Biographical Notes

General relativity and quantum theory are two fundamental disciplines of the twentieth century that, in spite of the enormous effort of many scientists, have remained separated from each other for many decades. It certainly seems very surprising that there should exist such a deep conflict between these two theories. In fact, their unification is now the most outstanding problem facing theoretical physicists. In view of the many attempts, carried out in the course of the past several decades, at a consistent unification of general relativity with quantum theory, no contemporary theoretical framework can lay claim, without qualifications, to the title of "quantum general relativity".

In all these attempts, Eduard Prugovecki is the one who has probably gone the furthest in achieving this final goal by formulating a very general quantum-geometric framework for general relativity capable of incorporating those fundamental premises of general relativity and quantum theory which are not mutually inconsistent, and can be therefore retained in the process of unification.

It is now already 37 years since Eduard Prugovecki left Croatia. In the meantime, he has visited Zagreb several times and particularly the Ruder Boskovic Institute, but we have never had an opportunity to meet him. Last summer he retired, and is now living about 150 km north of Toronto, in a small town, called Honey Harbour. Due to serious health problems he was not able to accept my invitation to attend the symposium.

2. Curriculum Vitae

Eduard Prugovecki was born on March 19, 1937 in Craiova, Romania. His mother, Helena (Piatkowski), was Romanian, but his father, Slavoljub, was Croatian; therefore, he always had Yugoslav citizenship.

He completed his primary education and first four years of secondary schooling in Bucharest, Romania. Because of a very strong anti-Yugoslav campaign in 1951, Prugovecki's whole family had to return to Yugoslavia where they chose to live in Zagreb, Croatia. In Zagreb he finished the last four years of secondary school and decided to study theoretical physics, a subject whose intellectual demands and rigour were a source of great fascination to him. In 1959 he earned his first class degree (diploma) in theoretical physics at the University of Zagreb. After completing his degree he joined the Department of Theoretical Physics at the Institute Ruder Boskovic, where he worked, as research assistant, until 1961 (except for one year of compulsory military service––note added to the original text).

In 1961 he was sent, as the best student of his generation, to Princeton University (N.J.) to work on a Ph.D. thesis under the supervision of Prof. A. S. Wightman, a world-leading theoretical physicist at that time. His stay in Princeton, where he received a Ph.D. in 1964, became the turning point in his professional career as he decided not to return to Zagreb. The reasons for this unexpected decision were numerous.

He left for the U.S.A. in 1961 because he had the opportunity to study under Prof. Wightman. Actually, in 1962-63 he helped with proof-reading of the Streater-Wightman monograph PCT, Spin and Statistics, and All That. I mention this because Prof. R. F. Streater was my supervisor during my work on a Ph.D. thesis at Imperial College in London.

At that time Prugovecki thought that the Wightman School stood for ideals to which he strongly subscribed (and still does): the kind of mathematical rigour and basic honesty in science that he found sadly lacking in contemporary theoretical physics, dominated since the mid-1940s by questionable "renormalization schemes" and other techniques that, he personally felt, were doing a lot of harm to a great tradition in science.

In the meantime, his interest in physics had diminished because of an interest in pure mathematics and philosophy of science for which, he thought, there was no possibility of study at the Institute Ruder Boskovic. At that time and later he was deeply attached to the principles of mathematical soundness and beauty in his work, as always advocated by Dirac. Many quotes from Dirac are used in his last two monographs.

By the mid-1960s it became clear to him that all that Wightman and his followers had to offer was simply another fundamentally unsubstantiated form of dogma, advocated by means of sheer techniques rather than by a truly critical analysis of the foundations of quantum theory.

So, he decided to move to Canada since he thought that there he would be far enough from the centers of power in US to pursue his own program unmolested, and yet close enough to be able to exert some influence once he began to effectively develop it. Unfortunately, it turned out that he was very wrong in those assumptions.

Once he began having some real measure of success with his program, things began happening to him. He got the feeling that competition in science in North America was not pursued in the same ethical manner as in Europe. Therefore, he gave me the following sincere and succinct advice for those young Croatian theoretical physicists for whom science is not just another way to acquire influence and power at any cost: stay in Europe!

