Felix Berezin

Мaquette de livre FELIX BEREZIN This book is about Felix Berezin, an outstanding Soviet mathematician who in the 1960s and 70s was the driving force behind the emergence of the branch of mathematics now known as supermathematics. The integral over the anticoomuting Grassman variables he introduced in the 1960’s laid the foundation for the path integral formulation of quantum field theory with fermions, the heart of modern supersymmetric field theories and superstrings. The Berezin integral is named for him as is the closely related construction of the Berezinian which may be regarded as the superanalog of the determinant. In a masterfully written memoir, Berezin’s widow, Elena Karpel, narrates a remarkable story of Berezin’s life and his struggle for survival in the totalitarian Soviet regime. This story is supple- Life and death of the mastermind mented by recollections of Berezin’s close friends and colleagues. Berezin’s accomplishments of Supermathematics in mathematics, his novel ideas and breakthrough works, are reviews in two articles written by Andrei Losev and Robert Minlos. edited by M. Shifman January 29, 2007 11:38 WSPC/Trim Size: 9in x 6in for Proceedings Foreword iii FELIX BEREZIN LIFE AND DEATH OF THE MASTERMIND OF SUPERMATHEMATICS Edited by M. Shifman January 29, 2007 11:38 WSPC/Trim Size: 9in x 6in for Proceedings Foreword iv Copyright page The Editor and Publisher would like to thank the American Math- ematical Society for their kind permission to reprint the articles of V. Maslov, R. Minlos, M. Shubin, N. Vvedenskaya and the first article of A. Vershik from the American Mathematical Society Translations, Series 2, Advances in the Mathematical Sciences, Volume 175, 1996. Paintings Alexandra Rozenman Black-and-white sketches Yuri Korjevsky Graphic design Leigh Simmons Photographs Elena Karpel’s collection FELIX BEREZIN: LIFE AND DEATH OF THE MASTERMIND OF SUPERMATHEMATICS January 29, 2007 11:38 WSPC/Trim Size: 9in x 6in for Proceedings Foreword FOREWORD M. SHIFMAN W.I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA [email protected] The story of this Memorial Volume is as follows. In the fall of 2005 Arkady Vainshtein mentioned in passing that he had received Elena Karpel’s essay on Felix Berezin from his friend Dmitri Gitman. Of course, every student and every practitioner of modern field theory knows the Berezin integral over the Grassmann variables, which constitutes the basis of the current approach to theories with fermions and quantization of gauge theories (introduction of ghosts). Without using the Berezin integral, string theory and supersymmetry stud- ies, which are at the focus of modern high-energy physics, would be extremely hard, if not impossible. That was about the only thing I knew about Berezin, in addition to the fact that he had died in an accident in 1980 at the age of 49. I met him in person only once or twice in the late 1970s at seminars in Moscow. My impression was that the Berezin integral had always existed, “from the days of antiquity.” I am sure, this is also the impression of young people who came to mathematical and theoretical physics in the 1990s and later. Given Arkady’s remark that the essay was very moving, and my interest in the history of high-energy physics, my immediate reaction was to get hold of this memoir as soon as possible and read it myself. I wrote to Gitman, and he kindly forwarded to me the original Russian text. I read it right away. Indeed, it was both moving and masterfully written. To say that it was impressive is an understatement. I was captivated by the depth and literary maturity of Karpel’s story, a story of the life and death of a remarkable man, an outstanding mathematician whose ideas were ahead of his time and in many in- v January 29, 2007 11:38 WSPC/Trim Size: 9in x 6in for Proceedings Foreword vi stances were not immediately appreciated by his contemporaries. It was only later that they shaped entire areas of mathematical physics and proved to be instrumental in the development of modern high- energy theory and string theory. What struck me most was an obvious similarity between Berezin’s life and fate and those of other Soviet scholars on whose biographies I had worked in the past. This volume is my fourth book of this type. In 2000 I published two books honoring Yuri Golfand,a the theoretical physicist who discovered supersymmetry in four-dimensional field theories. In 2005 I edited a book dedicated to the memory of Bella Subbotovskaya,b a pioneer in the mathematics of complex systems and the founder of the Jewish People’s University in Moscow in the late 1970s and early ’80s. These three people were contemporaries, but everything else was different — age, gender, family status, char- acter traits, general interests, areas in which they worked... And yet, they shared a common destiny: intellectual