mathemati-cal frameworks, designed specifi-cally for the needs of quantum physics.
    In Secs. 12.4 and 12.6 we shall argue that it is not only possible7, but even neces-sary, to combine the mutually consistent aspects of the “classical realism” (Folse, 1985) advocated by Einstein in his later years with the “existentialistic positivism” of Bohr, in order to arrive at an epis-temology capable of consistently dealing with relativistic quantum theory. Indeed, the fact that both protagonists in this great debate went to unwarranted extremes becomes evident as soon as we take a second look at the epistemology of classical general relativity. Thus, while Bohr kept insisting that the language of “classical” physics was absolutely essential to describe all experimental data, he arbitrarily restricted himself to nonrelativistic classical physics, even when discussing essentially relativistic phenomena, such as the purportedly instantaneous propagation of measurement effects in the EPR paradox. On the other hand, as we have seen in Secs. 11.1 and 11.4, the epistemological ques-tions in CGR concerning what is observ-able are by no means straightforward when viewed through the lense of nonrelativistic physics – as the confu-sion surrounding Einstein's “hole argument” vividly illustrates (Stachel, 1989).
    An implicit assumption of Bohr's epistemology was that the basic language of physics, required for the communication of experimental data, is static at the historical level. Thus, he asserted that: “Strictly speaking, the mathematical formalism of quantum theory and electrodynamics merely offers rules of calculation for the deduction of expectations about observations obtained under well-defined experimental conditions specified by classical physical concepts”. From his published debate with Einstein and his other writings (Bohr, 1955, 1962), it is clear that the “classical physical concepts” he had in mind were steeped in Newtonian classical physics, rather than reflecting those of classical general relativity – where point coincidences represent the most fundamental reflection of physical reality. Furthermore, it is an obvious and basic fact that every language, including our “everyday” language, constantly grows as it incorporates new concepts, that not only were not conceived, but might have been even unimaginable to earlier generations speaking that language8. It is therefore not unfair to conclude that it is dogmatic to insist that the language of Newton's classical mechanics, taken in conjunction with the “everyday” language of any given era in human history, is the one and only language capable of describing all con-ceivable “experimental conditions”.
    As pointed out in Sec. 1.3, the other leading proponents of the Copenhagen school were more than willing to look well beyond Bohr's “terminology of classical physics” in the search for solutions to the new problems raised by quantum theory, that were not shared by classical physics. Indeed, Heisenberg was one of the first proponents of the introduction of a fundamental length in quantum physics, whereas Born's maxims cited in Sec. 1.1 actually paved the way for the introduction of fundamentally indeterminate values of quantum observables, which underlie Principle 1 in Sec. 1.3.

12.2. Conventionalistic Instrumentalism in Contemporary Quantum Physics

While advocating, as the undisputed leader of the Copenhagen school, his peculiar mixture of positivism,  realism, and existentialism, Bohr unfortunately did not anticipate the long-range effects of his teachings on all those in the future generations of physicists who lacked the philosophical training or the sophistication required to distinguish between subtle philosophical nuances (Murdoch, 1990, Chapter 10) and their gross over-simplifications. Such physicists condensed Bohr's entire philosophy into simplified enunciations9 of the principles of complementarity, wave-particle duality and the purportedly “classical nature” of the “apparatus”, and simply ignored the rest. Indeed, what Karl Popper calls the “third group of physicists”, who emerged right after World War II, and soon became the overwhelming majority, is described by him as follows: “It consists of those who have turned away from  discussions [concerning the confrontation between positivism and realism in quantum physics] because they regard them, rightly, as philosophical, and because they believe, wrongly, that philosophical discussions are unimportant for physics. To this group belong many younger physicists who have grown up in a period of over-specialization, and in the newly developing cult of narrowness, and the contempt for the non-specialist older generation: a tradition which may easily lead to the end of science and its replacement by technology.” (Popper, 1982a, p. 100). Upon labeling the attitude of this “third group of physicists” a form of instrumentalism, Popper goes on to say: “But this instrumentalism, this fashionable attitude of being tough and not standing for any nonsense – is itself an old philosophical theory, however modern it may seem to us. For a long time the Church used the instrumentalist view of science as a weapon against a rising science ... [as can be seen in the] argument with which Cardinal Bellarmino opposed Galileo's teachings of the Copernican system, and with which Bishop Berkeley opposed Newton. ... Thus instrumentalism only revives a philosophy of considerable antiquity. But modern instrumentalists are, of course, unaware that they are philosophizing. Accordingly, they are unaware of even the possibility that their fashionable philosophy may in fact be uncritical, irrational, and objectionable – as I am convinced that it is.” (ibid., pp. 102-103).
    One does not have to subscribe to the tenets of Popper's realism – or, for that matter, of any of the various coexisting brands of philosophical “realism” (d'Espagnat, 1989) – to agree with these assessments. In fact, some of his observations not only receive support from the statements of the founders of quantum theory (Dirac, Heisenberg, Born, etc.), cited earlier in this monograph, but were unwittingly echoed by one of the most outstanding members of the “third group of physicists” in the following statement: “The post-war developments of quantum electrodynamics have been largely dominated by questions of formalism and technique, and do not contain any fundamental improvement in the physical foundations of the theory.” (Schwinger, 1958, p. xv). Unfortunately, this and other similar statements by one of the most outstanding and talented theoretical physicists of the post-World War II era, have not had any deeper impact on those of his contemporaries who belonged to the group of “younger physicists who have grown up in a period of over-specialization”. In fact, one cannot help but agree with Popper as he arrives at the following pessimistic assessment of the post-World War II developments in quantum physics:
    “A very serious situation has arisen. The general anti-rationalist atmosphere which has become a major menace of our time, and which to combat is the duty of every thinker who cares for the traditions of our civilization, has led to a most serious deterioration of the standards of scientific discussion. It is all connected with the difficulties of the theory – or rather, not so much with the difficulties of the theory itself as with the difficulties of the new techniques which threaten to engulf the theory. It started with brilliant young physicists who gloried in their mastery of the tools and look down upon us amateurs who struggle to understand what they are doing and saying. It became a menace when this attitude hardened into a kind of professional etiquette. But the greatest among the contemporary physicists never adopted such an attitude. This holds for Einstein and Schrödinger, and also for Bohr. They never gloried in their formalism, but always remained seekers, only too conscious of the vastness of their ignorance.” (Popper, 1982a, p. 156).   
    Historically, this “very serious situation” began with the wholehearted acceptance by the new post-World War II generation of physicists of an algorithmic scheme for removing “infinities” from the perturbation expansion for the S-matrix in quantum electrodynamics (QED) – the same QED that was founded by Dirac (1927), but in whose formulation he began to publicly express doubts already in the mid-1930s (cf. Sec. 9.6). Indeed, after coming upon certain experimental discrepancies, in his habitual forthright and decisively uncompromising manner, which he used even with regard to his own theories, Dirac stated the following: “The only important part [of theoretical physics] that we have to give up is quantum electrodynamics ... We may give it up without regrets ...; in fact, on account of its extreme complexity, most physicists will be glad to see the end of it.” (Dirac, 1936). However, Dirac invested an additional ten years of hard work aimed at trying to come to grips with the infinities in QED by studying classical electrodynamics, only to eventually come “to the view that the infinities are a mathematical artifact resulting from expansions in [the coupling constant] a that are actually invalid (Dirac, 1946).” (Pais, 1987, p. 106).
    Consequently, as opposed to the new post-World War II generation of physicists, Dirac remained totally unimpressed by the numerical successes of the renormalization pro-gramme in QED. As mentioned in the introductory remarks to this chapter, he declared from the outset: “Recent work by Lamb, Schwinger and Feynman and others has been very successful . . . but the resulting theory is an ugly and incomplete one.” (Dirac, 1951). And, as seen from the many quotations of Dirac's words in this monograph, and as extensively documented in his recent biography (Kragh, 1990), throughout the remainder of his life he never wa-vered in the verdict that “these [renormalization] rules, even though they may lead to results in agreement with observations, are artificial rules, and I just can-not accept that the present foundations [of relativistic quantum theory] are correct.” (Dirac 1978a, p. 20).
     That this verdict is a fair and correct one was confirmed by one of the main founders of the conventional renormalization programme, when he stated: “The observational basis of quantum electrodynamics is self-contradictory . . .  .  We conclude that a convergent theory cannot be formulated consistently within the framework of present space-time concepts.” (Schwinger, 1958, pp. xv-xvi).
    Indeed, how can one possibly arrive at any other verdict if one is rationally considering the following plain facts: 1) QED, as well as all other “renormalizable” conventional quantum field theories, are formulated in terms of quantum field operators which do not exist as functions of points in Minkowski space (cf. [BL], Sec. 10.4); 2) a list of renormalization rules are derived, however, as if those quantum fields at a point did have a mathematical meaning (cf. [IQ], Sec. 6-1); 3) the practitioner of the conventional renormalization is asked to implement them, without concern for mathematical consistency or epistemological validity, but by using them as algorithmic rules for subtracting divergencies from originally meaningless integrals10; 4) those finite expressions are then claimed to provide the terms of a perturbation “expansion” – but after more than forty years there is no proof that the objects which one “expands”, namely the S-matrix elements for various processes, actually exist in any well-defined mathematical sense; 5) in fact, not only does the “perturbation series” not converge, but the generally accepted conjecture in QED, as well as in other conventional quantum field theoretical models, is that this “perturbation expansion” is an asymptotic series (Dyson, 1952); 6) however, in the absence of a proof of the existence of an S-matrix – i.e., of functions in relation to which such series are supposedly asymptotic – the concept of “asymptotic series” is itself mathematically meaningless11; 7) after a protracted effort of more than twenty years, constructive quantum field theoretical attempts at imparting rigorous mathematical meaning to the conventional renormalization procedure has resulted merely in the conclusion that the S-“matrix” in QED, as well as in other “renormalizable” conventional quantum field theories in four spacetime dimensions, is most probably trivial, i.e., equal to the identity matrix (Glimm and Jaffe, 1987, p. 120).
    On purely rational grounds, it might have been expected that even before this last bit of distressing information became available in the 1980s, Dirac's public admonishments and Schwinger's remarks would have already been taken to heart in the early 1950s, and a concerted effort would have been mounted to investigate the foundations of relativistic quantum mechanics in general, and of quantum field theory in particular. However, as is well-known, that is not at all what took place. Rather, in the mid-fifties a parade of changing fashions began to unfold in elementary particle physics, and is still continuing unabated to the present day12. During most of this period the prevailing belief was that these developments led to predictions which were “in agreement with experiment” – which, in a gross over-simplification and distortion of Bohr's teachings, was viewed as the ultimate arbiter of the validity of the various (and many transiently) fashionable theories. However, not only has it been repeatedly demonstrated  that the analysis of experimental results can be wrong, theoretical computation can be incorrect, and the very comparison between theory and experiment can be faulty, but as Heisenberg acerbicly pointed out on one occasion, “if predictive power were indeed the only criterion for truth, Ptolemy's astronomy would be no worse than Newton's” (Heisenberg, 1971, p. 212).
    Indeed, in addition to Dirac, the only other founding father of quantum theory who lived to see these developments expressed his dismay and disapproval in an article which was published at the very same time that his death was announced to the professional world of physics. The second paragraph of this article contains the following declaration: “I believe that certain erroneous developments in particle theory – and I am afraid that such developments do exist – are caused by a misconception by some physicists that it is possible to avoid philosophical arguments altogether. Starting with poor philosophy, they pose the wrong questions. It is only a slight exaggeration to say that good physics has at times been spoiled by poor philosophy.” (Heisenberg, 1976, p. 32).
    As we mentioned earlier, Karl Popper very appropriately classified this type of “poor philosophy” as a form of instrumentalism, and described it as “the view that there is nothing to be understood in a [scientific theory]: that we can do no more than master the mathematical formalism, and then learn how to apply it.” (Popper, 1982a, p. 101).
    However, the cavalier manner in which mathematics itself has been treated after the inception of the renormalization programme, indicates that instrumentalism per se is not actually solely responsible for this considerable decline in the standards for establishing truth in science. Indeed, ever since the advent of renormalization theory in QED, in quantum theory (as opposed to CGR) “mastering a mathematical formalism” has meant developing the computational skills to algorithmically derive from the “theory” numerical “predictions”. Such practices require the uncritical acceptance of a series of computational “working rules”, or, at the very least, the acceptance of the most subjective types of criteria for their mathematical validity – even when those criteria run counter to all deductive norms accepted in contemporary mathematics. In Dirac's words, these practices represent “a drastic departure from ... logical deduction to a mere setting up of working rules.” (Dirac, 1965, p. 685). In fact, as we shall see from the examples cited in the next section, ever since the advent of renormalization theory, the prevailing attitude became to ignore objective mathematical criteria of truth and consistency, and to substitute instead conventionally acceptable mathematical procedures, i.e., formal computational rules conventionally deemed to produce valid results as long as they were declared as acceptable by those whom Dyson (1983) describes as the “mandarins” of the post-World War II generation of physicists14.
    It therefore seems appropriate to categorize this kind of approach to science by the more precise label of conventionalistic instrumentalism. This label is intended to reflect the fact that its general practices ignore or dismiss not only all the truth-values which scientific realism expects to be fulfilled by physical theories (cf. Murdoch, 1989, pp. 200-201), but even the most basic forms of mathematical truth – replacing them with mere conventions. As reflected by the activities of the “mainstream” in quantum theory, such conventions are primarily based on the consensus prevailing amongst the leading physicists of the present instrumentalist period in quantum physics as to what types of computational procedures are “acceptable”. As such, the conventionalistic aspects of this form of instrumentalism should be strictly distinguished from Poincaré's type of conventionalism, mentioned in Sec. 1.2 (which indeed viewed the choice of geometries suitable for the description of the physical world as being a matter of convention, but otherwise reflected a deep respect for objective mathematical truth, and a love for mathematical beauty on a par with that displayed by Dirac15), or from contemporary forms of conventionalism in philosophy, which view all statements in logic and mathematics as being purely analytic, and as such emerging from linguistic conventions (Quine, 1949). It should also be distinguished from logical positivism, especially in view of the fact that when some contemporary authors discuss the foundations of quantum theory, they tend to identify the term “instrumentalism” with the so-called “positivism of physicists” (d'Espagnat, 1989, p. 28). Indeed, although when viewed as general philosophies, as well as working philosophies applied to science, logical positivism and instrumentalism share some common points, they are fundamentally distinct in many aspects – as will become apparent from the considerations in the next section.
    Is conventionalistic instrumentalism an intrinsically unavoidable feature of contemporary quantum physics?
    The preceding nine chapters of this monograph are meant to prove that it is not. On the other hand, some other critics of instrumentalism in contemporary quantum physics seem very anxious to affirm that the brand of instrumentalism which has become “the foundation stone of contemporary physics” in the second half of this century, “has been astonishingly successful in various fields, from elementary particle physics to astrophysics, quantum optics, solid state physics” (d'Espagnat, 1989, p. 28). It is therefore argued that “it is hardly surprising that physicists should see this foundation stone as being very solid and providing a basis for objective reality, [so that merely] a number of philosophers of science are not of the same opinion. ... The truth here is perhaps that since there is a high level of instrumentalist technical sophistication which science apparently cannot legitimately avoid, there is a gap of some kind between the theoretical physicist's activities and his thinking. Either he thinks or he develops physics.” (ibid, pp. 28-31) – emphasis added.
    The above type of rationalization of the prevailing instrumentalist attitudes in quantum physics seems, however, to ignore a fact amply demonstrated by the founders of relativity and quantum theory, namely that a theoretical physicist can both think and develop outstanding physics – and, in fact, that the first activity is necessary for the second. This monograph is dedicated to the memory of P. A. M. Dirac, since he was the most outspoken and persistent of the critics of the values and practices of conventionalistic instrumentalism in quantum field theory. However, he most certainly was not alone in his critical attitude towards these types of developments in post-World War II physics (cf. Note 23 to Chapter 9). Indeed, it is an acknowledged historical fact that “the workers of the 1930s, particularly Bohr and Dirac, had sought solutions to the problems [of quantum field theory] in terms of revolutionary departures. ... The solution advanced by Feynman, Schwinger, and Dyson was at its core conservative: it asked to take seriously the received formulation of quantum mechanics and special relativity and to explore the content of [their] synthesis. A generational conflict manifested itself in the contrast between the revolutionary and conservative stances of the pre- and post-World War II theoreticians.” (Schweber, 1986, p. 299).
    In Chapters 3, 5, 7, 9 and 11 we have provided extensive evidence that no consistent “synthesis” of these two fields was ever achieved in the context of conventional theory – albeit a public relations campaign was launched after the advent of conventionalistic renormalization theory, meant to convince everybody that such a “synthesis” had already become fait accompli. In the next section we shall demonstrate that the problems that have been left open by this “renormalization” theory are deep rooted. Until recently this PR campaign had, however, by and large succeeded to gloss them over with a glittering veneer of formal manipulations, protected from closer scrutiny by the nurturing of a cavalier attitude towards all the basic tenets of mathematical truth and deductive validity. Indeed, amongst many “mainstream” quantum physicists, it only very recently became true that “suggestions that the foundations of quantum mechanics might be subject to critical scrutiny and alteration [are] no longer taken as signs of mental incompetence” (Stachel, 1989, p. 2). In the meantime, “old” unsolved problems remained deeply entrenched, but were left untouched, due to a systematic neglect of the foundations of quantum physics. That neglect can be clearly perceived (Bell, 1990) in the mainstream textbooks on quantum mechanics and quantum field theory. In particular, as will be illustrated in the next section, it is especially evident in the manner in which much of the required mathematics is treated in them.