At Princeton University he stayed for four years from 1961 to 1965; there he became a research associate. He emigrated to Canada in 1965 and became Postdoctoral Fellow at the Institute of Theoretical Physics in Edmonton, Alberta, 1965-67 and Lecturer at the University of Alberta 1966-67. Then he moved to Toronto to take the post of Assistant Professor of Mathematical Physics at the University of Toronto, 1967-69, Associate Professor 1969-75 and finally Professor 1975-97. Now he is Professor Emeritus at the University of Toronto.

In 1974 he was Visiting Professor at the Centre National de Recherches Scientifique, Marseilles, France. Around 1985-86 he resigned, on a matter of principle, from the International Association of Mathematical Physicists. He is also a member of Science for Peace.

He has written about a hundred research papers, published in refereed scientific journals, and about a dozen review papers, published in conference proceedings, on the mathematical foundations and methods in quantum mechanics, quantum field theory in flat and curved spacetime, and quantum gravity. Their conclusions and main results are systematically presented in the following four monographs:

–– Quantum Mechanics in Hilbert Space (Academic Press, 1971; 2nd edition 1981),
–– Stochastic Quantum Mechanics and Quantum Spacetime (Reidel, 1984; rev. printing 1986),
–– Quantum Geometry (Kluwer, 1992)
–– Principles of Quantum General Relativity (World Scientific, 1995).

The first two monographs have received quite a large number of reviews, and many of them were quite favourable; whereas only a few reviews appeared of his last two monographs, although he considers them personally to be much better and the culmination of his life's professional program: the consistent unification of quantum theory with general relativity. But, all that represents a very long story. Suffice it to say that, whereas only 30% of his first monograph was based on his own research, most of the material in his last three monographs was based on his own work.

The monographs explain a lot about the goals of his scientific work and his research activities over 35 years, which can be divided into five evolutionary periods:

–– Foundations of quantum mechanics (1961-67),
–– Functional analysis and quantum scattering theory (1967-75),
–– Quantum mechanics on phase space (1975-84),
–– Quantum field theory and quantum geometry (1984-89),
–– Unification of quantum theory and general relativity (1989-96).

3. Research into the foundations of general relativity and quantum theory.

A systematic and basically self-contained mathematical framework of a quantum-geometric unification of general relativity and quantum theory represents for Prugovecki the outcome of his thirty-five-year quest, which began in 1962 with the work on his Princeton University Ph.D. thesis entitled On the Empirical and Mathematical Foundations of Quantum Mechanics. In that thesis he tried to combine new epistemic ideas concerning the possible physical significance to quantum theory of the simultaneous unsharp measurements of position and momentum (which, as such, would not violate the uncertainty principle) with a new mathematical framework for quantum mechanics. This framework attempted to impart physical meaning to complex probabilities––an attempt which had already been made by Dirac in 1942.

That first attempt, which preoccupied him until 1968, eventually proved, at least from the point of view of physics, as unsuccessful as all the other "axiomatic" approaches to quantum theory. However, it made him aware of the intrinsic weakness shared by all such "axiomatic" approaches, whereby a fashionable mathematical discipline would be given preference on a priori grounds. He therefore came to the conviction that the correct path to follow should be the one adopted by the founders of relativity and quantum theory. He believed that the development of conceptual frameworks should be based on deeper foundational insights of the problems faced first by quantum physics, and then followed by the search for the kind of mathematics that would be most appropriate for their technical implementation.

In the 1968-75 period he therefore turned to attempting to truly understand conventional quantum mechanics at a foundational level. Some of the by-products of those attempts have been incorporated in the 1st (1971) and 2nd (1981) edition of his first monograph Quantum Mechanics in Hilbert Space, published by Academic Press, in which he improved the understanding of complete sets of observables and of various aspects of conventional quantum scattering theory.

