Felix A. Berezin (1931-1980)

commemorating the 50th anniversary of his paper on second quantization

Mathematical physics was the main pivot point of Felix Alexandrovich Berezin's scientific interest, embracing many complicated and sometimes extremely abstract theories and constructions designed to provide mathematical understandings of fundamental physical theories - including quantum physics, the theory of gravitation, and statistical mechanics. The fact that today mathematical physics is more widely and deeply understood by both mathematicians and physicists is undoubtedly due in no small measure to Berezin's role. In his youth he was enchanted by the beauty and significance of mathematical problems pertaining to quantum theory, and ever since, this remained the leitmotif of his creative work.

Berezin's first remarkable paper, completed before his conversion to mathematical physics, is related to the theory of linear representations of semisimple complex Lie groups. The paper contains the following result: every infinite-dimensional irreducible representation of such a group by operators in Banach space is isomorphic to a certain subfactor of a representation induced by some character of its Borel subgroup.

In the mid-1950s Berezin, on the advice of his teacher I. M. Gelfand, deepened his study of quantum field theory. From this time onward, his preoccupation with mathematical physics is evident. During the first period of this work, Berezin pondered questions of perturbation techniques, spectral theory and especially scattering problems. His papers of that time are related to the study of many-particle quantum systems with pairwise interaction. For a class of dissipative Schrödinger operators, he posed the problem of deducing their spectral expansion by making use of wave operators.

In 1960, Berezin carried out his work on second quantization - i.e., the formalism based on representing linear operators in Fock space as functions of "creation" and "annihilation" operators. Not only did he give an elegant form to this calculus, but he also extended our understanding the method itself. A complete exposition of the results he obtained was presented in his monograph, The Method of Second Quantization, which became widely known. According to A. S. Wightman, this book "was influential in two ways. First, it summarized several decades of applications of the formalism of second quantization. Second, by developing systematically the formalism of functions of anticommuting variables it displayed the parallelism between Bose and Fermi systems in a new way."

Berezin's work on second quantization, and the application of this method to the quantum theory of fields, were the starting-point for his series of papers on symbolic operator calculus. Here operators in Hilbert space are studied making use of their representations by various functional symbols. In many of its aspects, this direction of research was close to (and arose at the same time as) the theory of pseudodifferential operators, which now plays a remarkable role in mathematical physics. Thus, many important ideas of the latter theory emerged independently in Berezin's papers, although the significance of his contributions from this point of view was not immediately recognized.

In turn, there is a relation between symbolic operator calculus and Berezin.s work on the deformation quantization of classical dynamical systems. The basic idea in his papers is to give the following mathematical meaning to quantization: quantum observables are elements of a deformation of the algebra of classical observables, such that the commutator coincides with the Poisson bracket in first order of the deformation parameter (Planck's "constant"). From this point of view, Berezin considered the situation where the phase space of classical dynamics is a homogeneous symmetric domain in complex space. In this case, an interesting new effect emerges: the set of deformation parameter values can be discrete and bounded above.

Last but not least is the topic of supergeometry and analysis on supermanifolds - perhaps one of the most important subject areas for mathematical physics, whose origin also goes back to Berezin's fundamental work on second quantization. His ideas and results are exceptionally important in the elaboration of the mathematical machinery for constructing supersymmetric models in theories of elementary particles and gravity.

Berezin's scientific achievements, and their role in the development of mathematical physics, become still more striking when seen against the background of his activity as a teacher, understood in the broadest sense. His students, scientific disciples, and colleagues unanimously observe that their scientific conversations, talks, discussions, and consultations with Felix Alexandrovich were enlightening, stimulating, seminal, and inspiring.

To those of us who knew him, Berezin was absolutely unpretentious man, a person of exceptional modesty and great innate dignity.

(compiled by R. A. Minlos, Yu. A. Neretin and A. M. Stepin, March 2011)