12.3. Inadequacies of Conventionalistic Instrumentalism in Quantum Physics

In contemporary philosophy, the term “instrumentalism” is primarily applied to the theory about the nature of truth and falsehood advocated by John Dewey, which emerged on the North American continent as a natural extrapolation of the pragmatism of C.S. Peirce and William James (1970) – cf. (Mackay, 1961). As seen by a contemporary elementary particle physicist: “James argued at length for a certain conception of what it means for an idea to be true. This conception was, in brief, that an idea is true if it works.” (Stapp, 1972, p. 1103). In turn, John Dewey adapted this pragmatic criterion for truth in philosophy and science, as well as in everyday life, as being that which “works satisfactorily in the widest sense of the word”, and based his instrumentalist concept of “truth” on the achievement of consensus. Thus, in scientific applications: “The significance of this viewpoint for science is its negation of the idea that the aim of science is to construct a mental or mathematical image of the world itself. According to the pragmatist view, the proper goal of science is to augment and order our experience. A scientific theory should be judged on how well it serves to extend the range of our experience and reduce it to order.” (ibid., p. 1104).
    Such a principal criterion for judging a scientific theory can have some rather undesir-able social consequences. Indeed, in his “History of Western Philos-ophy” Bertrand Russell writes that Dewey “quotes with approval Peirce's definition: ‘Truth’ is ‘the opinion which is fated to be ultimately agreed to by all who investigate’.” (Russell, 1945, p. 824). Then, upon  demonstrating the logical untenability of the criterion that “an idea is ‘true’ so long as to believe it is profitable to our lives”16, he concludes the chapter on the philosophy of John Dewey with the following critical observations: “The concept of ‘truth’ as something dependent upon facts largely outside human control has been one of the ways in which philosophy hitherto has inculcated the necessary element of humility. When this check upon pride is removed, a further step is taken on the road towards a certain kind of madness – the intoxication of power which invaded philosophy with Fichte, and to which modern men, whether philosophers or not, are prone17. I am persuaded that this intoxication is the greatest danger of our time, and that any philosophy which, however unintentionally, contributes to it is increasing the danger of vast social disaster.” (Russell, 1945, p. 1828).
    Thus, the emergence of conventionalistic instrumentalism as the officially undeclared, but functionally prevalent philosophy amongst quantum physicists of the post-World War II generation, might indeed represent a manifestation18 of the “general anti-rationalist atmosphere which has become a major menace of our time” (Popper, 1982a, p. 156). And that in the eyes not only of such advocates of realism as Popper (1983), but also of those who accept the stan-dard criteria of truth and deductive validity in mathematics19, and yet believe that quantum mechanics and quantum field theory are very important and fundamental theories in science, in which the traditional standards of Truth should be preserved.
    Indeed, the initial indifference of the undeclared adherents to conventionalistic instrumentalism towards the criticisms from Dirac, Heisenberg, and other leading physicists of the pre-World War II generation (i.e., from the very founders of quantum mechanics and quantum field theory), ultimately proved to be only a preamble to the eventually prevailing institutional intolerance in the most active areas of quantum physics towards anything that was out of step with the prevailing instrumentalist conventions. This intolerance manifested itself most clearly in the new criteria for acceptance of papers in major physics journals – which began to favor those based on sheer formal computations at the expense of those emphasizing mathematically and conceptually sound arguments – as well as by the cavalier manner in which relevant mathematics was treated in the most popular textbooks on quantum theory. It also manifested itself as a breakdown of the close contact and communication20 between physicists and mathematicians, which, from Newton's era to Einstein's time, has been underlying all significant progress in theoretical physics21. In fact, it is only in the course of the 1980s that new channels of communication have reopened between some of the leading physicists of the younger generation and some leading mathematicians – cf., e.g., (Witten, 1988), (Atiyah, 1990), (Nahm et al., 1991). On the other hand, in addition to exhibiting foundational weaknesses (Bell, 1990), many of the mathematical standards exhibited by conventionally oriented quantum theoretical textbooks and practices are still rather distant from those acceptable in contemporary mathematics22.
    The most serious breaches of basic mathematical standards of consistency occur in relativistic quantum theory. However, telltale signs are already apparent in the nonrelativistic context. Since, in some of the preceding chapters, we have extensively discussed and analyzed the main failings of conventional relativistic quantum theory, let us now focus our attention for a while on the deficiencies of the conventionalistic approach to nonrelativistic quantum mechanics – illustrating in the process how, by violating the laws of standard mathematics, even some rather basic and crucial physics can be misrepresented.
    We shall devote most of that attention to the deficiencies exhibited by the treatment which this subject receives in mainstream textbooks. Indeed, such textbooks not only reflect prevailing standards, but also shape and instill them in the minds of new generations of physicists. We shall strive to provide by means of readily comprehensible, and therefore of necessity elementary examples, a demonstration of the fact that the indiscriminate use, in professional practice, of the instrumentalist idea of “truth” can lead to a poor understanding of fundamental issues. In everyday practice, such a misunderstanding is then maintained by institutionally reinforcing conformity (namely what Feynman (1954) colorfully called the “pack effect”) by a variety of means – ranging from the criteria used in the refereeing of research papers submitted for publication in leading professional journals, to the standards applied during the allocation of research grants and other forms of financial support23. Naturally, with such means of “persuasion”, the criterion that “truth” is “the opinion which is fated to be ultimately agreed to by all who investigate” is certainly “destined” to prevail.
     Two years after Dirac published his justly famous textbook entitled “Principles of Quantum Mechanics”, the German original of the “Mathematical Foundations of Quantum Mechanics” by von Neumann (1932) made its appearance. In it, von Neumann provided rigorous mathematical justification for many of the heuristic procedures used by Dirac – who, naturally, as a physicist totally involved with the various very rapidly expanding fields of quantum theory, was in no position to follow developments in functional analysis, which was emerging at that time as a new and separate discipline in mathematics. It might have been expected, however, that once the period of rapid growth in nonrelativistic quantum theory had came to an end – as it most certainly did by the end of the 1940s – all the subsequently written and published textbooks in quantum mechanics would begin to reflect at least the main lessons that could be learned from von Neumann's outstanding monograph – whose translation in English was eventually published in 1955.
    That, however, did not take place at that time – and has still not taken place even in the most recent mainstream textbooks on nonrelativistic quantum theory24. This clearly demonstrates how the instrumentalistic identification of mathematical and other forms of “truth” with “generally held opinion” and “professional consensus” can act as a bulwark against true progress in the understanding of the basic structure of quantum theories.
    An elementary but notable example of the deficient mathematical standards prevalent in main-stream textbooks is the treatment of those quantum mechanical observables which are represented by unbounded self-adjoint operators – such as is the case with the majority of important observables, namely energy, position, momentum, (external) angular momentum, etc. According to a theorem by Hellinger and Toeplitz25, no such operators can be defined on the entire Hilbert space of a quantum system, which as a rule is separable but not finite-dimensional. However, not only is this most basic mathematical fact, which was very clearly emphasized already by von Neumann (1932, 1955), not mentioned at all in any of the mainstream textbooks on quantum mechanics, but the student of quantum theory is as a rule left with the false impression that every state vector of the quantum system is in the domain of definition of these operators.
    While the failings of the conventionalistic approach to this type of problem might be deemed innocuous – as it rarely gives rise directly to physically incorrect conclusions – we shall see that there are other closely related problems which lead to physically questionable, and even to false physical conclusions. In fact, one of the sources of the foundational difficulties encountered by conventional relativistic quantum mechanics can be traced to its purely conventionalistic treatment of eigenfunction expansions for position and momentum operators in nonrelativistic quantum mechanics, which ignores some very essential mathematical as well as physical points. Let us therefore first examine the key aspects of this treatment on a few simple examples.
    As is well-known, in the configuration representation the elements of eigenfunction expansions for position and momentum are given by delta-“functions” and plane waves, respectively. Thus, in the simple case of a single nonrelativistic quantum particle of zero spin, one conventionally writes:

formula    (3.1)

It is clear, however, that neither the delta-“functions”, nor the plane waves, are Lebesgue square-integrable functions [PQ], so that they do not belong to the Hilbert space with the inner product defined in (3.1.1). For that reason, von Neumann (1932) avoided the use of delta-“functions”. Eventually their mathematical nature was, however, totally clarified by L. Schwartz (1945). The mathematically correct general treatment of the objects in (3.1) was subsequently supplied by the theory of rigged Hilbert spaces (Gel'fand et al., 1964, 1968), as well as that of equipped Hilbert spaces (Berezanskii, 1968, 1978). These mathematical frameworks pinpoint the objects in (3.1) as elements of eigenfunction expansions – and not as eigenvectors of Hilbert space operators. Adaptations of both these general frameworks to the needs of quantum physics have actually been in existence for quite a while (cf., e.g., Antoine, 1969, 1980; Prugovecki, 1973). Regardless of which one of these particular frameworks one adopts, they all underline the fact that

formula    (3.2)

where H  is, in general, a topological vector space which provides an extension of the Hilbert space H of state vectors. The space H+  is dense in H  in the norm topology of H, and it is equipped with a topology that is finer than the norm topology of H, and which makes H equal to the dual of  H+ (whereas H  can be identified with its own dual H*).
    The key point, that had become clear a couple of decades after the appearance in 1930 of Dirac's famous textbook, is that these eigenfunctions do not provide resolutions of the identity operator 1 in the Hilbert space H of state vectors, but, strictly speaking26, only of the identity operator 1+  in  H+, i.e.,
formula    (3.3)

Furthermore, the choice of  H+  is generally dictated by mathematical convenience, rather than by general physical principles. The use of the round brackets in (3.3) is, therefore, meant to emphasize that, although the theory of equipped Hilbert spaces allows us to write

formula    (3.4)

the sesquilinear form on the left-hand side of the above relation is not an inner product. In fact, the domain of definition for the variable on its right-hand side cannot be extended to the entire Hilbert space H  – as is the custom in all conventional literature which adopts an instrumentalist attitude towards mathematical truth. However, that this feature of the sesquilinear form in (3.4) is an unavoidable mathematical fact follows from another basic mathematical fact: the generic element of H is not a single function, but rather an equivalence class of almost everywhere (in the Lebesgue sense [PQ]) equal functions, which are such that one can change the value of any one of these functions at any given point x without leaving that equivalence class – namely, in physical terms, without changing the quantum state vector. Upon restricting oneself to mathematically convenient27 dense subspaces H+, one can choose representative functions for which (3.4) holds true – but that is not possible globally on H. Thus, strictly speaking, one can speak of the probabilities (3.1.7) for sharp position measurement outcomes within Borel regions B in configuration space, but not of probability densities for arbitrary wave functions at single points in configuration space. For that reason, von Neumann concentrated on the probability measures in (3.1.7), rather than on the probability densities in (3.5.1).
    This seemingly innocuous mathematical point has significant physical repercussions. Thus, although the conventionalistic custom is to refer to |x> “ as an “eigenvector” of the nonrelativistic position operators, and to consider the left-hand side of (3.4) a “transition probability” purportedly corresponding to a sharp measurement of position, we see that actually these “transition probabilities” are not generically well-defined at the mathematical level. Does that mean that they are not well-defined also operationally, at a physical level?
    That does not immediately follow, but the above points indicate that caution should be exercised even in nonrelativistic quantum mechanics, and that one should regard sharp localization as a limit of realistic measurement procedures, which necessarily entail only unsharp localizations. In fact, the adaptation to position measurements of the Wigner-Araki- Yanase (1952, 1960) arguments on the impossibility of arbitrarily precise measurements of quantities which do not commute with an additive conserved quantity (i.e., with momentum, in the case of position measurements), shows that sharp localization is unachievable not only in practice, but also in principle, even in the context of the nonrelativistic quantum theory of measurement (Busch, 1985b). Hence, the fundamental impossibility of sharp relativistic localization of quantum systems, discussed in Secs. 3.3 and 3.5, has its roots in nonrelativistic quantum mechanics – but that fact is conventionally ignored.  
    It might be believed that these rather elementary observations are of no deeper consequence, since the conventionalistically predisposed quantum theorist can in practice easily avoid all the ensuing pitfalls. We shall, therefore, now present two elementary examples which demonstrate that this is not always the case.
    First, it should be recalled that the EPR paradox was originally formulated (Einstein et al., 1935) in the language of sharp position and momentum measurements, based on the above interpretation of the quantities in (3.1) and (3.4) as bona fide transition probabilities, and that it was only later adapted by Bohm (1951) to measurements of spin – but with the original epistemic assumption of (an arbitrarily close) realizability of sharp measurement outcomes retained. This led to Bell's inequalities, whose first experimental tests were performed in the 1970s. However, it was only with the experiments of Aspect et al. (1981, 1982) that the basic issue of nonexistence of local hidden variables was settled in favor of quantum mechanics. On the other hand, the discussion of the consequences of those experiments for the concept of locality is still going on unabated as if the macroscopic concept of arbitrary precise localization could be transferred without major revisions to the microdomain, so that microscopic localizability could be identified with macroscopic separability (Selleri, 1990, p. 202). However, in Chapters 1 and 3 we reviewed conclusive evidence to the effect that such a transference leads to definite contradictions with the concept of Einstein causality – which is the hub of the ongoing disputes (van der Merwe et al., 1988; Tarozzi and van der Merwe, 1988; Kafatos, 1989) about the significance of the EPR paradox. Once the impossibility of such transference is generally acknowledged, the focus of these debates could be shifted to posing the EPR problem in an epistemologically correct manner – namely as a natural by-product of the need for using at the microlevel geometries specifically designed to take the fundamental quantum features of localizability into account from the outset, and dispense with the interpretation of (3.4) as a literal representation of a transition probability amplitude for “observing” a “quantum particle at x”.
    A second illustration of physical misconceptions that have resulted from the same type of in-terpretation of elements of eigenfunction expansions as “transition probability amplitudes” is provided by the conventionalistic derivation of such a most basic formula as that for the differential cross-section in two-body nonrelativistic scattering theory.
    First of all, it should be noted that the conventionalistic approach tends to favor the stationary, i.e., time-independent formulation28, despite the fact that the time-dependent approach comes much closer to reflecting physical reality by treating the scattering operator S as related to an idealization of a scattering process – namely as a process which evolves in Newtonian time t, but entails the physically unachievable limits of t tending to infinity. This preference of stationary methods is, however, not accidental, since the S-matrix program of the 1960s (cf. Notes 35-36) was headed by elementary particle physicists whose advocacy of instrumentalist standards in physics eventually led to the conjecture that the entire concept of space-time might be just a macroscopic “illusion” (Kaplunowski and Weinstein, 1985).
NOTE: The indented text below, which is missing the formulae that could not be reproduced in html, can be skipped.
    In keeping with such attitudes (which for a while threatened to prevail in all of quantum physics), in mainstream textbooks on quantum mechanics one typically begins the derivation of the aforementioned differential cross-section by considering the asymptotic expansion (cf., e.g., Messiah, 1961, p. 371)

formula    (3.5)
of an incoming distorted plane wave, which represents an eigenfunction (in the extension to H) of the total internal Hamiltonian of the two-body system (cf. [PQ], pp. 425-436 and 553-556). One then conventionalistically interprets the plane wave on the right-hand side of (3.5) as a “probability amplitude” that gives rise, in accordance with (3.5.7), to a current density k/m. This current density is again conventionalistically interpreted as representing the incident flux of an incoming beam; whereas the term between square brackets is similarly interpreted as a probability amplitude of an outgoing (scattered) spherical wave. Then, treating, again by convention, the plane wave and the spherical wave as if they were not superimposed, and hence neglecting the cross term resulting from that superposition – typically on grounds that it “oscillates very rapidly as a function of r as r  becomes large” (Joachain, 1975, p. 51) – one arrives at the well-known formula

formula    (3.6)

for the differential scattering cross section in the “center-of-mass reference frame” of the two-body system – where the expression on the right-hand side of the first equation in (3.6) is the so-called T-“matrix”.
    The physical meaning of the “center-of-mass reference frame” is not questioned in such derivations, as it is taken for granted that “somehow” classical concepts still apply. When some fundamental difficulties with this type of conventionalistic derivation of (3.6) were pointed out by Band and Park (1978), it was, however, acknowledged by the author of one of the leading mainstream textbooks on quantum scattering theory that: “The traditional derivation (as given, for example, by Goldberger and Watson, 1964, or by Newton, 1966) involves a bit of fakery that hides the issue of pure states versus mixed states. A correct derivation uses a beam represented as a mixed state of packets with different impact pa-rameters. Such a derivation (Taylor, 1972) is analogous to the classical one, in which it is also necessary to assume that the incident beam consists of particles whose impact pa-rame-ters are uniformly distributed.” (Newton, 1979, pp. 929-930).
    The response of Band and Park to the above statement was: “Newton's revelation of ‘fakery’ in orthodox pure-state collision theory and admission of an analogy with the coarse-graining device used classically to suspend basic mechanical laws are welcome confirmations of our main contention, that, if collision theory is followed consistently with quantum mechanical unitary evolution, it is impossible to explain thereby the approach to equilibrium in a gas.” (Band and Park, 1979, p. 938).
    It turns out, however, that an alternative to the “suspension of basic mechanical laws” is possible, on account of the existence29 of single-target differential cross-section, whose derivation does not involve coarse-graining. This type of cross-section is therefore given by a formula that is distinct from (3.6), since it involves a T-“supermatrix” (rather than a T-“matrix”), as well as the confidence function in (3.5.3) (cf. [PQ], p. 518; [P],  p. 170):

formula    (3.7)

    Indeed, it is not true that any of the rigorous derivations of (3.6), namely those based on wave packets, rather than on plane waves and spherical waves  (cf. [PQ], pp. 430-436; [Messiah, 1961], Ch. X, §§5-6; Taylor, 1972], Sec. 3-e; [Newton, 1979]), are “analogous to the classical” derivation. In fact, in the classical context it is not at all necessary to assume that the “incident beam consists of particles whose impact parameters are uniformly distributed” in order to derive the classical scattering differential cross-section formula in its most basic form, namely in the form (cf., e.g., Balescu, 1975)

formula    (3.8)