Eventually, armed with accumulated experience and insight, he managed to formulate what he believed to be the correct epistemology and mathematics for generalizing conventional non-relativistic quantum mechanics into a consistent framework for relativistic quantum theory. These considerations have led to a formulation of non-relativistic and special-relativistic quantum theory on phase space. From the point of view of quantum mechanics on phase space the ensuing geometries enable an extrapolation of their special-relativistic frameworks to the general-relativistic regime. The results and conclusions of these investigations were presented in a series of 1976-83 papers, which formed the basis for his second monograph Stochastic Quantum Mechanics and Quantum Spacetime, published by Reidel in 1984. The central aspect of this framework lies in a new group-theoretical method of quantization, later called geometro-stochastic. This method enabled him to impart an operational interpretation to the idea of quantum frame.

However, the ensuing framework ran into the same main difficulty as the more conventional approaches to quantum field theory in curved spacetime––namely, it did not succeed in properly adapting the equivalence principle of classical general relativity to the quantum regime.

Further foundational analysis of the classical theory of general relativity made him realize that conventional methods based on classical geometries and on the single Hilbert space frameworks for quantum mechanics could not reconcile the equivalence principle of classical general relativity with the uncertainty principle. New types of geometries were required. They were developed in a 1985-89 series of papers, which formed the basis of his third monograph Quantum Geometry, published by Kluwer in 1992. These proposed geometries are referred to as quantum geometries, since basic quantum principles were incorporated into their structure from the outset. The mathematical tools used in constructing these quantum geometries were drawn from functional analysis and fibre bundle theory, and in particular from Hilbert space theory, group representation theory, and modern formulations of differential theory. The base manifolds of these fibre bundles were Lorentzian manifolds, or their appropriate frame-bundle extensions; whereas their typical fibres are infinite-dimensional (pseudo-) Hilbert spaces or superspaces. The study of connections on such fibre bundles posed interesting mathematical problems, which have received only scant attention in mainstream literature.

From the physics point of view, the principal areas of application of this quantum geometry framework were to quantum field theory in curved spacetime and to quantum gravity. The proposed methodology was, however, distinct from that of other approaches to these disciplines in that it is derived from foundational measurement-theoretical considerations. These developments reflected the possibility of resolving long-standing quantum paradoxes by a careful analysis of old as well as new quantum-measurement schemes and experimental procedures. The novel features of these frameworks not only clarify some long-standing questions of quantum field theory in curved spacetime and of quantum gravity, but give rise to some new perspective in the world of elementary particles. Central to the application of the quantum geometry framework to quantum physics is the idea of geometro-stochastic propagation, developed as a mathematically and epistemologically sound extrapolation of the standard path-integral formalism. Fundamentally, this idea proposes a modification of the concept of quantum point particle (which according to the orthodox interpretation displays the behavior of either a wave or a particle) into that of geometro-stochastic exciton, whose microdynamics simultaneously embodies the classical attributes of "waves" as well as of "particles".

Thus, in his geometro-stochastic quantum general relativity the wave-particle dichotomy present in the orthodox approach to quantum mechanics is replaced, at the micro-level, with a unified physical picture of quantum behavior, which has no counterpart in classical physics. It is at the macro-level that, in complete accord with Bohr's ideas, geometro-stochastic exciton behavior manifests itself, in the context of certain experimental arrangements, as that of a "particle"; whereas, in the context of some other experimental arrangements, it manifests itself as that of a "wave".

Furthermore, in the non-relativistic context, the mathematical apparatus of quantum geometry allows the transition to a sharp-point limit, which mathematically corresponds to the transition to delta-like proper wave functions. In that limit the conventional quantum mechanics of point-like particles is recovered, thus demonstrating that geometro-stochastic dynamics is a natural outgrowth of conventional quantum theory. The quantum geometry framework is not a finished theory but only provides a viable possibility for the consistent unification of general relativity and quantum theory.

In his last monograph Principles of Quantum General Relativity, published by World Scientific in 1995, he had explained and analyzed the principles of a quantum-geometric unification of general relativity and quantum theory in a manner that can supply the foundations for a quantum counterpart of classical general relativity. By taking advantage of recent advances in certain areas of mathematics and theoretical physics he was able to show that such a framework is capable of incorporating theoretical models whose numerical predictions would be in agreement with those of experimentally well-supported conventional models in relativistic quantum theory.

