Prugovecki's Biographical Notes
It is with sorrow that we
report that Dr. Eduard Prugovecki passed away at his home in Lake
Chapala Mexico on October 13th
EDUARD PRUGOVECKI is a scientist who earned his Dipl. Phys.
from the University of Zagreb in 1959, and his Ph.D. from Princeton
University in 1964. After spending one year in post-doctoral studies
and research at Princeton University, and two years at the Institute
for Theoretical Physics in Edmonton, Alberta, he joined the faculty
of the University of Toronto, Canada. During the subsequent thirty years
that he was engaged in research and teaching at that university, he published
a multitude of research papers on quantum physics, as well as four monographs
based on that research: Quantum Mechanics in Hilbert Space (Academic
Press, New York 1971; 2nd edition 1981); Stochastic Quantum Mechanics
and Quantum Spacetime (Reidel, Dordrecht,
1984; rev. printing 1986); Quantum Geometry (Kluwer, Dordrecht, 1992)
and Principles of Quantum General Relativity (World Scientific,
Singapore and London, 1995).
The primary goal of that research was to achieve a framework
for theoretical physics that unifies its various disciplines into
a mathematically rigorous structure, which culminates with the consistent
unification of quantum theory and general relativity along the lines
proposed by Paul M. Dirac, Werner Heisenberg, Max Born and other well-known
founders of quantum mechanics. The final conclusions of this life-long
program are presented in Principles of Quantum General Relativity
(World Scientific, Singapore, 1995), and summarized in the review paper
Quantum Geometry and Gravity, published in Quantum Gravity: International
School of Cosmology and Gravitation XIV Course in Erice 1995, eds.
P.G. Bergmann, V. de Sabbata and H.-D. Treder (World Scientific, Singapore
and London, 1996). A concise survey of its main ideas and principles is
provided in an article published by Dr. Mladen
Martinis in Distinguished Croatian Scientists
in America, Part Two, Janko Herak, ed.
(Croatian-American Society, Zagreb, 1999)––which
is reproduced after these notes.
The most fundamental of these ideas is that spacetime
is quantum rather than classical in its basic structure, and that the
evolution of the universe is stochastic rather than deterministic at
the microscopic level. Therefore new physical ideas and mathematical
techniques have to be injected into theoretical physics in order to successfully
cope with these features of the cosmos, and arrive at understanding that
goes beyond the type of phenomenology that is so popular at the present
time. Thus, this program is very much in the spirit of Einstein, Bohr,
Born, Dirac, Heisenberg and other great physicists who emphasized the
need of combining various mathematical, philosophical and physical ideas,
instead of primarily relying on pragmatic arguments and phenomenology,
in order to arrive at a methodology and epistemology capable of coping
in the long run with the problems which Nature poses to us at the level
of the physical sciences.
Dr. Prugovecki also entertained a life-long interest in
the social role of science, and in the responsibilities of the scientist
towards building a better future for mankind. Hence, during his first
few years at the University of Toronto he joined a futuristic club
and became interested in the type of utopian literature that dealt
in a responsible manner with ideas extrapolating from contemporary
sociological, economic, scientific and political trends. He found
it distressing, however, that the best known contemporary novels dealing
with this subject were not utopias but dystopias, which emphasized
the negative aspects of the impact of science on society. Such an approach
leads to a distorted view of the social role of science, since science
actually affords mankind the freedom of choosing between positive
and negative applications of the natural laws it uncovers, and as such
it can liberate as much as enslave the human spirit.
Therefore, Dr. Prugovecki published in late 2001 the futuristic
of the Future
(Cross Cultural Publications, Notre Dame, 2001), whose original
draft he wrote in 1974, but
which he revised and improved after his retirement from
academic life in 1997. This novel deals with both a utopia as well
as a dystopia, and illustrates how the same scientific discoveries
and new technologies, with their roots in the present, can be used
for good as well as for evil purposes.
In his second futuristic novel Dawn of the New Man
(Xlibris, Philadelphia, 2002) he goes one step further and shows how
Terra––the enlightened society of the previous novel––is capable of fulfilling
some of the deepest aspirations of mankind, and how its protagonist is
able to steer FWF––the socially repressive counterpart of Terra––towards
a better future.
Dr. Prugovecki is at present Professor Emeritus at the University
of Toronto, but resides in the Chapala area of Mexico. He can be
reached by e-mail at firstname.lastname@example.org.
mathematical physicist, scientist and writer
Institute Ruder Boskovic, Zagreb
NOTE: The original of this
article had appeared in Distinguished Croatian Scientists in America,
Part Two, Janko Herak, ed. (Croatian-American Society, Zagreb, 1999)
– ISBN 953-97325-1-4. It is reproduced here with the permission of its author,
Dr. Mladen Martinis.
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
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
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
–– 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,
–– Principles of Quantum General Relativity (World Scientific,
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