On the other hand, if one does make the transition from classical mechanics to classical statistical mechanics, one obtains from (3.8) a formula which is the equivalent of (3.7), and not  of (3.6). This was actually proved by developing a common framework for classical as well as quantum statistical mechanics (Prugovecki, 1978a,b), in which it is possible to derive (3.7) and its classical counterpart within the same Liouville superspace. Under reasonable assumptions on the orders of magnitude of basic parameters in a scattering experiment, (3.6a) and (3.7) appear to be numerically very close, but they certainly are not equal!
    The above elementary example illustrates how “theory selection” is actually effected in the purely pragmatic approach to quantum theory, which has become the trademark of post-World War II conventionalistic instrumentalism in quantum physics. The type of attitude it reflects is aptly described in the following quotation (which, in its original context, concentrated on the modus operandi of the “new physics” from the 1960s to the present): “Having decided upon how the natural world really is, those data which supported that image were granted the status of natural facts, and the theories which constituted the chosen world-view were presented as intrinsically plausible.” (Pickering, 1984, p. 404).
    Thus, instead of relying on the uncovering of scientific truth based exclusively on analytic and rigorously formulated thought, combined with impartial observations vis-à-vis fashionable theories, post-World War II instrumentalism identifies “truth” with “consensus”, which, in turn, becomes a matter of institutionally enforced30 “convention”. Over the past four decades such practices have provided dramatic illustrations of the rea-sons for Heisenberg's deep concern (which we cited already in Sec. 1.5) about the “erroneous developments ... [that] are caused by a misconception by some physicists that it is possible to avoid philosophical arguments altogether”. That concern added to Dirac's deep distress about the “complacency” of contemporary “theoretical physicists [who are satisfied with the use of] working rules and not regular mathematics”. Clearly, in relativistic quantum field theory, both these concerns have to be addressed simultaneously – as demonstrated by the failure of the constructive quantum field theory program to establish the consistency of QED after more than a quarter century of effort (cf. Secs. 1.2 and 7.8, as well as Note 33 to Chapter 7). The lesson that might be learned from that failure is that it is not sufficient to try to impart mathematical respectability to the algorithms of the conventional approach in order to arrive at a mathematically consistent and yet physically nontrivial framework for relativistic quantum field theory. Rather, an epistemological analysis of its fundamental concepts is also required, and the implemented mathematically sound techniques have to reflect that analysis. In other words, “one must seek a new relativistic quantum mechanics and one's prime concern must be to base it on sound mathematics” (Dirac, 1978b, p. 6) – emphases added.
    We have already documented in appropriate sections of the preceding chapters many of the failings of the conventionalistic outlook on relativistic quantum theory. Hence, we shall only very briefly review the principal ones in the remainder of this section, and then indicate how the existence of the “cosmological constant problem” described in Sec. 11.12 totally vindicates Dirac's steadfastly critical attitude towards all the developments in the post-World War II renormalization program.
    Perhaps the most striking instance of a claim made in conventionalistic literature, which has been rigorously proved (Gerlach et al., 1967) to be false, is the assertion that the timelike component j0(x) of the Klein-Gordon current in (3.3.9) is positive definite if one restricts oneself to positive-energy solutions of the Klein-Gordon equation [SI]. This and other similar claims in otherwise respectable conventionalistic textbooks have influenced the thinking of generations of physicists, since they left them with the impression that “old” problems concerning relativistic quantum particle localizability have been “solved” by conventional relativistic quantum theory a long time ago, when actually the opposite is the case: not only have those problems not been solved, but proofs exist (Hegerfeldt, 1974, 1985, 1989) that they are not solvable within the conventionalistic framework – namely that all formulations of quantum particle localizability based on classical geometries give rise to violations of relativistic Einstein causality, albeit the opposite is maintained.
    To some of those predisposed to favor either the conventionalistic instrumentalism of the contemporary mainstreams in quantum theoretical physics, or the formal instrumentalism of the dominant contemporary school in quantum mathematical physics, the answer to this type of insurmountable difficulty with conventional concepts for particle localization appears to lie in the substitution of quantum field localization for quantum particle localization. However, not only does this substitution replace one set of difficulties with another – namely with the still unresolved fundamental problem of a mathematically cogent concept of (interacting) quantum fields, that can be mathematically localized in arbitrarily small regions of classical spacetimes, e.g., by using test functions of arbitrarily small supports in the Wightman formalism [BL] – but the following physical question is then not asked and answered: how does one operationally localize a classical or a quantum field?
    If, however, the above question is asked, then the only answer available is: by the use of massive test bodies. In their well-known papers on this subject, Bohr and Rosenfeld (1933, 1950) employed an analysis of the behavior of such classical test bodies, which therefore necessarily have to occupy macroscopic domains. Indeed, once regions of atomic and subatomic size are reached, the “consideration of the atomistic structure of measuring in-struments”, whose need they emphasized in their work, becomes unavoidable, so that one has come full circle: a consistent theory of localization of material quantum objects is needed in order to be able to formulate, in a physically meaningful manner, the concept of quantum field localization.
    Until the last decade, conventionalistic instrumentalism tended to ignore such foundational questions on the pragmatic grounds that the agreement of its theoretical predictions with experimental results is all that matters. However, it has been demonstrated in a number of recent studies (Cushing, 1990; Franklin, 1986, 1990; Pickering, 1984, 1989) that experimental technique is itself highly conditioned by theoretical outlook. Furthermore, as illustrated in an extensively documented sociological history of post-1960 developments in high-energy physics, “the idea that experiment produces unequivocal fact is deeply problematic. ... [Actual experiments] are better regarded as being performed upon ‘open’, imperfectly understood systems, and therefore experimental reports are fallible.” (Pickering, 1984, p. 6). Therefore, fundamental faults in theory can give rise to fundamental deficiencies in experimental design and technique, thus creating a vicious circle of feedbacks. In fact, as we have seen already in Sec. 9.6, when we discussed Dirac's critical attitude towards the experimental confirmation of QED predictions that are very highly acclaimed in conventional literature, in the absence of a mathematically sound theory it becomes a matter of subjective belief whether such apparent agreement represents confirmation of a theory intrinsically based on conventional “working rules”, or just mere coincidence.
    This becomes especially evident when closer scrutiny reveals that some such “coincidences” could be ascribed to fortuitous theoretical manipulation, since conventionalistic instrumentalism has facilitated the fine-tuning of theoretical computations to fit the experimental results by simply ignoring or discarding what is undesired, under the heading of such typical rationalizations as that it might be “naive”, or “irrelevant”, or “renormalizable”, or “compactifiable”, etc., etc. For instance, in the earlier cited carefully documented study of the development of the “new physics” in the 1960s and 1970s, we are provided with example after example of the following sociological high-energy phenomenon: “Discrepancies between prediction and data were taken as important results rather than serious problems: topics for further work rather than objections to the model.” (ibid., p. 266). Moreover, “fine-tuning” in such “further work” was greatly facilitated by the fact that theoretical error bounds were intrinsically unavailable in the computation of the “predicted” values of fundamental physical quanti-ties, such as the S-matrix elements of conventional quantum field theories. Indeed, what would be the possible use and meaning of such traditional theoretical tools to the theorist who deals with theoretical constructs whose very mathematical existence is not at all assured? Or to the theorist who can conveniently stop the summation of a “perturbation” series, for constructs of undecided mathematical existence, as soon as the desired agreement with experimental data is achieved? On the other hand, it might be asked: What if its summation were continued? And, in view of the presumed “asymptotic” nature (Dyson, 1952) of all “renormalized perturbation series”: Where should one stop the summation, from an objective point of view?
    With regard to measurements of spatio-temporal relationships at the microlevel, even the reliability of experimental results as a direct guide to the validity of fashionable theories deserves closer scrutiny. Indeed, as discussed and documented by Hacking (1983), Cartwright (1983), Ackerman (1985), Galison (1987), Franklin (1986, 1990), and others, contemporary experimental procedures are heavily theory-dependent. Hence, just as with Kaufmann's (1905, 1906) negative experimental verdicts on Einstein's special relativity, cited in the introduction to this chapter, and other similar historically well-documented cases, some experimental results might have to be critically reevaluated if Dirac's often repeated urgings for the use of “sound mathematics” in relativistic quantum physics are eventually heeded, and a mathematically sound31  reappraisal of some key theories is undertaken.
    The fundamental inadequacies of the conventionalistic outlook emerge with full force when quantum fields in curved classical spacetimes are considered: as described in Secs. 7.2 and 7.3, not only do the fundamental mathematical difficulties of the conventionalistic approach to quantum field theory become then more pronounced, but even old and very well established physical principles are sacrificed in order to maintain some particularly favored conventionalistic scheme. Thus, as can be seen from the review and analysis of conventional quantum field theory in curved classical spacetime presented in Secs. 7.1-7.3, some of the adherents to conventionalistic instrumentalism transform even the law of local conservation of energy and momentum into a matter of mere convention, which can be violated in order to save the formal aspects of conventional quantum field theories in curved spacetime. These aspects, in turn, are disregarded at the level of quantum gravity and cosmology, where concern with unitarity of the S-matrix seems to take precedence over formulating a concept of physical time based on a consistent theory of measurement. On the other hand, the existence of a unitary S-matrix solution for any realistic quantum theory of interacting relativistic fields has never been proved32 even in Minkowski space (cf. Sec. 7.6 as well as Note 31 to Chapter 9) – not to mention in any kind of curved spacetime. Thus, whereas conventionalistic instrumentalism has failed to meet in quantum physics even its own most basic criteria during the span of close to half-a-century of intense computational activities, its preoccupation with those criteria has derailed it on a sidetrack, where some of the most sensible and best established physical principles of quantum theory in the pre-instrumentalist era are ultimately ignored, or even violated.
    As if all these distressing inadequacies were not enough, the developments in particle physics and quantum cosmology over the past three decades indicate “a blurring of distinction between physical science and mathematical abstraction ... [reflecting] a growing tendency to accept, and in some cases ignore, serious testability problems” (Oldershaw, 1988, p. 1076). Thus, no less than twenty major effectively untestable problems are listed in (Oldershaw, 1988) – each one of which is of the type that would have been deemed a serious cause for concern in the pre-instrumentalist era. In view of Dirac's steadfast opposition to the renormalization program, from the time of its inception in the late 1940s until his death (cf. the introduction to Chapter 7), we shall discuss only one of those twenty issues. It is the one which shows that his criticism of the ad hoc nature of that program, and of the fact that it does not provide “a correct mathematical theory at all”, has been completely vindicated by some of the developments which took place after his death.
    First of all, let us remind the reader that one of the two main progenitors of the renormal-ization program has recognized from the outset that “the observational basis of quantum electro-dynamics is self-con-tradictory”, and that “a convergent theory cannot be formulated consistently within the framework of present space-time concepts” (Schwinger, 1958, pp. xv-xvi); whereas, the second one eventually acknowledged that “it's also possible that electrodynamics [namely conventional QED] is not a consistent theory” (Feynman, 1989, p. 199). Furthermore, in this regard, to the end of his life Dirac's main point had been the following: “Just because the results [of the conventional renormalization procedures in quantum field theory] happen to be in agreement with observation does not prove that one's theory is correct.” (Dirac, 1987, p. 196).
    The glaring observational inconsistencies (cf. Sec 11.12), to which the in-troduction of the Higgs boson in the offspring of conventional QED (namely in electroweak theory and in QCD) has led, have proven Dirac absolutely right in all respects, including the observational ones. Indeed, on one hand, despite a wide-spread search (cf., e.g., Harari, 1983; Weinberg, 1987) there is absolutely no observational evidence in favor of such a Higgs “particle”, introduced solely for the purpose of making QCD “renormalizable”; on the other hand, its assumed existence gives rise to an enormous cosmological constant – in blatant contradiction to the most basic observational facts. Of course, many “solutions” to the “cosmological constant problem” have been proposed (cf., e.g., the review article by Weinberg, 1989), but in the end one has to concur with the opinion that: “None of [these] attempts has succeeded. If anything matters have grown worse because theorists keep dumping more particles and fields into the vacuum.” (Veltman, 1986, p. 78).      
    In fact, ever since the advent of quarks, which after the failure to be observed were simply declared to be permanently “confined” (with no indubitable proof of confinement yet in existence), there has been such a proliferation of ad hoc theoretical devices, designed solely to remove flagrant disagreements between conventional theories and experimental facts, that the above cited leading researcher in the theory of quantum Yang-Mills fields figuratively exclaimed in a tone of utter exasperation: “Indeed, modern theoretical physics is constantly filling the vacuum with so many contraptions such as the Higgs boson that it is amazing a person can see the stars on a clear night!” (ibid., p. 76). The following are just a few examples of the “contraptions” that have highlighted the “progress” from the 1960s to the late 1980s: “Instead of one photon we have 12; three of them have acquired masses from spontaneous symmetry breaking, and eight of them are trapped. Instead of one electron, we have a whole menu of quarks and leptons defined by their representations with respect to the weak and strong gauge groups, and this menu is replicated three times: There are three generations.” (Weinberg, 1987, p. 7). It is therefore of no surprise that when faced with such a cornucopia of offerings from particle physicists, a noted astrophysicist felt compelled to remark: “Indeed I sometimes have the feeling of taking part in a vaudeville skit: ‘... You want massive weakly interacting particles? We have a full rack. You want an effective potential for inflation with a shallow slope? We have several possibilities.’ This is a lot of activity to be fed by the thin gruel of theory and negative observational results, with no prediction and experimental verification of the sort that, according to the usual rules of evidence in physics, would lead us to think we are on the right track of the physics of the universe at [a redshift epoch] z  > 1010.” (Peebles, 1987, p. 236).
    So, in the end one can ask, who was proven right by all these developments: Dirac, or the multitude of “dynamically acquiescent” (Pickering, 1984, p. 272) theorists, whom Dirac often described (cf. Sec. 9.6) as being too “complacent about the faults” of the renormalization programme instituted after World War II ?
    Keeping all of the above points in mind, we can summarize the situation by saying that, at the foundational level, contemporary conventionalistic instrumentalism is con-fronted with two fundamental types of problems.
    1) Mathematically, there is the one of logical consistency: as is well-known, from an inconsistent set of statements any other statement can be in principle derived. Thus, the deductive power of the scientific method can be in practice unwittingly undermined by ad hoc manipulations that are not dictated by logical necessity, but rather by the desire to achieve agreement with experiment – not to mention professional recognition. This was obviously central to Dirac's often expressed concern that the laws of “regular”, “sound” and “sensible” mathematics be followed in contemporary relativistic quantum field theory.
    2) Physically, there is Heisenberg's concern with posing the epistemologically correct questions: the use of formal analogies can lead to the introduction and development of concepts in a new context where such concepts no longer have a legitimate physical meaning, and lead to physically meaningless “scenarios”. Perhaps the most extreme example of this type is provided by the ex nihilo “scenario” of the creation of our Universe. Indeed, the concept of a wave function, representing a quantum particle, “tunneling through” the potential barrier to which another system of existing quantum particles gives rise, is operationally well-defined, and it makes physical sense; however, what is the possible physical  meaning33 of Nothing tunneling through a potential barrier produced by Nothing, in order to “create” our Universe in some present-day cosmological “scenarios”? Even though such a “phenomenon” can be formally described (Tryon, 1973; Vilenkin, 1982, 1988), and certain features of inflationary cosmological models that are currently in fashion can be then reproduced by the mathematics employed, does that physically validate such a “scenario”? The fact that there are some features of the inflationary model that can be “deduced” from such a “scenario” cannot establish its physical meaning and validity any more than the existence of Santa Claus can be established by the mock argument of Bertrand Russell, cited in Note 16, which was aimed at demonstrating the utter fallacy of the principal instrumentalist criterion of “truth” for a hypothesis – namely that “an idea is true if it works” (Stapp, 1972, p. 1103). Indeed, if that were so, then as Bertrand Russell pointed out with refined irony, the application of this most basic instrumentalist doctrine would allow us to infer that “Santa Claus exists” from the obviously correct statement that “the hypothesis [of the existence of] Santa Claus ‘works satisfactorily in the widest sense of the word’”!
    It would appear that one of the basic methodologies of conventionalistic instrumentalism is to pick fundamental techniques and results from a domain of quantum physics, where those results have a consistent and well-defined physical and mathematical meaning, and then transfer them to some new area of quantum physics, where both those types of meanings might be lost, and where only entrenched conventionalism provides the thread that holds together a thus newly created theoretical framework. Of course, as long as  “truth” is to be found in the “wide acceptance of a theoretical idea”, which can be secured by a variety of means (such as skillful promotional techniques, which in pre-instrumentalist times would have been more characteristic of practices in business and commerce, rather than in science), then there is nothing wrong with such an approach.
    On the other hand, we have seen from the numerous quotations presented in this monograph, that Dirac and Heisenberg have criticized in print many of the post-World War II developments in conventional relativistic quantum theory which, as we approach the end of this century, have become entrenched in “pragmatic” attitudes towards what constitutes “truth” in many key areas of what Pickering (1984) and others have described as the “new physics”. Popper ascribes such attitudes to “a tradition which may easily lead to the end of science and its replacement by technology”34, and which is based on a “fashionable philosophy [which] may in fact be uncritical, irrational, and objectionable” (Popper, 1982a, pp. 100-103).
    These are unequivocal and strong statements. They have to be weighed, however, against the fact that the protracted and practically unchallenged dominance of conventionalistic instrumentalism in quantum theory has given rise to a situation without exact precedent in the history of science. One commentator, who finds some of the latest manifestations of this phenomenon to be “a cause for concern”, rhetorically asks: “If the empirical foundation of the new physics is so insecure, and if it is still an axiom of science that without an empirical foundation a paradigm is dangerously adrift in a sea of abstraction, then why is there an unquestioned faith in the new physics? How can we understand the remarkable optimism and credulity demonstrated by theorists, experimentalists, peer reviewers, editors, and science popularizers?” (Oldershaw, 1988, p. 1080).
    As illustrated in this section, and as demonstrated in some other specific instances discussed in appropriate previous sections of this monograph, to this “insecure empirical foundation” has to be added the fact that the mathematical and epistemological foundations of this “new physics” are at least as “insecure”. So, instead of answering the above two questions, let us merely pose a counter-question: Sociologically speaking, what else can be expected when traditional standards of epistemological soundness and mathematical truth have been uprooted, and replaced by purely instrumentalist standards of “truth” which encourage, and in many key institutional settings even enforce, the type of conformity whose manifestations Feynman (1954) has so colorfully described as the “pack effect”?
    As witnessed by the earlier cited public statements of Dirac, Einstein, Heisenberg, Popper, Russell, and many other outstanding physicists and philosophers of this century, those men of vision have given proper and timely warnings as to what can be expected to happen. And what they foresaw and feared has been happening with increasing frequency and intensity ever since “World War II altered the character of science in a fundamental and irreversible way” (Schweber, 1989 – cf. also Note 47).
    Perhaps it is time that those warnings were heeded.