He believes that such a task can be achieved if advantage is taken of key geometric aspects and results in the theory of gauge fields and of coherent states developed during the last two decades, by incorporating them into quantum mechanics and quantum field theory together with the idea of fundamental length along the epistemic lines first suggested by Born and Heisenberg, and by extrapolating the classical concept of local frame of reference into a concept of quantum local frame of reference, into which such a fundamental length can be embedded from the outset.

The introduction of a fundamental length in quantum physics was advocated already in the 1930s by several of the best known founders of quantum mechanics. However, for various historical reasons, it was paid scant attention by the subsequent generation of theoretical physicists. Only during the last decade string theory revived interest in this idea by proposing the Planck length as a basic parameter related to the extension of superstrings.

On the other hand, as opposed to the historical development of general relativity as well as of quantum theory, string theory itself emerged from phenomenological rather than from foundational considerations. As a consequence, its underlying physical principles are still obscure.

In all his research papers and monographs Prugovecki puts great emphasis on the principle of mathematical beauty and simplicity. This principle was championed by Einstein throughout his life. As a methodological guide, it had its advocates also in Poincaré and Weyl, but its strongest champion was Dirac. Indeed, as stated in a recent scientific biography of Dirac: "For Dirac this principle of mathematical beauty was partly a methodological 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." (H.S. Kragh, Dirac: A Scientific Biography, Cambridge University Press, 1990, p. 277.)

These twin methodological principles of mathematical soundness and beauty have extensively guided the approach to quantum general relativity presented in his monographs. The adoption of these principles influenced the choice of quantum-geometric propagators, and the formulation of quantum-geometric causality. This had a profound effect on the mathematical form of the implementation of the concept of quantum-geometric propagation of fields, which is central to quantum general relativity.

Prugovecki has best described his great appreciation of Dirac's scientific work and philosophy in the preface to his monograph Quantum Geometry:

"When I was a graduate student, I had the privilege of seeing Dirac, and of listening to him, as I attended seminars at which he was present during an extended visit which he made to the Institute of Advanced Studies in Princeton in the early 1960s. By that time he was, of course, already a legendary figure – one of the revered, great physicists of this century, whose professional stature overshadowed that of all the other distinguished physicists who were in regular attendance at those weekly seminars. However, while many of those well-known luminaries liked to impress the audience with comments which confirmed their intellectual brilliance, as a rule Dirac kept his counsel, and limited himself to only an occasional question, which would be pertinent but always unprepossessing. It was only many years later, after I began to read his critical assessments of the very foundations of some of the theories presented at those seminars, that I came to appreciate the understated greatness of his genius, and the depth of his commitment to an ideal of truth in physical theories, which manifested itself in mathematical beauty – a beauty totally at odds with the ungainliness inherent in the ad hoc “working rules” of conventional renormalization theory. Indeed, some of his characteristic comments, such as the one that “people are ... too complacent in accepting a theory which contains basic imperfections”, reveal that, already by that time, he was all too painfully aware that his goals and ideals had become decidedly “unfashionable” in the world of perpetually changing fashions in elementary particle physics.

"Considering that Dirac is the universally acknowledged founder of quantum field theory, the very title of his last paper, The Inadequacies of Quantum Field Theory, bespeaks of lofty professional ideals and ethical standards, maintained throughout his entire life with a steadfastness and uncompromising integrity that has almost no parallel in this century. Although his critical comments might not have reached the fashion-conscious amongst the theoretical and the mathematical physicists, his words and deeds have inspired and provided moral support to me, as I trust it did to many other researchers who share his ideals and basic values with regard to the principles and aims of theoretical physics. It is therefore with genuine reverence and overwhelming spiritual gratitude that I dedicate this work to the memory of Paul A. M. Dirac – a great physicist, as well as a truly great man."

4. Concluding Remarks

Eduard Prugovecki is certainly our best theoretical/mathematical physicist who had the courage to systematically and critically re-examine the very foundations of general relativity and quantum theory––physical theories which manifested themselves in extraordinary mathematical beauty. His remarkable research achievements in mathematical physics have made him a world-wide known scientist.