NOTE: The text below basically consists of Chapter 12 in the monograph

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

Historical and Epistemological Perspectives on

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 clear^{1}
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 reversed^{2},
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 stated^{3}:
“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 verdict^{4},
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 chapter^{5}
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.

**1****2.1.
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 contributed^{6}
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 possible^{7},
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 language^{8}.
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 enunciations^{9}
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* integrals^{10};
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 meaningless^{11};
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 day^{12}.
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 physicists^{14}.

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 Dirac^{15}),
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 prone^{17}.
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 manifestation^{18}
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 mathematics^{19},
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 communication^{20}
between physicists and mathematicians, which, from Newton's era to Einstein's
time, has been underlying all significant progress in theoretical physics^{21}.
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 mathematics^{22}.

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*mis*understanding 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 support^{23}. 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 theory^{24}. 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 Toeplitz^{25},
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 eigen*function* expansions – and not as eigen*vectors*
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 speaking^{26},
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
convenient^{27} 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 formulation^{28},
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).

*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 enforced^{30}
“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 *j*^{0}(*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*sound*^{31} 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 proved^{32} 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 *meaning*^{33}
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*micro*reality, 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 color^{35}
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 institutionalized^{36}.
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 phenomena^{37} –
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 *micro*level
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 mechanics^{38}:
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.

*f*_{l} 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 *f*_{l} 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*f*_{l} 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 *f*_{l} 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 *f*_{l}.
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*non*perturbative 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 technology^{41}.

The fundamentally*non*perturbative 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 exist*^{42}.
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 *f*_{l}
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, Pauli^{43},
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 questions^{44}.
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 *primarily*^{45} 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 Schrödinger^{46}, 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 War^{47}. 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 powers^{48}, 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 War^{49}. 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 opposition^{50} 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 cosmology^{51}, then Schwinger's 1958
assessment of post-World War II developments in relativistic quantum physics
can be, by and large, extrapolated to the present time^{52}: 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 conclusions^{53}. 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
suggested^{54} 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 traditional^{55} 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.

Einstein himself made clear

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 reversed

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 chapter

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.

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

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.” (

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 contributed

“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

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

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 possible

An

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.

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 enunciations

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

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

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

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

However, the cavalier manner in which mathematics itself has been treated after the inception of the renormalization programme, indicates that instrumentalism

It therefore seems appropriate to categorize this kind of approach to science by the more precise label of

Is conventionalistic instrumentalism an intrinsically unavoidable feature of contemporary quantum physics?

The preceding nine chapters of this monograph are meant to

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

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

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.” (

Such a

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 manifestation

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 communication

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

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 theory

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

While the failings of the conventionalistic approach to this type of problem might be deemed innocuous – as it rarely gives rise

As is well-known, in the configuration representation the elements of

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

where

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

Furthermore, the choice of

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

This seemingly innocuous mathematical point has significant physical repercussions. Thus, although the conventionalistic custom is to refer to |

That does not immediately follow, but the above points indicate that caution should be exercised even in nonrelativistic quantum mechanics, and that

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

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 formulation

The above elementary example illustrates how “theory selection” isNOTE: 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 toH_{–}) 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 densityk/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 ofrasrbecomes 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-calledT-“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 existence^{29}ofsingle-targetdifferential 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 aT-“supermatrix” (rather than aT-“matrix”), as well as the confidence function in (3.5.3) (cf. [PQ], p. 518; [P], p. 170):

formula(3.7)

Indeed, it isnottrue 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 isnotat 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!

Thus, instead of relying on the uncovering of scientific

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

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

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

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

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

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

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

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

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

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

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

2) Physically, there is Heisenberg's concern with posing the

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”

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

As witnessed by the earlier cited

Perhaps it is time that those warnings were heeded.

The quantum geometry framework described in the present monograph grew out of a systematic effort at trying to see whether the

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

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

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

It could be said that as a conceptual and mathematical

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

On the other hand, by introducing the concepts of

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

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

Thus, “strongly held beliefs” can color

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

Consequently, one of the key questions from the point of view of a

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 mechanics

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

For this very reason, these basic constants are embedded, in the form of the fundamental length

“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

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

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.

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

As mentioned earlier, much more compelling than the above string-motivated type of heuristics is the adoption of the quantum spacetime form factorNOTE: 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 theQ's and theP's, Born (1949) adopted (5.1a) as the basic eigenvalue equation for hisquantum 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 inR^{8}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 regime^{39}, 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 factorfin (5.5.5)._{l}

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 (Greenet 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 offundamentalquantum 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 incorporation^{40}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 bundleT*M +T*Mover the Lorentzian manifoldM. In such a model for GS excitons the proper wave function for a graviton at any base locationxinMcan be identified with the spin-2 ground state of the quantum metric operatorD^{2}(x) =Q^{2}(x) +P^{2}(x) at that location.

From a semiclassical point of view, this treatment envisages a stringlike GS exciton atxinMto be an excited eigenstate of a relativistic harmonic oscillator at that location. At such a heuristic level, a GS exciton above the base locationxinMcan be visualized as a string of pointsqinT_{x}Mexecuting, in general, vibratory as well as rotational motions with respect to a local Lorentz frame {e_{i}(x)}. The ground modes of such stringlike GS excitons would correspond to stochastic vibrations in the direction of motion specified by its 3-momentumk, transversal oscillations in the polarization planes orthogonal tok, and rotations around the direction in whichkpoints. 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 inT*_{x}Mby (k_{0},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 motionk. 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 vectorsf_{B,A}describing these higher exciton states satisfy the relativistic harmonic oscillator equation

formula(5.5)

in the variablesu=_{i}p–_{i}k, representing relative internal 4-momentum components with respect to the dual of the local Lorentz frame {_{i}e_{i}(x)}. The rest massesm_{B,A}carried by these excited modesf_{B,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 leaveskinvariant. Consequently, they can be factorized as follows

formula(5.9)

where eachZ(s_{A}) is constructed from polarization frames, such as those in Chapters 9 and 11, so that they can be grouped into sets {Z(s_{A})} providing integer-spin frames. The spins_{A}=1 ands_{A}=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 lengthl_{A}= 2 in Planck units, and supplying the fundamental quantum spacetime form factor in (9.2.14) upon settingu=_{i}v–_{i}k, and then renormalizing as_{i}m_{A}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 leftkinvariant. 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 operatorD^{2}(x) at eachxinM(cf. [P], p. 205):

formula(5.10)

The proper state vectors describing their internal stochastic motion with respect to the local Lorentz frame {e_{i}(x)} can be then computed as in (Brooke and Prugovecki, 1984).

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

The adoption of the quantum spacetime form factor

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 ∆

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

The central observation here is that, in the absence of a

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

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

The fundamentally

These GS probability amplitudes are superimposed in a

The existence in the GS approach of

The latter type of approach based on “internal statics” has an essential bearing on the epistemological significance of the concept of

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

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

In this type of GS conceptualization, any phrase providing the “probability of detection of a GS exciton within a region

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”.

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, Pauli

This concern is understandable: the loss of dedication to a fundamental notion of truth in science – namely

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

“The dispersion-theory and

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 questions

Some sociologists of science have documented these features in their studies of the “big science” that emerged after the Second World War

On the other hand, there are various publications which try to

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.

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 cosmology

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 conclusions

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

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

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 traditional

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

During the course of most of the post-World War II developments in relativistic quantum physics, concentrating one's attention on anything but the

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.

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