12.4. General Epistemological Aspects of Quantum Geometries

The quantum geometry framework described in the present monograph grew out of a systematic effort at trying to see whether the numerical successes of the conventional approach to relativistic quantum theory could be explained from a mathematically and physically cogent point of view. It appeared obvious from the beginning that, at the epistemological level, such a point of view would have to reexamine the very foundations of relativity and quantum theory. It was also clear that, in so doing, it would have to reconcile Einstein's “realism” with Bohr's “positivism”, by concentrating on the epistemological issues that united those two giants of twentieth century physics, and possibly ignoring the others – or, if absolutely necessary, even contradicting them on those issues that separated their distinct but not at all totally irreconcilable points of view.
    Indeed, it was pointed out in Sec. 12.1 that the basically operationalist attitude of Bohr was very much shared by Einstein during the period when he created special as well as general relativity. On the other hand, it should be obvious to readers who have read most of Chapters 3-11, that the operationalism of Bohr, as well as that of the pre-1920 Einstein, is retained in the formulation of the quantum geometries studied in those chapters. The concept of frame of reference, already so crucial to Einstein in the formulation of special relativity, and of “event”, defined as a spacetime coincidence, and viewed as the fundamental building block of all our observational constructs, namely all measurable physical quantities, were instrumental in those formulations. Such formulations are, therefore, also in agreement with Bohr's point of view – except that Bohr might have in-sisted on a classical description of all frames of reference.
    On the other hand, a form of quantum realism decidedly manifests itself in the present  framework in the form of the, until now, implicit premise that there is a physical reality, which is independent of any operational or linguistic conventions which any group of individuals happen to adopt. In other words the present work is founded on the belief that there is a single reality, which is quantum in its manifestations at the most fundamental level, and totally independent of any theoretical or experimental conventions. Hence, the quantum geometry framework presented in this monograph strives to remove the artificial dividing line which Bohr imposed between “system” and “apparatus”: there is only one reality, and that reality is quantum; ergo, any apparatus should be described at the most fundamental level in purely quantum terms. In particular, that conclusion is applied to frames of reference, which are viewed as quantum “objects”. However, as we have seen in Secs. 3.7 and 3.9, that does not preclude in some such frames the possibility of approximations of classical behavior: as we discussed in Sec. 3.9, such behavior is indeed manifested by sufficiently massive quantum frames. Thus, Bohr's teachings on the significance of classical concepts in the quantum theory of measurement are not ignored, but rather modified.
    Bohr's insistence on the importance and the role of language is not ignored either. In this respect the present approach is at odds with Popper's (1976, 1982, 1983) type of classical realism, which downgrades that role. However, there is absolutely no contradiction in maintaining that, on one hand, there is a microreality, and that the purpose of quantum theory is to reflect that reality as closely as possible, but that, on the other hand, in so doing it should employ the type of language best suited for that task, by incorporating all essential aspects of microreality, and at the same time avoiding, in accordance with Born's second maxim cited in Sec. 1.1, the introduction of redundant theoretical notions with no empirical counterpart. Consequently, the fundamental stance of quantum realism is epistemologically totally opposed to that of a “microrealism, according to which entities such as electrons, quarks, and the like, to which the name ‘particle’ is ascribed, are deemed to have a specific position at all times (and in terms of this conception, should also have, ‘for reasons of symmetry’, a specific velocity)” (d'Espagnat, 1989, p. 83).
    Indeed, the type of “microrealism” defined by d'Espagnat tries to understand the behavior of such “objects” as molecules, atoms, elementary particles, etc. exclusively in terms of concepts that have grown out of the fertile soil of our experiences with the macroscopic world, which we routinely encounter in our everyday lives. Of course, such concepts are perpetually nurtured by those experiences, so that they are our principal source of physical intuition – as rightly emphasized by Bohr. On the other hand, that does not mean that they have to remain our only source of such intuition, and that the human mind cannot grasp concepts and relationships that transcend the most immediate types of sense-impressions that reach it. Hence, the quantum realism underlying the present work tries to understand the microworld on its own terms, by developing the conceptual, linguistic and mathematical tools best suited for that task – irrespective of whether or not they are in accordance with the commonsensical ideas rooted in our everyday experiences.
    It could be said that as a conceptual and mathematical framework, rather than as a family of quantum theories, the purpose of quantum geometry is to supply a precise operationally-based mathematical language, as well as a metalanguage, for the description of quantum phenomena in purely quantum mechanical terms. In this context, the concept of informational completeness (cf. Sec. 3.7) emerges as fundamental, and it supersedes the EPR-type of classical realism, as applied to the quantum domain: a quantum theoretical description is not considered complete “if, without in any way disturbing the system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity” (Einstein et al., 1935, p. 777); on the contrary, at the most fundamental quantum level, Wigner-Araki-Yanase types of arguments (discovered long after the advent of the EPR paper) indicate that  in quantum theory there is no place for sharp stochastic values (i.e., for values predictable “with probability equal to unity”), so that the EPR formulation cannot possibly lead to valid criteria of completeness for the theoretical description of any quantum reality. Thus, since even in principle, and not only in practice, all values of physical quantities are unsharp at the quantum level, one of the basic principles adopted in quantum geometry is that of informational completeness (cf. Principle 2 in Sec. 1.3) at the local  level, i.e., in the quantum fibres above the points of a base spacetime manifold (cf. Principle 3 in Sec. 1.3). In other words, any quantum state in those fibres is completely determined by the measurement of its Fubini-Study distance from the elements of an informationally complete quantum frame in that fibre, which in turn is given in terms of operationally directly measurable “transition” probabilities – cf. Eqs. (3.7.10) to (3.7.15).
    This fundamental feature also dispenses with the need for von Neumann's questionable postulate about the identifiability of the set of quantum observables with the set of all self-adjoint operators in a Hilbert space (cf. Note 27 to Chapter 7). Furthermore, in the presence of quantum frame analyticity, only measurements of Fubini-Study distances of the local quantum state of a system to frame elements within arbitrarily small neighborhoods of the point of contact between tangent space and base manifold are required for the complete determination of that state. Therefore, such measurements are in principle implementable in the presence of arbitrarily strong gravitational fields. Thus, quantum realism is operationally based only in the context of measurement theoretical concepts  (cf. Sec. 12.5).
    On the other hand, by introducing the concepts of proper quantum state vector and of quantum frame as fundamental, it clearly recognizes that not all basic elements in its theoretical superstructure can possess direct operational counterparts, which, as such, would be simply groupings of our sense-experiences. Indeed: “In order to be able to consider a logical system as a physical theory it is not necessary to demand that all of its assertions can be independently interpreted and ‘tested’ ‘operationally’; de facto this has never yet been achieved by any theory and can not at all be achieved.” (Einstein, 1949, p. 679, ). Rather: “Although [theoretical] conceptual systems are logically entirely arbitrary, they are bound by the aim to permit the most nearly possible certain (intuitive) and complete co-ordination with the totality of sense-experiences; secondly they aim at the greatest possible sparsity of their logically independent elements  (basic concepts and axioms), i.e., undefined concepts and underived (postulated) propositions.” (Einstein, 1949, p. 13) – emphases added.
    The fundamental role played by measurement theoretical aspects brings, however, to the fore the question of where the present quantum geometry framework stands in the on-going realism–anti-realism dispute over the ontological status of the measured quantities. The following quotation succinctly reviews the issues in question:
    “Anti-realism with respect to measurement can assume a variety of forms. The simplest is an austere operationalism [expressed by the idea that measurable quantities] derive their meaning entirely from our measurement practices. ... This outlook is a species of a more general and widespread view, according to which the fundamental facts about measurement are grounded in conventions ... . A much more sophisticated conventionalism ... [is the] carefully qualified development of the idea that measurement operations can be said to measure the same thing if they give rise to the same ordering of objects under the same conditions. By contrast, I take realism with respect to measurement to be the view that in many cases measurement can give information about objective features of phenomena that is tinged with interesting elements of convention. ... The realist's thesis is that there are objective facts about what the length of something is, facts that are – within precisely specifiable limits – independent of our linguistic and scientific conventions, the particular theories we happen to accept, and the beliefs we happen to hold. Length can be measured on a ratio scale, and that means once a unit (e.g., the meter) is conventionally selected, there will be an objective fact as to how many meters long any given object is (since this will just be a fact about the ratio of its length to that of the meter bar). The realism–anti-realism dispute over measurement is not usually cast in terms of semantic issues, but it is important to realize that they are just there beneath the surface.” (Swoyer, 1987).
    This and other publications (Bergmann, 1960; Reichenbach, 1961) on these issues in contemporary philosophy of science reveal that “semantic issues” are indeed at stake in much of the ongoing polemic. For those concerned with more substantive issues, there are merits and demerits in both the operationalist as well as in the realist points of view. It is, therefore, tempting for a scientist to completely ignore such polemics, and dismiss them as totally “irrelevant” to the actual practice of science.
    The history of science teaches us, however, that utterly erroneous opinions were sometimes held because certain beliefs as to what is actually measurable, and how it is to be measured, were uncritically held in the face of existing strong evidence to the contrary. For example, since quantum geometry is applicable, amongst other fields, to quantum cosmology, the following comments might be of interest: “On various occasions in the history of cosmology the subject has been dominated by the bandwagon effect, that is, strongly held beliefs have been widely held because they were unquestioned or fashionable, rather than because they were supported by evidence. As a result, particular theories have sometimes dominated the discussion while more convincing explanations were missed or neglected for a substantial time, even though the basis for their understanding was already present.” (Ellis, 1989, p. 367) –  emphasis added.
    Thus, “strongly held beliefs” can color35 one's perception as to what observational evidence supports and what it does not, and, in fact, even shape one's beliefs as to what is observable and what is not. For example, in the heyday of S-matrix theory in the 1960s the opinion that the description of quantum phenomena did not require any concept of space-time was not only widely held in elementary particle circles, but became thoroughly institutionalized36. In fact, opinions to this effect were heralded at international conferences and in review articles as the only acceptable approach to the physics of fundamental quantum phenomena37 – without such basic questions being asked and answered, as to how such a belief could be reconciled with the fact that a spacetime background was essential to the rest of physics. In fact, even nowadays, residues of that belief condition research in conventional quantum gravity and instrumentalistically motivated quantum cosmology, where the question of “renormalizability” of the so-called “perturbation” series for the S-“matrix” still occupy center stage. However, in such cosmological pursuits, the following elementary question is not asked: what is the possible literal physical  meaning of the concept of S-matrix in the real universe in which we live, namely in a universe in which, according to all evidence, asymptotic flatness of spacetime is certainly not present in the “cosmic” past, and, by all accounts, will never become realized in the “cosmic” future.
    This is not to say that, if one subscribes to the point of view of quantum realism, according to which spatio-temporal relationships have an objective existence, which is independent of prevailing theories and operational procedures, then those operational procedures are automatically provided by the quantum reality. Rather, the opposite is true in practice: operational procedures are heavily theory-dependent, even to the extent that modifications of the underlying theories entail radical modifications in the measured values.
  Consequently, one of the key questions from the point of view of a quantum realist, concerned with empirical reality (rather than with so-called “intrinsic reality” – cf. d'Espagnat, 1989), is what are the truly fundamental units for the measurement of space-time separations in Nature. In other words, special relativity was grounded in an operationalist attitude, which stipulated that spatial distances are to be measured with “rigid” rods, and temporal separations with “standard” clocks (Einstein, 1905). Although the concept of strictly rigid rod is actually untenable in relativity (Stachel, 1980), that of standard clock suffices under the assumption of the constancy of the speed of light with respect to all Lorentz frames. That raises the question, however, as to what choice of clock should be made for that standard; and, even more importantly, why would Nature abide even at the microlevel by any particular choice of macroscopic clock, made on technological or other anthropic grounds? In other words, except if real (as opposed to operational) time is somehow an intrinsic property of all matter in existence, it would be unrealistic to expect that Nature would abide by any purely conventional (Jammer, 1979) choice at all of its levels of magnitude, from the very smallest subnuclear processes, to the large-scale structure of our Universe. Indeed, in practice, totally different units and operational procedures are used at the two ends of this scale of magnitudes, as well as at many particular stages in between.
    The present quantum geometry framework is based on the premise that a fundamental choice, independent of all conventions, does exist for the specification and measurement of spatio-temporal relationships, and that, therefore, it has to be inscribed in every single bit of matter in existence. That natural choice can be found by simply tracing the origins of de Broglie's idea, which heralded the emergence of quantum mechanics38: namely that, on account of its rest mass m, each massive elementary quantum object represents a natural clock with period T = 2(pi)/m in Planck natural units. The universal constancy of the ratios of the observed rest masses of elementary particles vouches that all the elementary particles in Nature keep the same local time, so that any geometro-stochastic propagation can take place under well-specified spatio-temporal conditions. Without that assumption, the proposed idea of any quantum geometry would make no sense at all as a candidate for a physical geometry. But, without the hypothesis of cosmic constancy of the ratios of the rest masses of all “elementary particles”, elementary particle physics would not make any sense either!
    So, those in elementary particle circles who argue that at a fundamental level the concept of spacetime might not be meaningful (Chew and Stapp, 1988), or that it might be a mere illusion (Kaplunowski and Weinstein, 1985), are simply ignoring the most fundamental evidence in their own field: the existence of quantum entities which conventional terminology has labelled as “elementary particles”. The fact that it might eventually turn out that all of these massive “objects” are neither “elementary” nor “particles” is irrelevant: the main point is that they do possess rest masses, and therefore they are localizable in reality, and that they do keep their own proper time. It is, therefore, a matter for theoreticians to display enough imagination in the creation of theories which properly reflect these quantum facts. In particular, this intrinsically fundamental physical significance of the concept of spacetime has to reflect the measurement- theoretical limitations imposed by the existence of the Planck length and of the Planck time.
    For this very reason, these basic constants are embedded, in the form of the fundamental length l ( = 1 in Planck natural units), into the very structure of the fibres of quantum geometries. This is very much in keeping with Einstein's epistemology (albeit it would not have been in keeping with his predilection for classical realism):
    “The relations between the concepts and propositions [of a theoretical framework] are of a logical nature, and the business of logical thinking is strictly limited to the achievement of the connection between concepts and propositions among each other according to firmly laid down rules, which are the concern of logic. The concepts and propositions get ‘meaning’, viz., ‘content’, only through their connection with sense experiences. The connection of the latter with the former is purely intuitive, not itself of a logical nature. The degree of certainty with which this connection, viz., intuitive combination, can be undertaken, and nothing else, differentiates empty phantasy from scientific ‘truth’.” (Einstein, 1949, pp. 11-13) – emphases added.
    Finally, the retention of the equivalence principle in the relativistic quantum regime is the last, but certainly not the least, of the epistemological cornerstones in the formulation of the quantum geometries in the preceding seven chapters. In fact, the simplest type of experimental test, helping to choose between theories formulated within the present quantum geometry framework and those based on conventional frameworks (cf. Secs. 7.2 and 7.3), lies in the verification of this very principle in the quantum regime: is there, or is there not, actual (as opposed to conventionally agreed upon) Rindler particle production in Nature? Is there spontaneous particle production ex nihilo in Nature, that as such can be observed by inertial observers under very different free-fall conditions? Is there, therefore, local energy-momentum violation that such observers can witness?
    The answer of the present GS framework, based on the application to general relativity of ideas anchored in the epistemology of quantum realism, is a firm: No! Some of the papers cited in Secs. 7.2 and 7.3 (cf., e.g., Unruh, 1976; Unruh and Wald, 1984), based on conventional instrumentalist conceptualizations of relativistic quantum theory in curved spacetime, claim: Yes! Hence, this is a very clear-cut case where experiments, performed under carefully and properly controlled conditions (cf. p. 203), should decide the issue.

12.5. The Concept of Point and Form Factor in Quantum Geometry

At the most fundamental epistemological level, the distinction between classical geometries and the quantum geometries treated in this monograph lies in the treatment of the concept of “point”. From a purely mathematical perspective, the distinction does not appear that great: the points of classical geometries belong to finite-dimensional manifolds; whereas, those of quantum geometries belong to fibre bundles which constitute infinite-dimensional manifolds or super- manifolds. However, physically, the distinction is considerably greater. It can be described by saying that the points of classical geometries are “sharp” and “structureless”; whereas, those of quantum geometries are “unsharp” and can possess an internal structure. In the quantum geometries that describe quantum spacetimes, that structure is embedded in their quantum spacetime form factors. It therefore seems mandatory to single out a fundamental quantum spacetime form factor, which distinguishes itself by an outstanding simplicity of its internal structure, as well as some very special physical characteristics vis-à-vis some model of universal significance in quantum physics.
NOTE: The indented text below, which is missing the formulae that could not be reproduced in html, can be skipped.
   At the very foundations of quantum physics lie the canonical commutation relations between position and momentum. The harmonic oscillator is the simplest as well as the most fundamental physical model that embeds the constituents of those canonical relations into the eigenvalue equation for its energy spectrum. In the case of the relativistic harmonic oscillator that equation assumes the form

 formula    (5.1a)

formula    (5.1b)

into which the Minkowski metric enters intrinsically, and into which the relativistic canonical commutation relations are also intrinsically embedded:

formula    (5.2)

For that reason, as well as on account of the formal symmetry played in (5.1) by the Q's and the P's, Born (1949) adopted (5.1a) as the basic eigenvalue equation for his quantum metric operator.
    Naturally, as they stand, (5.1) and (5.2) do not constitute a well-posed eigenvalue problem without the stipulation of boundary conditions on the eigenfunctions. Such boundary conditions can be imposed in the traditional manner by the requirement that the eigenfunctions be square-integrable in R8 with respect to the Lebesgue measure. However, such a stipulation cannot be justified from the point of view of a relativistic “quantum metric operator”, since it is obviously related to the Euclidean regime39, and, moreover, it leads to an eigenvalue spectrum which is unbounded from below. On the other hand, if (5.1a) is interpreted as an eigenvalue equation for quantum metric fluctuation amplitudes which result in local exciton propagators (cf. Sec. 7.4),  then it turns out (cf. [P], Sec. 4.5) that its spectrum consists of eigenvalues bounded from below by a unique minimum eigen-value, which corresponds to the fundamental quantum spacetime form factor  fl  in (5.5.5).
    In view of the close connection between oscillator states and the realizations of Virasoro algebras emerging from some of the older treatments of string quantization (Green et al., 1987, Sec. 2.2), a treatment of the eigenvalue problem in (5.1) can be devised which results in an entire family of “stringlike” quantum metric fluctuation amplitudes. Of course, although such possibilities of interpretation of excited states of the quantum metric operator in (5.1b) are intriguing, they are not particularly compelling, since the conjectures that excited string states might have occurred only during the “Planck era after the Big Bang” represent sheer speculation, which is unlikely to receive any direct experimental support in the foreseeable future. Nevertheless, in view of some still prevailing popularity of string theories, we shall briefly review them, before turning in the last part of this section towards the much firmer ground which underlies the choice of fundamental quantum spacetime form factor in this monograph. Hence, this review is intended primarily as an illustration of the fact that, although there are many other technical as well as conceptual differences between string theory and the present geometro-stochastic framework, there is also a certain underlying affinity of heuristic physical ideas, which could be used to establish closer theoretical links.
    The incorporation40 of massless oscillatory exciton states into a previous adaptation (Prugovecki, 1981b) of Born's (1949) quantum metric operator to GS quantum theory leads to a quantum relativistic harmonic oscillator, whose eigenstates display some of the features of string modes that are present in the fibres of a prequantum bundle over a ten-dimensional base space embedded in the bundle T*M + T*M over the Lorentzian manifold M. In such a model for GS excitons the proper wave function for a graviton at any base location x in M can be identified with the spin-2 ground state of the quantum metric operator  D2(x) = Q2(x) + P2(x) at that location.
    From a semiclassical point of view, this treatment envisages a stringlike GS exciton at x in M to be an excited eigenstate of a relativistic harmonic oscillator at that location.  At such a heuristic level, a GS exciton above the base location x in M can be visualized as a string of points q in TxM executing, in general, vibratory as well as rotational motions with respect to a local Lorentz frame {ei(x)}. The ground modes of such stringlike GS excitons would correspond to stochastic vibrations in the direction of motion specified by its 3-momentum k, transversal oscillations in the polarization planes orthogonal to k, and rotations around the direction in which k points. As a result of all these motions, its suitably renormalized probability wave amplitudes
formula    (5.3)

satisfy the string equation

formula    (5.4)

in the frontal localization frame (Prugovecki, 1978c) determined in Tx*M by (k0,0) and (0,k). In general we can also expect, however, more complex internal motions, involving additional rotational degrees of freedom that are not around the axis provided by their direction of motion k. If it is assumed that all GS exciton transition amplitudes (cf. [P], Sec. 4.5) to excited modes for such motions are eigenfunctions of Born's quantum metric operator, and that they satisfy the equation proposed in the context of Born's reciprocity theory by Yukawa (1953), then the proper state vectors fB,A describing these higher exciton states satisfy the relativistic harmonic oscillator equation

formula    (5.5)

in the variables ui = piki, representing relative internal 4-momentum components with respect to the dual of the local Lorentz frame {ei(x)}. The rest masses mB,A carried by these excited modes  fB,A are then given, in Planck natural units, by the following equation,

formula    (5.6)

relating them to the eigenvalues in (5.5), whose explicit values will be provided in (5.12).
    Indeed, the eigenvalues and eigenstates of the relativistic harmonic oscillator equation in (5.5) can be computed by the standard use of raising and lowering operators, provided in the present context by the following expressions:

 formula    (5.7)

In the present context these operators satisfy relativistic canonical commutation relations that are equivalent to those in (5.2):

formula    (5.8)

However,  the ground state is degenerate since it corresponds to zero mass, so that various polarization modes exist that give rise to a great variety of internal gauges – as exemplified in Chapter 11 in the case of the graviton. Indeed, these ground states display invariance under the SO(2) group of motions that leaves k invariant. Consequently, they can be factorized as follows

formula    (5.9)    

where each Z(sA) is constructed from polarization frames, such as those in Chapters 9 and 11, so that they can be grouped into sets {Z(sA)} providing integer-spin frames. The spin  sA =1 and sA =2 cases provide ground exciton states that are capable of representing photons and gravitons, respectively.
    All ground GS exciton states share the common form factor

formula    (5.10)

reflecting a string length  lA = 2 in Planck units, and supplying the fundamental quantum spacetime form factor in (9.2.14) upon setting ui = viki, and then renormalizing as  mA tends to 0. The higher exciton modes can be obtained from the solutions for the eigenstates in (5.5) in the following simple manner (cf. [P], p. 204):

formula    (5.11)

Since by (5.6) these states are massive, they reflect a breaking of the SO(2) symmetry that left k invariant. However, in order to be physical GS exciton modes, they have to display an SO(3) invariance that reflects the presence of specific internal spin value. Thus, they correspond to the following eigenvalues of the quantum metric operator D2(x) at each x in M (cf. [P], p. 205):

 formula    (5.10)