potential wasted in vain, suppressed talents, unacknowledged skills, unfulfilled expectations, everyday humiliations, neglect and outright persecution by the au- thorities... Their lives were broken, just as the lives of so many other people whose only “fault” was to be born in the wrong place at the wrong time. Officially, they were supposed to enjoy the “advantages of real socialism,” Soviet style. And instead?.. How many discoveries could have been made? How many good deeds could have been done? We will never know. My involvement in Berezin’s story became a personal issue, even though I never knew this man personally. Soon my investigations led me to a group of mathematicians c who in 1996 edited the F.A. Berezin Memorial Volume issued by the American Mathemat- ical Society.d In addition to mathematical treatises this Volume aThe Many Faces of the Superworld, Ed. M. Shifman, (World Scientific, Singapore, 2000); The Supersymmetric World, Ed. G. Kane and M. Shifman, (World Scientific, Singapore, 2000). bYou Failed Your Math Test, Comrade Einstein, Ed. M. Shifman, (World Scientific, Singapore, 2005). cR.L. Dobrushin, R.A. Minlos, M.A. Shubin, A.M. Vershik. dAmerican Mathematical Society Translations, Series 2, Advances in the Mathematical Sciences, Volumes 175 and 177 (AMS, Providence, Rhode Island, 1996). January 29, 2007 11:38 WSPC/Trim Size: 9in x 6in for Proceedings Foreword vii contained recollections by some of Berezin’s friends and colleagues. The reader will find these articles reprinted below, in Part II of the present book. I wrote to Misha Shubin, Anatoly Vershik and Nikita Vvedenskaya, who replied to my letters immediately promising their enthusiastic support. Professor Vershik went as far as to prepare a new essay on Felix Berezin, written expressly for this book, which you will also see in Part II. He also informed me of a recent Russian publication honoring Felix Berezin.e In November of 2005 I got in touch with Elena Karpel, and in the summer of 2006 I met and interviewed her in La Rochelle, the French city where she now resides. In 1990 she and her daughter Natalie Berezin left the Soviet Union in a bid to start a new life in France. She told me of the fifteen years that had elapsed since then, a dramatic story of a long struggle for survival in an unfamiliar and often hostile environment. Being quite remarkable by itself, this story does not belong to this book, however. It may or may not be published elsewhere in the future. Karpel’s memoir, which constitutes the core of this book, is very dear to Elena Grigorievna. She tried first to publish the original Russian version in Russia. For various reasons this attempt failed, as did her subsequent attempt to publish a French translation in France. Thus, the English translation which opens Part II presents the first publication. Elena Grigorievna connected me with Victor Palamodov who now resides in Israel and, somewhat later, with Dimitri Leites in Germany. They took this project close to their hearts. Not only did they con- tribute original articles, they also offered plenty of advice and com- ments as to possible improvements. I managed to implement some of them. From day one I was in permanent contact with Dmitri Gitman. His story of a few encounters with Felix Berezin presents a clear picture of the atmosphere we lived in. It nicely matches and, in a sense, supplements Karpel’s memoir. This book consists of three parts. Part I contains two surveys eZapiski Nauchnyh Seminarov POMI, Vol. 331, Representation Theory, Dynamical Sys- tems, Combinatorial and Algorithmic Methods, Ed. A. Vershik, (St. Petersburg, 2006), pp. 1–235. January 29, 2007 11:38 WSPC/Trim Size: 9in x 6in for Proceedings Foreword viii written by Andrei Losev and Robert Minlos summarizing the scientific contribution of Felix Berezin. They show Berezin’s discoveries and innovative ideas in mathematical physics in a proper perspective, that of today’s researcher. Part II presents recollections by Elena Karpel and Berezin’s colleagues, friends and students. Finally, in Part III of this Volume I publish the English translation of two documents from Berezin’s archive: letters to the Rector of the Moscow State University and to the Governing Board of the Moscow Mathematical Society. Needless to say, currently they are only of historic interest. They acquaint the reader with the work environment to which Felix Berezin was confined. For the Western reader it will be hard to understand what he was fighting for. Ironically, the suggestions that Berezin made in his letters are standard procedure at Western universities. In the process of conducting this work, I shared my excitement with many friends. Some of them had the opportunity to read Karpel’s memoir either in its original form or at various stages of translation.

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