The proper state vectors describing their internal stochastic motion with respect to the local Lorentz frame {ei(x)} can be then computed as in (Brooke and Prugovecki, 1984).
    As mentioned earlier, much more compelling than the above string-motivated type of heuristics is the adoption of the quantum spacetime form factor  fl  in (5.5.5) as fundamental to any model of quantum spacetime – regardless of whether it manifests itself as the ground state of a quantum metric operator, or simply as the only quantum spacetime form factor in existence. Indeed, as we pointed out in Sec. 1.5, quantum geometries do not require the existence of physical “objects” and test “bodies” which exactly “fit” into their points, any more than classical geometries require truly pointlike test particles that exactly fit into theirs: in either case, the concept of point can be viewed as an abstraction, suggested by an empirical reality which is quantum in the former case, and classical in the latter, but without necessarily faithfully reflecting those respective realities. On the other hand, the adoption of fl as the quantum spacetime form factor can be justified purely on grounds of mathematical simplicity and aesthetics, combined with the fact that, as demonstrated in Sec. 11.4, it assures the informational completeness of the ensuing quantum frames.
    Indeed, it is well known that, as a methodological guide to uncovering new physical laws and features of Nature, the principle of mathematical simplicity was already advocated by Newton, and that Einstein championed it throughout his life. The idea of mathematical beauty as methodological guide had its recent advocates in Poincaré and Weyl, and perhaps its strongest champion in Dirac: “For Dirac the principle of mathematical beauty was partly a method-ological moral and partly a postulate about nature's qualities. It was clearly inspired by the theory of relativity, the general theory in particular, and also by the development of quantum mechanics.” (Kragh, 1990, p. 277).
    Of course, both these principles should be used only sparingly and judiciously, as they have been (justifiably) criticized on the basis that not all mathematicians or physicists share the same idea of either mathematical simplicity or beauty. In other words, mathematical beauty as well as simplicity might exist only “in the mind of the beholder”. But then, we have seen in many previous examples that, to a certain extent, the same can be said even of the appraisals of the degree of support received by a very popular theory from various experiments. In fact, there are cases in which a compelling simplicity and beauty can be even more universally “obvious” in a theory than its purported agreement with experiment, since in the latter case, one often merely tries “to make sense of the mass of data provided by the experimentalists” (cf. Note 28); whereas, the former might almost be “able to speak for it-self”, on account of elegant features in its appearance as well as in its underlying ideas – as, most certainly, is the case with the Dirac equation. Hence, it is not at all surprising that Dirac “asserted that mathematical-aesthetic considerations should (sometimes) have priority over experimental facts and in this way act as criteria of truth” (Kragh, 1990, p. 284).
    The adoption of the quantum spacetime form factor  fl  in (5.5.5) as fundamental embodies the criterion of mathematical simplicity in a most direct and evident form. It also incorporates one of Dirac's favorite paradigms of mathematical beauty – namely the theory of functions of one or more complex variables. Indeed, upon adopting fl as being the fundamental quantum spacetime form factor, the following straightforward substitution can be carried out in all local quantum fluctuation amplitudes (cf., e.g., (9.2.22), or (9.6.3) and (9.6.4)), whereby real Poincaré gauge variables are replaced with complex ones:    

formula    (5.13)

It thus solves one of the “many problems left over concerning particles other than those that come into electrodynamics: ... how to introduce the fundamental length to physics in some natural way” (Dirac, 1963, p. 50). It also mediates in a most natural way the strongly- advocated-by-Dirac replacement in quantum theory of real with complex variables. Indeed:         “As an interesting mathematical theory that fulfilled his criteria of mathematical beauty, Dirac emphasized in 1939 the theory of functions of a complex variable. He found this field to be of ‘exceptional beauty’ and hence likely to lead to deep physical insight. In quantum mechanics the state of a system is usually represented by a function of real variables, the domains of which are the eigenvalues of certain observables. In 1937, Dirac suggested that the condition of realness be dropped and the variables be considered as complex quantities so that the representatives of dynamical variables could be worked out with the powerful mathematical machinery belonging to the theory of complex functions. If dynamical variables are treated as complex quantities, they can no longer be associated with physical observables. Dirac admitted this loss of physical understanding but did not regard the increased level of abstraction as a disadvantage. ... Dirac never gave up his idea of mathematical beauty, to which he referred in numerous publications, technical as well as nontechnical.” (Kragh, 1990, pp. 282-283).
    The GS interpretation of the components of the complex variables in (5.13) not only removes any possibility of some “loss of physical understanding”, but it also harmonizes very well with Born's (1938, 1949) reciprocity ideas about the symmetric role played in nature by the position and momentum variables. At the same time, the introduction of the complex variables in (5.13), mediated in a most natural manner by the choice of the  fundamental quantum spacetime form factor fl  in (5.5.5), also ensures that the GS quantum fluctuation ampli-tudes (i.e., local GS propagators such as (±) and S(±) in Secs. 7.4 and 8.1, respectively) are analytic extensions (in the sense of distributions) of their conventional counterparts. In view of the status of contemporary experimental high energy technology, which is still far from being able to probe energies and distances of “Planckian” orders of magnitude, this feature is bound to secure numerical agreement at the formal perturbative level, and within the domains experimentally reached thus far, between conventional quantum field theoretical models and their GS counterparts that are based on the fundamental quantum spacetime form factor fl. Hence, the choice between conventional models and their GS counterparts is not one that could be made, at the present technological level, on the basis of experiment alone. Rather, it is one that involves criteria for mathematical and epistemic soundness, which reflect a long-range view of the role of a quantum theory that incorporates gravity, rather than the immediate gratification of some simple-minded instrumentalist criterion of “agreement with experiment”.

12.6. The Physical Significance of Quantum Geometries

The framework for quantum geometries presented in this monograph enables the embedding of fundamental measurement-theoretical limitations directly into the very structure of relativistic quantum field theories formulated in terms of such geometries. We have pointed out in the last section of Chapter 9 that the formal manipulations characteristic of conventional quantum field theoretical models can be duplicated in the context of GS models, and their “perturbation expansions” could be then recovered term by term in the Minkowski regime by taking the limit in which the fundamental length l tends ot zero. There appears to be no point, however, in such formal manipulations, except as paradigms in the study of the fundamental question of relativistic microcausality.
    The central observation here is that, in the absence of a proof of the existence of the S-matrix in the quantum field theoretical models, from QED to QCD, that are currently in vogue in elementary particle physics, no test of the formulation of microcausality based on “local” (anti)commutativity can be said to have been performed thus far. Furthermore, even if we grant the existence of the S-matrix in such quantum field theoretical models, the fact that certain well-known properties of the S-matrix can be formally derived (cf., e.g., Blokhintsev, 1973) by the use of “local” (anti)commutativity does not prove that such (anti)commutativity is a necessary (and not just sufficient) condition for those properties to hold. For example, the violations of “local” commutativity for asymptotic fields in QED (Fröhlich et al., 1979) provide one of the many indications that no such necessity is, in fact, present even within the conventional quantum field theoretical framework. Furthermore, as discussed in Sec. 7.6, the mere postulation of “algebras of observables” which purportedly satisfy “local” commutativity neither proves their mathematical existence for physically nontrivial conventional models, nor does it settle any fundamental measurement-theoretical questions as to the operational feasibility of associating actual observables with arbitrarily sharply delineated domains in classical spacetime manifolds.  
    In fact, in Secs. 7.6 and 9.6 we have pointed out that the conventionalistic identification of “microcausality” with “local” (anti)commutativity has no bearing on the GS approach, since such (anti)commutativity has no physically truly meaningful relationship to the question of Einstein causality any more than it would in classical relativistic theory. Indeed, in classical special relativistic theory, the commutativity of all observables is trivially satisfied, since all classical fields and their observables commute. On the other hand, in a classical general relativistic theory such commutativity for non-scalar fields is undefined at distinct spacetime points. Of course, in the special relativistic regime, the concept of locality that emerges from the “naïve” realism predating modern quantum theory makes such a concept “plausible”. However, there has never been any serious attempt in the literature to rigorously prove that the identification of “local” (anti)commutativity with some form of Einstein causality follows from any cogent quantum theory of measurement. Rather, from the earliest days this idea was introduced by postulation  in the LSZ formulation (cf. Note 31 to Chapter 9), as well as in axiomatic quantum field theory (Streater and Wightman, 1964).
    On the other hand, in the GS approach microcausality is directly related to the mode of propagation, i.e., to the realistically posed question as to which stochastic paths are followed in GS propagation: are only those paths allowed which can be approximated by piecewise smooth curves, whose smooth segments are strictly causal in the classical sense, as in strongly causal GS propagation, or are certain types of noncausal smooth arcs also allowed, as is the case in weakly causal GS propagation?
    In developing a framework within which such questions can be meaningfully posed, the quantum geometry framework assigns total priority to geometric over variational principles. This is in contradistinction to Feynman's path-integral formulation of quantum propagation, which assigns the most prominent role to Lagrangians, and underplays the fact that each “sum-over-paths” is fundamentally a geometric concept, which can be formulated in a Lagrangian-independent manner. Hence, in the GS approach the entities of direct physical significance are the GS propagators themselves, which describe propagation between base spacetime points along causally ordered 3-manifolds, rather than being the conventional “propagators” in momentum space representations, whose introduction is motivated by the computational expediency imposed by conventional “perturbation” theories.
    The ultimate question of choice between strongly and weakly causal GS propagation will have to be obviously answered by experiments based on properly formulated theoretical predictions of measurable effects that can distinguish between these two modes of propagation. Such predictions will have to take advantage of the fundamentally nonperturbative formulation of GS propagation. Indeed, clearly specified error bounds would have to be computed at those base spacetime points where probability transition amplitudes for the two modes might be observationally distinguishable by means of present-day technology41.
    The fundamentally nonperturbative nature of GS propagation is a reflection of the fact that the quantum reality envisaged by the GS approach is based on quantum stochasticity. The manifestations of this kind of stochasticity are in their most essential aspects totally different from those assumed in classical physics. This fundamental distinction emerges from the fact that in quantum GS formulations the concept of probability measure for quantum stochastic paths does not exist42. Hence, of necessity, GS propagation has to be formu-lated in terms of probability amplitudes over broken paths, with a subse-quent specification of limits – the same type of limits as in Riemannian integration – rather than in terms of prob-ability measures over stochastic paths that employ Lebesgue integra-tion, as is the case in the theory of classical stochastic processes.
    These GS probability amplitudes are superimposed in a coherent manner, due to the intrinsic proper time kept by proper state vectors, represented by local coherent states, as they propagate along such paths. As discussed in Sec. 1.4, the process of observation corresponds to decoherence, so that the “classical path” would be the most likely one to be “observed” in the sense that it might provide the best fit for the discrete set of base spacetime locations where actual macroscopic registrations have taken place. On the other hand, the existence of proper state vectors permits, by the application of the superposition principle, the possibility of weak relativistic GS microcausality – a concept that makes absolutely no sense for point particles whose behavior is governed by classical diffusion processes.
    The existence in the GS approach of proper state vectors also enables the formulation of new types of quantum models based on the adopted structures of quantum spacetime form factors – such as those briefly mentioned in Sec. 1.5. Thus, in the GS context the problem of strong interactions can be approached from two very distinct angles: 1) with an external dynamics perspective in mind, which would lead to a GS counterpart of QCD, and in which the fundamental quantum spacetime form factor  fl  in (5.5.5) would be the only quantum spacetime form factor, while the interactions between quantum fields creating and annihilating quarks would take place by means of the external exchange of gluons; 2) from an internal statics point of view, whereby new quantum spacetime form factors would be “shaped” either by the presence of a quantum metric operator (such  as the one discussed in the preceding section, or by an internal “Hamiltonian” based on fundamental oscillator and rotator models – cf. Bohm et al., 1988), or on account of having a ground exciton “trapped” in some internal geometry – such as in the de Sitter types of quantum geometries adopted in (Drechsler and Prugovecki, 1991) and in (Drechsler, 1991).
    The latter type of approach based on “internal statics” has an essential bearing on the epistemological significance of the concept of congruence in physical geometries. As discussed in (Jammer, 1969), pp. 208-211, various conceptions of geometric congruence were advanced in this century by Russell, Whitehead, Eddington, Bridgeman and others, and their significance to the empirical role of metric in CGR was debated by Einstein, Reichenbach and Robertson in (Einstein, 1949). The possibility of the existence of a quantum metric operator that “shapes” the points of quantum spacetimes obviously opens new possible perspectives on these “old” issues, by showing that the foundations of the physical geometries used in the description of spacetime do not reside in any kind of geo-chronometric conventionalism, such as that advocated by Reichenbach and Grünbaum, but rather in the intrinsic quantum features of spacetime.
    In general, it can be said that quantum geometries throw new light on some of the “old” problems, that were raised in earlier times in the context of classical geometries, and that at the same time they give rise to new physical concepts of a geometric nature, whose very meaning would be nonexistent in their absence.
    The quantum geometry framework also opens new possibilities in the theory of quantum measurement. In fact, as presented in Chapters 3–10, the GS approach has been in this respect quite conservative: the development of the GS theory of measurement was based on a very gradual and very careful extrapolation of the orthodox approach – so as to avoid any of the needless epistemological excesses encountered in some other non-orthodox approaches to the quantum theory of measurement (DeWitt and Graham, 1973; Barrow and Tipler, 1986). Thus, as described in Chapter 3, at the measurement theoretical level the GS program began with the assumption that the existence of (previously unsuspected) Galilei and Poincaré covariant and conserved probability currents, such as the ones in (3.5.1)–(3.5.9) and (3.5.13)–(3.5.15), respectively, were not due to sheer coincidence. The validity of this conjecture was reinforced by the striking similarity in external appearance of those currents, and by the fact that the non-relativistic ones merge in the sharp-point limit into the conventional ones in (3.5.7). As we have seen in Chapter 3, in the nonrelativistic regime, this assured the possibility of a gradual transition from the orthodox to the SQM theory of measurement; whereas, in the special relativistic regime it enabled the straightforward extrapolation of the basic formal aspects of the conventional theory, and the avoidance of the difficulties created in that theory by the absence of bona fide probability currents. In the presence of gravity, an interpretation that was still very close to the orthodox one was adopted in the semiclassical approximation, in which the gravitational field is treated only as an external field (cf. Sec. 5.5).
    The application of GS quantum gravity in Chapter 11 to quantum cosmology led, however, to the introduction of a “universal GS wave function”, which represents a GS counterpart of the “wave function of the universe” (Hartle and Hawking, 1983; Barrow and Tipler, 1986), since it is meant to describe all the matter and gauge fields in existence. This necessitated the consideration of very fundamental epistemological questions, traceable in the history of philosophy to the mind-body problem and to the question of free will (cf., e.g., Weyl, 1949). Historically, these questions have impinged upon the epistemology of quantum mechanics in the form of von Neumann's (1932, 1955) ”psycho-physical parallelism”, and Wigner's (1962) subsequent analysis of the thesis that the “reduction of the wave packet” might take place in the mind of the “observer”.
    Whereas the empirical significance of such a thesis in ordinary quantum mechanics is very much open to debate, the general questions that it implicitly raises in cosmology are related to the issue of the freedom of the experimenter to locally change physical conditions, rather than act as merely a passive “observer”. Indeed, such measurement-producing actions can give rise to “reductions of the universal wave function” that would not have occurred otherwise. Hence, in any theory describing a single universe (as opposed to “scenarios” based on any form of “parallel universes” – cf. Barrow and Tipler, 1986), they give rise to profound questions concerning the nature of fundamental causality – namely of the forms of causality in the traditional philosophic sense (Weyl, 1949; Bunge, 1970), some of which predate by millennia the notions of microcausality and of Einstein causality, which we discussed earlier.    
    Thus, as we saw in Chapter 11, the quantum geometry framework based on a GS conceptualization of quantum reality reverses Bohr's epistemic outlook, and asks us to envisage how macroscopic phenomena appear from a microscopic point of view. In other words, it poses from a microscopic perspective the questions: What is an “observation”? What is an “apparatus”? All of this provided, of course, that we grant as a basic methodological feature that the latter must be, in some unambiguously prescribed sense, a macroscopic object whose behavior can be approximately described in classical terms.
    In this type of GS conceptualization, any phrase providing the “probability of detection of a GS exciton within a region B in M” is merely a short-hand for the descriptively more accurate, but cumbersome and tediously long phrase, asserting the provision of the “probability of a macroscopic manifestation, within a region B in M, of a given form of perturbation in a particular type of conglomeration of local state vectors, that constitute a GS wave function primarily localized in some vicinity of B”. Furthermore, any such provision has to be supplemented by an unambiguous and detailed description of all the “well-defined experimental conditions specified by quantum physical concepts” under which such manifestations are to be observed.
    In paraphrasing, in this last stipulation, one of Bohr's principal dictums (cf. Sec. 12.1) by the simple expedient of replacing in the original text the term “classical” with the term “quantum”, we wish to underline the fact that much of the essence of Bohr's philosophical outlook can, and must, be retained in the future developments of any quantum theory of measurement. It is only what might be termed “epistemological dead wood” that has to be trimmed away, in order to arrive at a better understanding of the foundational issues whose study was initiated by the Copenhagen school, as well as by many of its outstanding contemporaries in the “opposing camp”.  

12.7. Summary and Conclusions

As we have seen in the first section of this chapter, from the perspective of philosophy of science, the development of quantum theory during the first half of this century was marked by the confrontation between classical realism and logical positivism. This confrontation was personified by Einstein and Bohr, respectively – although neither of them fully and consistently embraced the philosophies they were supposed to represent.
    During the second half of this century the arena of such confrontations changed radically. As we discussed in the second and third section, most of the basic issues which were the focus of attention in the historic confrontation between Bohr and Einstein became either irrelevant or largely forgotten soon after World War II, while a new philosophy took over, and has dominated, practically unchallenged, the world of quantum physics for the last four decades. That publicly unacknowledged, but in everyday practice of quantum physics all-pervasive, philosophy was identified by Popper (1976, 1982), as well as other authors, to be a form of instrumentalism. This label was further qualified in this chapter as conventionalistic instrumentalism, in order to descriptively incorporate and characterize by it also the new attitudes towards mathematical standards of truth and deductive validity that emerged in quantum physics during the first decade of the post-World War II era.
    Bohr, Dirac, Heisenberg, Pauli43, Popper and many others of the pre-World War II “older generation” of physicists and philosophers of science have reacted with distinct disapproval towards the most prominent aspects and practices of this tacitly but widely accepted philosophy of the new generation of physicists – especially towards its computational opportunism, and its lack of commitment to the rational and objective mathematical and/or epistemological standards, which in previous eras represented the traditional hallmark of the scientific outlook. In fact, disapproval became a general sounding of the alarm when Popper described the professional atmosphere created by the functionally unconditional adoption of instrumentalism in contemporary quantum physics as “a major menace of our time”, and when he stated that “to combat it is the duty of every thinker who cares for the traditions of our civilization”.
    This concern is understandable: the loss of dedication to a fundamental notion of truth in science – namely truth which stands above any of the temporary fashions reflected in whatever “conventional wisdom” might prevail in a given era in the history of mankind –  is a very serious matter to anybody who believes that basic science is one of the very last bastions of rationality and integrity in contemporary civilization. Indeed, in similar cautionary words aimed at instrumentalism in general, Bertrand Russell pointed out that “the intoxication of power [reflected by the advocacy of an instrumentalist notion of ‘truth’], which invaded philosophy with Fichte, and to which modern men, whether philosophers or not, are prone ... is the greatest danger of our time, and any philosophy which, however unintentionally, contributes to it is increasing the danger of vast social disaster” (Russell, 1945, p. 1828). Hence, one does not have to subscribe to all, or even to some, of the tenets of Popper's form of realism in order to share his misgivings about the uncontested prevalence of an instrumentalist attitude in contemporary quantum physics.
    The reasons for this practically unchallenged dominance of instrumentalistic philosophy in quantum physics cannot be attributed to either its use of carefully and rigorously reasoned arguments, or to the revelation of some deeper and previously unsuspected funda-mental truths. The development of the S-matrix program, which enjoyed overwhelming popularity in the 1960s, only to fall out of favor in the 1970s, provides a good illustration:     
    “The dispersion-theory and S-matrix theory programs of the late 1950s and early 1960s had great appeal initially because they worked (i.e., they successfully related many directly measurable experimental quantities to each other). Of course, some of this success was ‘arranged’ (or greatly aided) since needed results (such as dispersion relations for massive particles and for nonforward directions, Regge asymptotic behavior, etc.) were assumed long before they could be proved (and many never were). ... These programs were characterized by a desire to ‘get on with things’, to ‘do something’. Cini (1980) and Pickering (1989a) have stressed the pragmatic aspect of these approaches and Schweber (1989) has suggested that this was a hallmark of much of theoretical physics after the Second World War (as contrasted with the period before the War).” (Cushing, 1990, p. 214).

    Thus, the trademark of conventionalistic instrumentalism was, and in a large measure still is, computational facility based on formal manipulations that disregard deeper physical, mathematical or epistemological questions44. Since its basic appeal is not to critical faculties, or to a sense of mathematical beauty, or to the desire to truly understand the workings of Nature, the reasons for its dominance must be primarily45 sociological. Indeed, a pragmatism reflecting primarily the desire to “get on with things”, even at the price of ignoring foundational issues, would not have surfaced to such an overwhelming degree in a science based on a quantum theory founded by individuals, such as Bohr, Born, Heisenberg, Pauli and Schrödinger46, with a deep concern with fundamental philosophical issues, were it not for a specific type of change in the social climate brought about in science, as well as in other spheres of human activity, by the Second World War47. Clearly, this social change has shaped a new generation of quantum physicists with a strong predisposition to conform, and “to follow the very latest fashion”. This was reinforced to a considerable extent by a tight control of institutional powers48, and by the exercise of those powers to shape mental attitudes and professional opinions, in a manner which systematically rewarded conformity and discouraged critical appraisals of the status quo.
    Some sociologists of science have documented these features in their studies of the “big science” that emerged after the Second World War49. However, since this sociological phenomenon lies outside the scope of the present monograph, it was pursued in the present chapter only in the context of specific instances, which dealt with the history and development of the pertinent ideas in quantum theory. Readers interested in it at a general level are referred to the work of Mitroff (1974), Pickering (1984), Savan (1988), and other sociologists of science cited in Sec. 12.3, who have written and published on this subject.
    On the other hand, there are various publications which try to rationalize the reasons and origins of the domination exercised by conventionalistic instrumentalism on contemporary quantum theory by presenting them as the natural outgrowth of the philosophy of the Copenhagen school. We hope that the brief historical survey in this chapter has convinced the reader that, although the Copenhagen school may have unwittingly created a fertile soil for the seeds of such ideas, the post-World War II conventionalistic instrumentalism is not in the least its brainchild – as witnessed by the publicly stated opposition50 of Dirac and Heisenberg, as well as others (cf. the quotation at the end of Sec. 12.2), to some of its practices, since its inception in the second half of the 1940s.
    Of course, some instrumentalists might argue that Bohr was on their side throughout his life. However, as pointed out in the most recent expository analysis of Niels Bohr's philosophy of physics, “it would be quite wrong to describe Bohr as a weak instrumentalist, because for the latter the truth, as distinct from empirical adequacy, of a physical theory is of no concern whatever.” (Murdoch, 1989, p. 222). Another recent analyst of Bohr's philosophical ideas has, independently, arrived at the same conclusion: “As there are various forms of realism, so there are different forms of anti-realism. The dominant one during Bohr's career was that of ‘instrumentalism’, the view that theoretical terms serve only as constructs enabling correct inferences to predictions concerning phenomena observed in specified circumstances. Many defenders of anti-realism also hold the view of ‘phenomenalism’, the assertion that the only reality of which we can form an idea with any content is that of phenomena, and that therefore statements about a reality behind phenomena are meaningless. Both of these views have been imputed to Bohr quite incorrectly.” (Folse, 1985, p. 195) – emphasis added. Indeed, some key correspondence between Bohr and Born is reproduced in (Folse, 1985), p. 248, which conclusively demonstrates that both these great physicists and founders of quantum mechanics were very decidedly opposed to the “instrumentalist standpoint”.
    There also are hundreds of publications, ranging from textbooks to popularizations of quantum theory in general, which are aimed at convincing their readers that giant strides were made by post-world War II physics not only in the realm of technology (which is indisputable), but also in the realm of fundamental ideas in quantum physics. Explicitly or implicitly, these publications ascribe all those purported successes to the conventionalistic outlook. The fact is, however, that if one leaves aside various extreme ideas in quantum cosmology51, then Schwinger's 1958 assessment of post-World War II developments in relativistic quantum physics can be, by and large, extrapolated to the present time52: at a fundamental level all post-World War II developments “have been largely dominated by questions of formalism and technique, and do not contain any fundamental improvement in the physical foundations of the theory” (Schwinger, 1958, p. xv).
    As discussed in Secs. 9.6 and 12.2, other physicists and historians of science, who took a careful look at those developments, have arrived at similar conclusions53. In particular, Dirac believed that the type of renormalization theory that became fashionable soon after the end of World War II represents “a drastic departure from logic. It changes the whole character of the theory, from logical deduction to a mere setting up of working rules.” (Dirac, 1965, p. 685).
    Thus, from an informed and purely rational point of view, the case in favor of adopting conventionalistic instrumentalism as a valid and fruitful philosophy for quantum theory rests exclusively with its systematically and widely advertised successes in the production of numerical predictions, which are purportedly in good to excellent agreement with the experiment results. When this claim is assessed, it should be recalled, however, that instead of deeming them as clear-cut confirmations of the advocated theories, Dirac suggested54 that even the agreements between the numerically most successful of models in quantum field theory (namely conventional QED) with the experimental results might be due to coincidence, and backed this observation with similar previous occurrences that took place in Bohr's semi- classical quantum theory of the 1910s.
    Indeed, when the theoretical manipulations are based on simply “discarding” undesired terms, and on “asymptotic” series in which the summation is carried out only as far as it is necessary for “agreement with observation”, the possibility of repeated occurrences of “coincidences” is not that easy to rule out. Furthermore, as discussed and documented in Sec. 9.6, as well as in this chapter, the analysis of the raw experimental data is prone to various types of systematic errors, whose likelihood increases dramatically once a strong predisposition exists to confirm a highly acclaimed theory (cf. Sec. 12.3, as well as Note 6). Healthy skepticism is therefore called for until the theoretical underpinnings of present-day fashionable theoretical models in high-energy physics are considerably strengthened, and the basic mathematical standards are fundamentally improved. It is only when all such theories become founded on sound mathematics – namely mathematics based on well-established canons of logical deduction, rather than on the “mere setting up of working rules” – that those believing in the rationality of science can attain the confidence that such theories provide a reliable account of quantum reality. And even for those who do not believe that there is a quantum reality, but that quantum theories are mere “instruments, which enable us, on the basis of the observed facts, to predict either with certainty or probabilistically the results of observations” (d'Espagnat, 1989, p. 27), such mathematical legitimacy can still provide the needed assurance of anthropic objectivity and reliability.
    The present quantum geometry framework has been formulated during the span of many years, with the above type of healthy skepticism in mind, but with the otherwise constructive and progressive type of attitude that is suggested by the quotation of Bertrand Russell heading this chapter. Thus, as opposed to other types of stochastic approaches to quantum theory (cf. Note 2 to Chapter 1), it was never the intention of the GS program to try to turn back the clock of history, and impose in quantum theory values derived from some kind of “physical realism” (Bunge, 1967; d'Espagnat, 1989) with its roots in classical physics. Rather, the challenge met was to try to understand the numerical successes of post-War War II relativistic quantum theory by developing mathematically sound methods, that would enable “successive approximations to the truth, [and] in which each new stage results from an improvement, not a rejection, of what has gone before”. On the other hand, another one of the principal aims of the program that eventually matured into the framework presented in this monograph, was to systematically reapply to quantum physics the traditional55 pre-World War II criteria of “scientific truthfulness” (Russell, 1948), rather than to rely exclusively on instrumentalist criteria based on “conventional wisdom” and on “general consensus”, that have become entrenched in the conventional relativistic quantum mechanics and quantum field theory of the post-World War II era.
    In order to have any chance at achieving such a goal, it became mandatory to dig deep into the foundations of relativity and quantum theory in general, and to appeal not only to physical insights and intuition, but also to a wide range of ideas and techniques of contemporary mathematics, as well as to carefully formulated epistemological studies of those foundations. The central conclusions reached in this manner, and which pertain primarily to foundations, were discussed in Secs. 12.4–12.6. Those sections also contain, sometimes in an explicit form, but mostly implicitly, the main tenets of a quantum realism which is distinct from both classical realism as well as from logical positivism, and yet incorporates key epistemological ideas from both these very fundamental philosophies of the twentieth century. Naturally, the acceptance of a philosophy that envisages a quantum reality which exists independently of whether we “observe” it or not, is not necessary for the understanding and application of the present quantum geometry framework any more than the comprehension and adoption of the philosophy of the Copenhagen school is necessary for acquiring a working knowledge of nonrelativistic quantum mechanics. However, as Heisenberg has emphasized in his last (1976) paper, in the long run, philosophical assumptions can play a decisive role in the formation and development of physical theories.  
    During the course of most of the post-World War II developments in relativistic quantum physics, concentrating one's attention on anything but the conventional formalism of quantum field theory has been very unfashionable, not only amongst theoretical physicists, but also in the dominant mathematical physics circles. Fortunately, the last decade has witnessed, at least amongst certain types of theoretical physicists and mathematicians, a growth of interest in deeper mathematical questions, that call for the development of advanced nonperturbative mathematical tools in relativistic quantum theory. It has also witnessed, amongst a relatively small number of yet another type of physicist, a gradual revival of professional concern with the deeper epistemological questions pertaining to the foundations of relativity and quantum physics. As a result: “Physics finds itself in recent years in an exciting and revolutionary phase of development: after a long intermission – and despite practical successes – critical questions about the proper foundations are being asked, and far-reaching attempts are being made to gain a deeper understanding of the whole structure of the theory of our time.” (Bleuler, 1991, p. 304).
    It is hoped that the epistemological ideas and mathematical techniques expounded in the present monograph will contribute to the future merging of the above mentioned two very healthy trends in contemporary relativistic quantum theory, and to their joint subsequent development.


  1 As related by Heisenberg in a 1968 talk delivered at ICTP in Trieste, during the early stages in the development of quantum mechanics, he himself thought that the most important philosophical idea was that of “introducing only observable quantities”. Then Heisenberg went on to say: “But when I had to give a talk about quantum mechanics in Berlin in 1926, Einstein listened to the talk and corrected this view. ... He said ‘whether you can observe a thing or not depends on the theory which you use. It is the theory which decides what can be observed’. His argument was like this: ‘Observation means that we construct some connection between a phenomenon and our realization of the phenomenon. There is something happening in the atom, the light is emitted, the light hits the photographic plate, we see the photographic plate and so on and so on. In the whole course of events between the atom and your eye and your consciousness you must assume that everything works as in the old physics. If you change the theory concerning this sequence of events then the course of observation would be altered’. ... Einstein had pointed out to me that it is really dangerous that one should only speak about observable quantities. Every reasonable theory will, besides all things which one can observe directly, also give the possibility of observing things more indirectly. ... I should also add that when one has invented a new scheme which concerns observable quantities, the decisive question is: which of the old concepts can you really abandon?” (cf. Salam, 1990, pp. 98-101).
  2 As will be discussed in Sec. 12.3, this reversal was basically unrelated to any deeper epistemological considerations – as is illustrated most dramatically by the rapidly changing “fashions” in the elementary particle physics of the post-World War II era. Thus, the reasons for this rather dramatic change in basic attitudes were purely social and sociological, and closely related to the global aftereffects of World War II, whereby the focus of advanced research in basic science was shifted to societies in which pragmatic and instrumentalist attitudes towards science were already generally entrenched (cf. Note 12).  
  3 The text of this talk (cf. Note 36 to Chapter 7) has been reprinted in (Salam, 1990), pp. 125-143, and the present quote (to which italics have been added for emphasis) can be found on pp. 137-138.
  4 An amusing and yet enlightening anecdote that illustrates Feynman's reaction to Dirac's verdict on renormalization theory is reported by A. D. Krisch (1987). It ends with the following observation: “What I concluded from this incident was that either Feynman shared Dirac's concerns [that an inelegant theory, such as QED, could not possibly be correct] or that there may be levels in the Theoretical ‘pecking order’ that are not easily observable to an experimenter.” (Krisch, 1987, p. 50).
  5 Most of the sections in this chapter provide, primarily in the form of additional notes, an elaboration and further documentation of those previously presented in an article entitled “Realism, Positivism, Instrumentalism and Quantum Geometry”, due to appear in an issue of Foundations of Physics dedicated to the ninetieth birthday of K. R. Popper. Much of this documentation is in the form of quotations from historical and sociological studies. In particular, pertinent quotations are provided from a recent scientific biography of Dirac by H. S. Kragh (1990), from a sociological study of theory selection in contemporary physics by J. T. Cushing (1990), and from a historical study of developments in particle physics in the 1950s by S. S. Schweber (1989). We would like to thank Cambridge University Press, which holds the exclusive copyrights to these publications, for the permission to extensively quote from them.
  6 The somewhat shaky experimental status of CGR might have been a contributing factor, as witnessed by the following observations: “Before [Eddington's famous account of the eclipse observations in] 1919 no one claimed to have obtained spectral shifts of the [by CGR] required size; but within one year of the announcement of the eclipse results several researchers reported finding the Einstein effect. The red-shift was confirmed because reputable people agreed to throw out a good part of their observations [emphasis added]. They did so in part because they believed in the theory; and they believed in the theory, again at least in part, because they believed that the British eclipse expeditions had confirmed it. Now the eclipse expeditions confirmed the theory only if part of the observations were thrown out and the discrepancies in the remainder ignored; Dyson and Eddington, who presented the results to the scientific world, threw out a good part of the data and ignored the discrepancies.” (Earman and Glymour, 1980, p. 85). Indeed, from the 1920s to the present time, a great many observations, as well as disputes over the validity of a number of experimental results that were originally claimed to support Einstein's CGR and its basic principles, seemed at times to almost invalidate it. It was only in the summary of the 1989 General Relativity and Gravitation conference that it could be finally stated with confidence that: “In view of the now quite manifold and accurate empirical evidence it seems, then, that there is no reason, at least in the macro-domain, to look for an alternative to Einstein's theory.” (Ehlers, 1990, p. 493).
  7 In fact, a recent analyst of Bohr's philosophy of physics has arrived at the following conclusions: “Thus Bohr was indeed a foe of the realistic understanding of particle and wave as viewed from within the classical framework. He was, in other words, against the realism that Einstein seemingly wanted to defend, what might be called ‘classical realism’. However, to conclude from this fact that he embraced an anti-realist understanding of science would require to assume that there is no other interpretation of science other than that which operates from the viewpoint of the classical framework. ... The reason for the common misreading of Bohr as an anti-realist lies not only in his attack against classical realism but also in his lack of any criticism of such an interpretation in quantum physics. But Bohr never attacked anti-realism not because he embraced this view but simply because he considered it foreign to the basic presuppositions of natural philosophy. ... He took it as empirically demonstrated that atomic systems were real objects which it is the goal of acceptable atomic theory to describe. At least as Bohr understood it, the debate was joined over the nature of the framework within which the description of such objects is to be understood.” (Folse, 1985, p. 22).
  8 Naturally, Bohr was aware of this fact, but chose to underestimate its significance. Thus, according to Petersen (1985, p. 305), Bohr once said jokingly: “Of course, it may be that when, in a thousand years, the electronic computers begin to talk, they will speak a language completely different from ours and lock us in asylums because they cannot communicate with us.” In the same spirit, it should be noted that instead of a thousand years, computers have started to “talk” less than fifty years after this statement might have been made, so that Bohr was wrong on that count; hopefully, he will also turn out to have been wrong on the rest of his prediction!
  9 For example, an expert on the complementarity principle has the following to say about this topic: “In spite of [the] dominance [of the complementarity principle] during this period of the awesome growth of atomic physics, the 'textbook' presentations of complementarity which introduced most physics students to Bohr's views hardly could be considered to do the subject justice.” (Folse, 1985, p. 27).
10 Because of the extensive use in this context of the term ‘regularization’, it might be thought that these procedures can be made mathematically legitimate by the use of the theory of distributions (Schwartz, 1945) and of generalized functions (Gel'fand et al., 1964-68). However, a distribution or generalized function is a continuous linear functional, so that the theory of distributions cannot handle nonlinear expressions, which are characteristic of interacting fields. This is precisely the difficulty encountered by the constructive quantum field theory program, whose ad hoc methods of trying to by-pass the need for defining directly the nonlinear terms in interactions of quantum fields represented by operator-valued distributions have eventually led to the conclusion that the interactions of primary interest in physics lead to trivial quantum field theories – cf. (Glimm and Jaffe, 1987), p. 120.
11 As opposed to a convergent series, whose partial sums approach in the limit well-defined values in its domain of convergence (so that they can be used to define a function in that domain as equal to the sum of that series), that is not the case for an asymptotic series. Consequently, in rigorous mathematics, an asymptotic series is always a series for an analytic function f(z), which has to be defined independently of that series. Indeed, in general, for a given value of z, the partial sums sn(z) of such a series approach with increasing n the value of f(z) up to a given optimum distance, reached for some n0(z), but then they get to be further and further away from f(z) as n becomes larger and larger. Borel summability can be used to reconstruct a function from a divergent power series (Hardy, 1949; Sokai, 1980), but the pre-conditions for its applicability are not satisfied in QED.
12 According to Dyson, who was one of the principal contributors of the very first of those fashions, namely renormalization theory, this is a general and necessary phenomenon in science. He says of his own experience: “When I first came [to the Institute of Advanced Studies in Princeton] as a visiting member 34 years ago, the ruling mandarin was Robert Oppenheimer. Oppenheimer decided which areas of physics were worth pursuing. His tastes always coincided with the most recent fashions. Being then young and ambitious, I came to him with a quick piece of work dealing with a fashionable problem and was duly awarded with a permanent appointment. ... I am now, after 30 years, one of the mandarins. I try in a vague and feeble way to encourage young physicists to work outside the fashionable areas, ... [but] the young people are compelled today to follow fashion by forces stronger than wording of contracts and the authority of mandarins.” (Dyson, 1983, p. 48). On a more general level, he states: “It has always been true, and it is true now more than ever, that the path of wisdom for a young scientist of mediocre talent is to follow the prevailing fashion. ... To find and keep a job you have to be competent in an area of science which the mandarins who control the job market find interesting. ... Anybody doing fundamental work in mathematical physics is almost certain to be unfashionable.” (ibid., pp. 47 and 53). However, while these assessments are most certainly very accurate for the post-World War II era, their degree of accuracy diminishes very rapidly when we look at the past history of science. The main work of most outstanding mathematicians and physicists in the preceding two centuries (e.g., Euler, Laplace, Lagrange, Gauss, Maxwell, etc.) was in mathematical physics, and yet they enjoyed status and recognition in their own times, whereas that would not be the case in the post-World War II era  – cf. also (Popper, 1982a), as well as the next note.
13 Cf. Note 5 as well as the introductory paragraphs to this chapter. Another interesting example is provided by the developments in cosmology in the late 1920s, when “it was taken for granted that the universe must be static – despite data being available that would shortly be taken to prove the contrary [emphasis added], with at least three published papers proposing the idea of an expanding universe” (Ellis, 1989, p. 379). More recent examples, related to the theory of parity violations as well as to electroweak interactions, are extensively documented by Franklin (1986, 1990). The philosophy behind the development of the S-matrix program, which enjoyed overwhelming popularity in the 1960s, has been recently analyzed by Cushing (1990).  
14 The most significant influence on the developments of post-World War II physics was the shift of central focus from Europe to North-America, and the concurrent entrance into the era of “big science”. As Bertrand Russell pointed out as early as 1945, “it is natural that the strongest appeal of [John Dewey] should be to Americans [since all men's views] are influenced by [their] social  environment.” It is therefore not surprising that, as documented by many studies in the sociology of science, such as those by Mitroff (1974) and Savan (1988), one of the corollaries of shift development was, on one hand, the introduction of Madison Avenue techniques in the promotion of scientific ideas, and, on the other hand, of the elimination of ideas that challenged whatever fashionable orthodoxy the “mandarins” (cf. Note 10) chose to enforce at a given time. This was achieved not by means of public debates, as in the pre-World War II era, but rather by the tight control exerted by the North American scientific establishment over the publication and dissemination of scientific ideas. “I can't find any fundamental difference between the scientific method and the procedures for making progress in business and the arts” says one of the North American physicists interviewed by Mitroff (1974, p. 65). As a natural consequence, the “selling of ideas” (Polkinghorne, 1985) became a generally accepted practice in much of theoretical and mathematical physics: “Some [people] are very successful in pure science but it really isn't pure; nobody is pure. ... People want to sell their point of view, beat down the other guy because it means more glory, more ego satisfaction, more money” says another of the physicists interviewed by Mitroff (1974, p. 70).
15 Cf. (Kragh, 1990), p. 278. On the other hand,  there is no doubt that as far as mathematical rigor is concerned, Dirac's standards were rather lax, as witnessed by the critical remarks of Birkhoff' that are cited in (Kragh, 1990), pp. 279-280. However, traditionally the standards of mathematical argumentation and proof used in physics, even in the pre-instrumentalist era, were not as demanding as those in pure mathematics – and Dirac was certainly no exception to that rule. Hence, he insisted only on mathematically sound arguments, by which he obviously meant arguments in which not all details are carefully spelled out (as is, in fact, the case with many arguments in this monograph, as well as in [P], as opposed to those in [PQ]); or even arguments in which the entire deductive approach requires modification in order to produce a rigorous version, but which are at least capable of such modifications. Such mathematical “soundness”, rather than rigor, was exhibited by many of Dirac's own mathematical concepts and arguments, which eventually received a mathematically rigorous treatment – the theory of distributions of Schwartz (1945), that emerged from Dirac's introduction and use of the delta-“function”, being the best known example. In contradistinction, in conventional renormalization theory the Feynman rules reflect only a physical heuristics whose end products (namely the “perturbation series” for various processes in QED and in other conventional quantum field theoretical models) are not capable of receiving mathematically rigorous justification – as was pointed out in Sec. 9.6. In fact, an acknowledgedly asymptotic series cannot serve as the basis for any kind of reliable computation, in the absence of an independent proof of existence of the function whose expansion it is supposed to represent, capable of producing independent estimates for the values of that function. For example, “Dyson estimates in quantum electrodynamics the terms of the [perturbation] series will decrease to a minimum and then increase again without limit” ([SI], p. 644). However, how does one know that, first of all, the S-matrix exists in QED (according to Glimm and Jaffe (1987), it probably exists, but it is trivial!); and second, even if an independent existence proof is produced, how does one estimate how far a “perturbative” partial sum for a given process is from the actual total S-matrix theoretical prediction for that same process?   
16 Cf. (Russell, 1945), p. 816. With characteristic wit, Russell also writes: “With James's definition [of truth], it might happen that ‘A exists’ is true although in fact A does not exist. I have found always that the hypothesis of Santa Claus ‘works satisfactorily in the widest sense of the word’; therefore 'Santa Claus exists' is true, although Santa Claus does not exist.” (Russell, 1945, pp. 817-818). On a more serious note, the following is a common reaction amongst those philosophers who are critical of the instrumentalist criteria for truth in everyday life, as well as in mathematics, science and philosophy: “To say that the truth of a belief or judgement depends on its practical consequences was to debase truth to considerations of personal profit or to other mercenary aims, while the attempt to enlarge the scope of the practical so as to include the abstract results of mathematical analysis or theoretical conclusions in pure physics was to deprive the word, practice, of any distinctive meaning.” (Mackay, 1961, p. 393). On a loftier plane, a systematic case against instrumentalism in science is made in (Popper, 1983), pp. 111-131, where the positions of some other well-known philosophers with instrumentalist leanings are analyzed and criticized.  
17 One might be prepared to believe that modern scientists are immune to the effects of such “intoxication of power”, even if they subscribe, either explicitly and openly, or only in the manner in which they conduct their professional activities, to instrumentalist doctrines. A few nagging doubts might surface, however, in one's mind as one reads some of the over-confident claims following the emergence in the 1980s of superstring theory as the “Theory of Everything” – claims totally opposite in spirit to the humility exhibited by Newton, Einstein, Dirac and many other truly great physicists, as they contemplated the limitations of their own theories, when confronted with the mysteries of Nature. For example, quotations in Note 24 to Chapter 11 might be compared with the following prophetic quotation from the article “The Evolution of the Physicist's Picture of Nature” by P.A.M. Dirac: “There are a good many problems left over concerning particles other than those that come into electrodynamics: ... how to introduce the fundamental length to physics in some natural way, how to explain the ratio of masses of the elementary particles and how to explain their other properties. I believe that separate ideas will be needed to solve these distinct problems and that they will be solved one at a time through successive stages in the future evolution of physics. At this point I find myself in disagreement with most physicists. They are inclined to think one master idea will be discovered that will solve all these problems together.” (Dirac, 1963, p. 50).    
18 The reader who desires illustrations of such an “anti-rationalist atmosphere which has become a major menace of our time” can easily find many examples even amongst the references cited in this monograph. Unfortunately for the future of some areas of quantum physics, this “menace of our time” represented by the practice and imposition of unadulterated instrumentalism is not only figurative, but a very real threat to all researchers who oppose this trend. Indeed, after taking control of funding agencies in many countries that are in the foreground of pure research (cf., e.g., Note 47), instrumentalists have in some instances prevailed on the bureaucracies in those agencies to support their own research at the expense of research dedicated to traditional values in science. As a rule, the argument offered is that traditionally-oriented research, based on goals and values that used to be the hallmark of all basic research in the pre-instrumentalist era, is no longer “in the mainstream”. It is largely due to such practices that contemporary instrumentalists have succeeded to eliminate in almost all areas of theoretical and mathematical physics the very last traces of significant opposition to their doctrines, which set “belonging to the mainstream” and “following the general consensus” ahead of the search for actual truth, and a deeper understanding of Nature. Of course, by the systematic use of such means of “persuasion”, the assertion that “truth is the opinion which is fated to be ultimately agreed to by all who investigate” regrettably becomes mere self-fulfilling and self-serving prophecy. And, unfortunately, by rewarding conformity, such professional practices thwart initiative and stultify the spirit of free inquiry in science.  
19 Admittedly, it cannot be said that everything is satisfactory in the world of contemporary mathematics for those in search of objective truth, rather than solutions to fashionable problems. A trend decreeing, under the banner of “mathematics for mathematics' sake”, that the ultimate arbiter of what is valuable in mathematics lies exclusively in the opinion of “leading mathematicians”, rather than in its potential of solving problems of the real world around us, had emerged and became dominant in this century soon after the confrontation between Hilbert and Brouwer in the late 1920's. As most mathematicians uncritically sided with Hilbert (van Dalen, 1990), the aftermath practically destroyed the intuitionistic school, and set the cause of constructivism in mathematics back by decades – albeit such a great mathematician as H. Weyl, generally deemed to be the successor of Hilbert in both depth and stature, remained predisposed to the intuitionistic as well as the applied point of view (“Mathematics with Brouwer gains its highest intuitive clarity. ... [I]t is the function of mathematics to be at the service of the natural sciences.“ – cf. Weyl, 1949, pp. 54 and 61). This unfortunate state of affairs was compounded by the difficulties which Hilbert's ambitious program (Weyl, 1949; Reid, 1986), aimed at establishing the consistency of all the major areas of mathematics, encountered after the discovery of Gödel's (1931) incompleteness theorem – cf. (Kline, 1980), pp. 260-264. On the other hand, in contemporary mathematics the pursuit of fashions is considerably more subdued than in quantum physics in general, and in elementary particle physics in particular. Furthermore, although even such a fundamental question as the consistency of arithmetic remains unresolved, at least as long as that consistency is accepted together with the law of excluded middle, an objective state of affairs prevails with regard to the criteria for mathematical truth. Hence, although what is at present deemed to be “deep” mathematics is very much a function of fashions dictated by the prevailing circles of “mandarins” (cf. Notes 10, 16, 18, 20 and 22), at least what is deemed to be valid and valuable mathematics is not exclusively a function of their pronouncements.
20 The well-documented (Kline, 1980) isolation of modern pure mathematics from all applications was most definitely a contributing factor to this counterproductive breakdown. Courant is cited to have remarked as early as 1927: “The predominant characteristic of American mathematicians seemed to be a tendency to favor abstract and the so-called areas of pure mathematics. ... Applied mathematics was treated as a stepchild in America.” (Reid, 1986, p. 382). With the post-World War II shift of focus in mainstream research from Europe to North America, this fact no doubt became one of the major factors that enlarged the chasm which emerged between the physics and mathematics communities during the second half of this century – a chasm which has begun to be bridged to a significant degree only in the course of the last decade.
21 Indeed, all physical theories, from Newtonian mechanics onwards, were developed by individuals who either simultaneously introduced the required mathematical tools at a level commensurate with the prevailing mathematical standards of their generation, or worked in close contact and collaboration with competent mathematicians, who helped steer them away from deductive mathematical errors that might have affected the physical content and predictions of their theories. In recent history, the best example is the 1913-1915 period in the development of general relativity by Einstein (Norton, 1989). That period started with Einstein's collaboration with his mathematician-friend M. Grossmann, and culminated in the triumphant final version of classical general relativity systematically presented by Einstein in 1916. The painstaking historical research by Norton (1987, 1989), Stachel (1980, 1989), and others, vividly illustrates how Einstein's superb physical intuition for once led him astray, so that in his Entwurf paper (Einstein and Grossmann, 1913) he discarded, on the basis of physical misconceptions, the requirement of general covariance. Then, for two full years he expressed in public as well as in private satisfaction with non-generally covariant equations – cf. (Cattanin and De Maria, 1989), p. 179. The constructive criticism and suggestions of such outstanding mathematicians as Hilbert and Levi-Civita emerges as a major, and perhaps even as the decisive factor (cf. Cattanin and De Maria, 1989, p. 185), which eventually enabled Einstein to publicly present to the Berlin Academy the correct field equations, in their final form (2.7.3), on November 25, 1915 – namely five days after, unbeknownst to him, Hilbert had already presented the same equations to the Göttingen Academy (Norton, 1989, p. 150).
22 Some of the examples which we shall provide in this chapter lead to straight contradictions, so that they would be unacceptable even to those mathematicians who accept the following evaluation: “There is no rigorous definition of rigor. A proof is accepted if it obtains the endorsement of the leading specialists of the time or employs the principles that are fashionable at the moment. But no standard is universally accepted today.” (Kline, 1980, p. 315). In this context, it should be noted that in this monograph the term “mathematically rigorous” is used as a short version for “mathematically acceptable by generally agreed upon contemporary standards in the mainstream areas of mathematics”. The present author is highly sympathetic to the constructivist school in mathematics, and hopes that one day it will prevail and supply constructive proofs for all theorems of relevance in applications – but that day seems to be still far off in the future. For the present, however, existence proofs relying on the law of the excluded middle are certainly preferable to no proofs at all, and to the blind application of “working rules” for arriving at “theoretical results in agreement with experiments” (cf. Note 6 on this last score).
23 Cf. (Savan, 1988) for a general analysis and documentation of this phenomenon in modern science. As for the situation in high-energy physics specifically, S. S. Schweber asks the following pertinent questions, and then provides some answers: “Did the involvement of many of the leading American high-energy theorists in defense matters reinforce a particular kind of theoretical orientation – pragmatic, phenomenological, with ‘S matrix theory’ as its most impressive statement – to the exclusion of others? Did it affect developments in theoretical high-energy physics? I would suggest that the fragmentation of interests by these leading theorists, stemming from their consulting and their involvement in defense matters, hindered – and to a certain extent prevented – their maintaining a sustained effort on fundamental theory. Also, in their capacity as reviewers of research proposals, and by virtue of their dominance in the funding process, they tended to reinforce their dominant view.” (Schweber, 1989), p. 681.
24 Naturally, a number of textbooks aimed at would-be “mathematical physicists” eventually made their appearance – of which the original 1971 edition of [PQ] represented perhaps the first attempt at rederiving all the major results presented in a typical “mainstream” textbook on nonrelativistic quantum mechanics in a mathematically acceptable manner. Unfortunately, although by the 1980s a score of very good textbooks and monographs published by various mathematical physicists made it easy for potential authors of “mainstream” textbooks on quantum theory to raise their mathematical standards to an acceptable level, that opportunity was ignored – and it is still being by and large ignored. Clearly, the underlying feeling must be that, since “truth” can be identified  with the “professional consensus as to what works”, and since that “consensus” was reached and firmly established within the profession by the working practices of the leading physicists of the post-World War II generation (namely Dyson's “mandarins” – cf. Note 12), which “work well” due to that very same consensus, there is no room left for further doubts, or for any critical reconsiderations of those practices.  
25 Cf. [PQ], p. 195. The Hellinger-Toeplitz theorem is a special case of the (to mathematicians) very well-known closed-graph theorem (which is stated and proved on p. 210 of [PQ] for the case of closed operators in Hilbert space), but which holds for much more general cases of topological vector spaces.
26 Naturally, after the resolutions in (3.3) are applied to all pairs of vectors from H+ , the resulting Lebesgue integrals can be extended to all of H by virtue of the fact that H+ is dense in H. But such a procedure merely reproduces the formula (3.1.1) for the inner product in H, and still leaves open the question as to how to specify a wave function at every single point x in R3 so that an extension to wave functions in H+ of the formal inner product in (3.4) would be achieved. It should be noted that, for the sake of notational simplicity, in all these considerations Berezanskii's (1968) notation for equipped Hilbert spaces is being used, but all the presented arguments apply equally well to rigged Hilbert spaces.
27 Such as to Schwartz S-spaces in the rigged Hilbert space approaches (Gel'fand et al., 1964; Antoine, 1969). In the equipped Hilbert space approaches (Berezanskii, 1968; Prugovecki, 1973)  H+  and H  are both Hilbert spaces, with H+ being the domain of an unbounded equipping operator, and H the range of its extension to H.
28 In Chapter V of [PQ] it is shown how one can start, in a physically legitimate and meaningful manner, from the time-dependent approach to scattering theory, and then derive from it in a mathematically rigorous manner all the main results of the stationary approach that are formalistically derived in typical textbooks on conventional quantum scattering theory exclusively in the stationary context.
29 This formula was originally derived in (Prugovecki, 1978b), pp. 240-247, in the context of developing a quantum mechanical counterpart of the well-known Boltzmann equation – cf., e.g., (Balescu, 1975). An alternative derivation was subsequently provided by Turner and Snider (1980), which corresponds to applying the sharp-momentum limit in (3.5.5) to (3.7). However, the physical significance of such a limit is rather questionable, and a problem of mathematical existence also emerges.
30 Cf., e.g., Note 25 to Chapter 9. The documentation of such cases could easily fill volumes of the same length as the present monograph – which is, however, chiefly concerned with the constructive task of setting the worthwhile aspects of present-day conventional theories on a solid foundation, and only incidentally (as well as rather reluctantly) with their critical evaluation. The fact that not many such volumes are in existence is not primarily due to lack of material, or even to a total absence of dedicated individuals willing to engage in such a thankless task. The chief explanation is that, from the point of view of the individual researcher striving for professional survival – not to mention professional recognition – it can be a professionally self-destructive enterprise to collect and document fallacies of institutionally strongly sponsored points of view, in the face of the multifarious devices for pressure and control (Savan, 1988) that are exerted in the modern era of “big science” in order to enforce and preserve conformity. Of course, such professional pressures would have been ineffective against such outstanding and well-known physicists as Dirac and Heisenberg, but their use against those who have heeded their open and justified criticisms has until recently made that criticism virtually counterproductive. Hence, as will be pointed out in Sec. 12.7, it was only in the course of the last few years that open and critical inquiry into the foundations of quantum mechanics and quantum field theory has begun to reassert itself at the international level.
31 We use Dirac's own stipulation of “mathematical soundness”, rather than the stronger condition of mathematical rigor (cf. Note 12). Eventual mathematically rigorous justification, rather than mere reliance on “agreement with experiment” is, however, especially essential at the present technological frontier of measurements in the microdomain, where the independent and repeated verifiability of experimental results becomes ever more questionable. Indeed, in contemporary elementary particle physics, very expensive experiments are carried out with costly experimental equipment, which, of necessity (Yaes, 1974), has become the monopoly of a handful of teams of experimentalists, who work in close contact with leading proponents of the theories they are verifying. Although such contact is in many respects desirable, it can be also conducive to erroneous analyses of experimental results, produced under various types of conscious or unconscious professional influence. A senior elementary particle physicist, who clearly perceives only the positive aspects of such close contact, and wholeheartedly approves of it, describes its effects as follows: “Constructing modern theories also means constructing new concepts and abandoning old ones ... [as it] would be obvious to all if all had a chance to experience life in a great research center in fundamental physics. In such places ... a permanent exchange of views is observed to take place between the two teams of people [namely experimentalists and theorists]; they seem both  to understand and to need each other. When we see all  this going on, it is not hard to appreciate that in order to make sense of the mass of data provided by the experimentalists, the theorists have to create new concepts.“ (d'Espagnat, 1987, p. 40) – emphasis added. Thus, in practice, the empirical verification of a theory is not all a mere matter of comparing “theoretical predictions” with “experimental results”. And, once officially sanctioned by being accepted for publication in professional journals, such results tend to become uncritically accepted as unconditionally valid – with occasionally published retractions proving the existence of occurrences of faulty analyses of data, but not giving any ideas as to the frequencies of such occurrences. Hence, the impossibility of their routine reproducibility in practice, and therefore of plentiful and independent verification, makes the acceptance of the remaining ones to a considerable degree a matter of subjective faith. Relevant examples of well-documented cases where that faith might have been misplaced can be found in Note 25 to Chapter 9.
32 The fact that such proofs are absolutely necessary is indicated already by mathematically rigorous results in nonrelativistic quantum scattering theory. Thus, contrary to assertions made in some textbooks on quantum mechanics that the unitarity of the S-“matrix” (i.e., of the scattering operator S = W*W+) is a consequence of the “conservation of probability” (i.e., of the fact that the time-evolution governed by the Schrödinger equation is represented by a family of unitary operators, which, as such, preserve all transition probabilities), such assertions are actually false: a mathematically rigorous theorem (cf. Thm. 2.5 in [PQ], p. 443) shows that, even if the initial domains, M+ and M , of the partial isometries W+ and W (representing Møller wave operators) are equal, a necessary as well as sufficient condition for the unitarity of S is that the ranges R+ and R of these wave operators be equal. An early but physically artificial model by Kato and Kuroda (1959) has shown that it can happen that R+R even when time-evolution is unitary, whereas later Pearson (1975) rigorously demonstrated that it can happen that R+R even in physically acceptable models. Pearson's model employs a local potential that oscillates ever more rapidly as one approaches the origin of the center-of-mass reference system, so that a quantum particle in certain asymptotically free incoming states becomes forever trapped in that potential, and does not produce corresponding asymptotically free outgoing states.     
33 J. Gribbin recounts the amusing circumstances of Tryon's creative spark, which triggered his “creation ex nihilo” idea, whereby at one of Sciama' seminars, “Tryon blurted out, to his own surprise as much as everyone else's, ‘maybe the Universe is a vacuum fluctuation!’” (Gribbin, 1986, p. 376). The sober evaluation of  its meaning and significance, outside the realm of religion or science fiction, actually does not encounter any problems with the superficial appearance that such a “concept of the universe being created from nothing is a crazy one” (Vilenkin, 1982, p. 26). Indeed, the underlying mathematics is rudimentary and well understood by any student of quantum mechanics (Vilenkin, 1982, 1988); whereas, at first sight, Einstein's ideas on relativity theory appeared much “crazier” to some of his contemporaries. The crucial difference is that Einstein's ideas were operationally well-founded, empirically directly testable, and ontologically sensible. But what is it that is supposedly “tunneling”, and through a barrier of what does that purported “tunneling” take place in the Tryon-Vilenkin “scenario”? The fact that such basic questions are not even asked (not to mention answered) by conventionally-minded instrumentalists in quantum cosmology forcefully illustrates the general grounds for the type of concern voiced in Heisenberg's last paper (cf. Sec. 1.5). Indeed, instead of being provided with such questions and answers, we are simply authoritatively told that “the only relevant question seems to be whether or not the spontaneous creation of universes is possible”, and that, after all, “obviously, we must live in one of the rare universes which tunneled to the symmetric vacuum state” (ibid., p. 27). Of course, the most conventionalistic of instrumentalists argue that we are to judge any “theory” exclusively by its “observational consequences”. Does that mean, however, that if ancient Greek mythology, present-day religious fables, or even ordinary fairy tales make “predictions” which are indeed in accordance with certain “observations“, then we are to accept them as serious contenders for “valid” scientific theories?
34 Perhaps Dirac's determination to publicly condemn the replacement of science by technology can be traced to his experiences as a student in an engineering college: “In [the] engineering courses [which he then attended] the emphasis was on mathematical rules with the help of which problems could be solved, without giving strict proofs or asking how or why the rules worked. Dirac remembered that there always remained a kind of magic about these rules, and frequently he had a strange feeling about how he ever got answers out of them.” (Mehra and Rechenberg, 1982, vol. 4, p. 11). However, in engineering those rules were at least deduced from an underlying consistent physical theory based on sound mathematics, but that is not the case in conventional renormalization theory. Hence, six decades later he was to say: “Working with the present foundations [of conventional quantum field theory], people have done an awful lot of work in making applications in which they find rules for discarding the infinities. But these rules, even though they may lead to results in agreement with observations, are artificial rules, and I just cannot accept that the present foundations are correct.” (Dirac 1978a, p. 20).
35 A well-documented example of systematic propagation of errors in the analysis of the raw experimental data in measurements of large-scale spatio-temporal separations is provided by the changes in the estimates of Cepheid distances by a factor of 2.6 in 1952, and by another 2.2 factor in 1958 (Ellis, 1989, p. 391). This led to radical changes in estimates of the age of our universe, as well as in estimates of all intergalactic distances.
32 In a recent monograph, Cushing (1990) provides an exhaustive account and analysis of the rise and fall of the S-matrix program in elementary particle physics. It began as follows: “Cosmic ray showers (or ‘explosions’) and the divergence of cross-sections beyond a certain energy in a classical (nonlinear) field theory version of Fermi's beta-decay formalism were taken by Heisenberg (1936, 1938a) to indicate the existence of a fundamental length and the need for a profound revision of elementary particle dynamics. Not knowing what the future theory might be, he proposed the S-matrix theory as an interim program.’ (Cushing, 1990, p. 33) – emphasis added. Later on Landau (1955) expressed his conviction that the only directly observable physical quantities were those associated with asymptotically free particles, such as their initial or final momenta before and after a scattering process. He therefore concluded that any quantum fields interpolated between asymptotic states were physically meaningless, and  advocated a break with quantum field theory, while supporting a program similar to the one of Heisenberg. However, it was Chew who ultimately “made a radical break with quantum field theory” (ibid., p. 167). The role of sociological factors in the adoption of theories in high-energy physics receives separate attention in Sec. 10.2 of (Cushing, 1990), where it is pointed out that “the very nature of scientific practice has changed significantly with the advent of ‘big science’ after the Second World War.”  
37 For example, in his talk delivered at the Twelfth Solvay Conference in Physics, M. L. Goldberger said: “My own feeling is that we have learned a great deal from field theory as we shall see, even dispersion theory came from it; that I am quite happy to discard it as an old but friendly mistress, who I would even be willing to recognize on the street if I should encounter her again. ... It is perhaps correct to say that much of the deeper philosophy of the S-matrix approach held by some of us, in particular Chew, who believe that there are no elementary particles, and that there are no undetermined dimensionless constants in the theory, has not yet been put to a test.” (Goldberger, 1961, pp. 179-180). According to G. F. Chew: “The capacity for experimental predictions is the only reliable measure of a physical theory. ... No suggestion is being made that space and time do not continue to be the basis of macroscopic physics; ... Does this mean that there can be no continuous connection between the microscopic and macroscopic worlds? The situation is no more uncomfortable than it has always been for quantum theory, where the conventional explanation of the relation between the classical observer and quantum laws leaves most people feeling queasy.” (Chew, 1963, pp. 533, 538). Of course, after the revival of interest in quantum field theories in the 1970s, this “uncomfortable situation” was forgotten: “The development of S-matrix theory was characterized by a certain degree of sectarian strife. ... It was not so much a question of its being expedient to be on the mass-shell as of its being sinful to be anywhere else. In particular, [the advocates of the S-matrix program] proclaimed the demise of quantum field theory. ... [However] the S-matrix endeavor looks a good deal less beguiling [now, in the late 1970s] than it did in those brave early days [namely the 1960s].” (Polkinghorne, 1979, p. 87).
38 Louis de Broglie recalls this emergence as follows: “When, in 1922-1923, I had my first ideas about wave mechanics, I was guided by the vision of constructing a true physical synthesis, resting upon precise concepts, of the coexistence of waves and particles. I never questioned then the nature of the physical reality of waves and particles. ... I also noticed that if the particle was regarded as containing the rest energy M0c2 = hv0, it was natural to compare it with a small clock of proper frequency v0.” (de Broglie, 1979, p. 7).
39 Cf. (Kim and Noz, 1986), Chapter V, where such a mathematical treatment is provided in the context of a quark model for mesons. However, a wave function that is square integrable in space and in time is of questionable physical significance even in the nonrelativistic regime, since it suggests that whatever entity is described by it spontaneously disappears from existence the further we look into the distant past or into the distant future.
40 The mathematical heuristics of this procedure was presented in (Prugovecki, 1988a), and further elaborated in Appendix A of (Prugovecki, 1989b), which the present review basically reproduces. However, as mentioned in Sec. 1.5, originally such considerations were used in the treatment of hadrons as GS excitons. That led to a mass formula (Prugovecki, 1981b), which produced Regge trajectories that were found (Brooke and Guz, 1984) to be in otherwise good agreement with the experimental data available at that time.
41 The very rough estimates in (Greenwood and Prugovecki, 1984) do not indicate that the prospects are very good. However, the intrinsic non-linearity of GS models for interacting quantum fields might produce new and unexpected results if numerical computations are performed even by the use of existing lattice approximation methods.
42 It is the neglect of this most crucial aspect of quantum propagation that causes Popper's (1967, 1982, 1988) “propensity interpretation” of quantum mechanics not to come even close to an adequate depiction of quantum phenomena – as persuasively demonstrated by a number of critics (cf. Jammer, 1974, pp. 448-453). A similar neglect also makes Nelson's (1986) “stochastic mechanics” depiction of the two-slit experiment to be totally at odds with quantum reality.
43 Due to lack of space, in this chapter we have concentrated most of the attention on the principal protagonists in the historical drama in the quantum physics of this century, that began with the confrontation between Einstein and Bohr in the 1920s, and after World War II developed into a historically most paradoxical situation, in which the usual stereotypes about the conservatism of “older” generations vs. the radicalism of “younger” generations in all walks of life are totally reversed: during the second half of this century, the “older generation” of theoretical physicists, incorporating all those who founded quantum theory, remained “revolutionary” in its outlook; whereas the “younger generation” turned out to be deeply “conservative”, as well as strikingly conformist in all the principal facets of its regular professional practice. For example, in a recent article entitled “Wolfgang Pauli: His Scientific Work and His Ideas on the Foundations of Physics”, the following was pointed out by Bleuler (1991, pp. 306-307): “I would like to conjecture that Pauli himself already had, in an early stage of the development of quantum field theory, profound doubts as to its adequacy and mathematical consistency as a general theory of elementary particles. He refused to be ‘renormalized’, as he expressed himself in relation to that famous principle of renormalisation, thus setting himself (together, however, with Dirac, Bohr and others) for a time into strict opposition with an enthusiastic, and at first very successful, younger generation. A (partial?) concession in this respect came only after Pauli's death, firstly in connection with the (unsuccessful?) attempts to ‘save’ local quantum field theory by seeking recourse to ‘strings’, and secondly in the recent profound and far-reaching attempts at a ‘non-commutative geometry’ of A. Connes, D. Kastler and others. It seems to me that his most recent (as yet not generally recognized) development is perhaps the fulfilment of a ‘vision’, which Pauli expressed in a long and unforgettable discussion a few weeks before his death: ‘For a real solution of the problem of singularities (i.e. the question of renormalization) a step of the same size and significance as that which was taken once before in the twenties might be necessary’.”
44 Ironically, it appears that only when a given instrumentalistically motivated theory begins to “fall out of fashion” that the concern of its still faithful adherents starts to shift to this type of questions. For example: “At the same time that the predictive fertility of the S-matrix program waned, it continued to have considerable philosophical appeal (Cushing, 1985). ... The duality and superstring models also became theory-driven, having little contact with experiment (Schwarz, 1975, p. 67; 1982, p. 7). Consistency, potential scope, and hoped-for contact (in a limited regime) with an empirically adequate theory (such as QCD) remained the major motivations for pursuit.” (Cushing, 1990, p. 215).
45 We italicized “primarily” since, of course, sociological factors have always played a role in science: “Pickering (1984) has presented the process of choice (or judgment) as a largely social exercise. In the tradition of the radical relativist-constructivist program in the sociology of knowledge, he has attempted to show that not only the form, but even the very content, of scientific knowledge is sociologically determined. ... However, he (1989b) has recently argued for a pragmatic realism in which not just anything goes. ... Galison (1987), in his study of the change in experimental practice in high-energy physics during the twentieth century, argues convincingly that it is not by deductive reasoning alone that scientists pass from the raw data of an experiment to a conviction that an effect has been seen.” (Cushing, 1990, p. 217). Thus, it is a matter of the degree to which sociological factors have become predominant in contemporary quantum physics.
46 A vividly drawn portrait of Schrödinger can be found in (Bernstein, 1991, pp. 32-33, where the following is pointed out: “All the inventors of the quantum theory, as it happened, were men of broad culture, perhaps attributable to their European gymnasium educations, but even in this group Schrödinger stood out.” This “broad culture” is in sharp contrast to the “newly developing cult of narrowness” exhibited by the “many younger physicists who have grown up in [the] period of over-specialization” (Popper, 1982a, p. 100) of the post-World War II era.
47 This change in social climate is described and documented by Schweber (1989, 1991), who states the following: “World War II altered the character of science in a fundamental and irreversible way: the importance and magnitude of the contribution of scientists and engineers, particularly physicists, to the American war effort changed the relationship between the scientists and the military, industry, and government. The Department of Defense, realizing that the security of the nation depended on the strength and creativity of the scientific community, invested heavily in both their support and control.” (Schweber, 1989, p. 670). Later on he describes how widespread this phenomenon soon became: “This ‘American’ style of doing physics was characteristic of the great wartime laboratories: the Radiation Laboratory at MIT, the Met Lab in Chicago, and Los Alamos. It was in these wartime laboratories that many of the outstanding theoreticians of the 1950s were molded: Feynman, Goldberger, Chew, Robert Marshak. It is a style that became institutionalized at all the leading departments during the fifties and became the national norm. ...  At the leading high-energy centers, [the] fortunes and future [of talented young people] were often determined by their skill in explaining experimental results, and more generally by their usefulness to their experimental colleagues; the latter had invested enormous energy, skills, and government resources in building their high-energy machines.” (ibid., p. 672). In the end this phenomenon became “hegemonic worldwide”: “The defense connection during the 1950s reinforced the pragmatic, utilitarian, instrumental style so characteristic of theoretical physics in the United States. The successes of this mode of doing theoretical physics help explain its diffusion to Europe and elsewhere. The pragmatic ideal of American physics that had been visible from early on now became not only the national norm but in fact hegemonic worldwide.”  (ibid., p. 673).
48 Indeed, “the structure of the scientific community is that of a pyramid, the apex being occupied by the relatively few creative people [cf. the description of Dyson's ‘mandarins’ in Note 13] who can invent and sustain successful theories. They ultimately make the rules of the game.” (Cushing, 1990, p. 253). Another source points out the following: “It is as though most of the members of the community consider it worthwhile to work out the approach suggested by the intellectual leader of the moment (e.g., Gell-Mann, Mandelstam, Chew) than work on their own ideas or on longer-range programs of research. As early as 1951 Feynman called it the ‘pack’ effect. The work on dispersion relations after Gell-Mann's and Goldberger's initial papers is an example of this phenomenon; the almost wholesale adoption of Chew's S-matrix program is another. The community at one time or another seems to be dominated by a single individual. Gell-Mann, Goldberger, Lee, Yang, and Chew were the dominant figures from the mid-1950s to the mid-1960s, a role Steven Weinberg assumed in the late sixties.” (Schweber, 1989, p. 673).
49 Cf. Schweber (1989, 1990). The controversial writings of Kuhn (1970) and Feyerabend (1975) might make it appear that such sociological phenomena are universal in the history of science, being part and parcel of its very methodology. This view is disputed, with ample documentation, by Franklin (1986), and implicitly also by Cushing (1990) – cf., e.g., the last quotation in Note 25 to Chapter 9.    
50 As a recent biographer of Dirac has stated: “Dirac firmly believed that a new revolution was needed. His lack of sympathy for the new quantum electrodynamics involved a lack of appreciation for the values of the new [i.e., post-World War II] generation of physicists.” (Kragh, 1990, p. 184). A similar “lack of appreciation” was displayed by Heisenberg (1976).
51 We refer here to such “predictions” as those discussed in Secs. 7.2-7.3, to the effect that particles are created ex nihilo in violation of energy conservation laws. Even more “daring” is the idea  that our entire universe was created by a tunneling of Nothing through a (potential?) barrier of Nothing (Vilenkin, 1982, 1988) – which might indeed appear very innovative to physicists, but perhaps not so to scholars of various religious scriptures dealing with the creation of our world, or to science fiction writers specializing in “alternate realities” and “parallel universes” (Wolf, 1990).
52 This applies even to the idea of gauge fields, which was initiated by Weyl in 1918, and extended by him to quantum mechanics in 1929. The idea of supersymmetry was new, but in addition to not receiving any experimental confirmation, it was also “dominated by questions of formalism and technique”. The theory of superstrings (which at the present time is suffering a sharp decline on the popularity scale) would have been an exception, had it been originally derived from clearly stated first principles, rather than in a manner which left “the fundamental physical and geometric principles that lie at [its] foundation . . . still unknown” (Kaku, 1988, p. viii). However, the most recent developments in topological quantum field theory are of definite mathematical interest. They are especially intriguing since they are based on the study of knot theory, which has its roots in the nineteenth century physics of the ether – cf. (Atiyah, 1990), Sec. 1.3. Perhaps, after all, the history of science does move in circles!
53 For example the following quotation of M. Moravczik can be found on pp. 279 and 280 of (Cushing, 1990): “[A]s far as strong interactions are concerned, we have not made any substantial physical progress since Yukawa in 1935. ... [O]ld fashioned field theory, then dispersion theory, then Regge poles, and now QCD are simply reincarnations of the same Yukawa idea.”
54 Dirac made many public statements to this effect. Those statements quoted in Sec. 9.6 are amongst the most representative. It should be also observed that Note 25 in that chapter provides additional circum-stantial evidence supporting Dirac's point of view.
55 It is in this respect that the motivating factors behind the present work are consonant with the spirit of Popper (1963, 1968, 1982, 1983) and his followers (Lakatos, 1976; Watkins, 1984; etc.), who maintain that there is rationality in science, and that the goal of science is the pursuit of an objective truth that is independent of transitory fashions and other social factors. Such Truth can be arrived at by the traditional methods characteristic of the science of Newton, Maxwell, Einstein, Dirac, and other truly great physicists. The attitude reflected in this monograph is therefore very much at odds with the view of science advocated by some contemporary sociologists and historians of science, according to whom: “Although philosophy of science may once have set truth as a goal to which science aspires (Watkins, 1984, p. 155), ... closer examination of the historical record of actual scientific practice has shown that things are not as simple as we might hope them to be. ... A relativist or irrationalist (perhaps better, arationalist in contrast to the rationalist) school sees scientific knowledge as contingent, being determined by social and historical factors, so that the specific laws of science become arbitrary.” (Cushing, 1990, pp. 282-283). All that a practising scientist, who is dedicated to Truth in science, can provide in the way of a retort to such an assertion is the following: Whereas it might be, unfortunately, true that, during the present instrumentalist era, many of the advocated “laws” of “big science” have become indeed rather arbitrary, that is a phenomenon which characterizes the activities of many, and perhaps even most scientists in some areas of contemporary science – but not Science itself. As with any social phenomenon, this particular one will turn out to be only transitory if there is enough determination and dedication to reverse the trend amongst all those who take the opposite point of view as to what makes Science worth pursuing: the concerted search for Truth, and ultimately its revelation.

References for Quantum Geomtry

Key references. A number of titles have been singled out from the following list of references, since they contain material essential for the understanding of the mathematical framework presented in this monograph, and therefore are frequently cited. These citations are primarily in the form of the first letter of the surname of their (first) author set in between square brackets. In those cases where the same first letter occurs in the surnames of the authors of two or more of these key references, the first letter in the surname of the second author, or the first letter in the title proper (i.e., ignoring definite or indefinite articles) is appended to avoid confusion – as it would be the case with: [B], [BD], [BG], [BI], [BL] and [BR]; [N] and [NT]; [P] and [PQ]; [SC], [SI] and [ST]; [W] and [WQ]. Furthermore, a number of these key references overlap in some of their subject matter. Hence, for most of the considerations in the present monograph, it is sufficient to choose and consult only one of the references from each of the following six groups: [BI] and [D]; [BD] and [N]; [C], [I] and [NT]; [IQ] and [SI]; [K] and [SC]; [M] and [W].

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