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 con- stitutes 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 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 . 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 master- fully 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-

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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 theoreti- cal 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 discover- ies 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

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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 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

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written by Andrei Losev and Robert Minlos summarizing the scien- tific 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 col- leagues, 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 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. One of the first readers was a well-known artist, Alexan- dra Rozenman, whose paintings are in high demand among the col- lectors and galleries. Captivated by the sad melody of this story she created the painting One-Way Ticket expressing this melody in a vi- sual form. The reader will find a small reproduction in the beginning of Part II.

General acknowledgments I am grateful to Ashleigh Geib who was responsible for typesetting a part of this volume in LATEX. My deep gratitude goes to Anne Barthel for expert advice on English grammar. Her effort at improving the English translation of Karpel’s memoir is impossible to overestimate. I want to thank Iman Benmachiche for her assistance with French, and Anya Dashevsky, Joel Giedt and Stephen Gasiorowicz for proof- reading some parts of the manuscript. I am thankful to Roxanne Weymouth who handled all the financial aspects of this project. My friend and colleague Dimitry Leites kindly agreed to proofread Losev’s review and Part III of this book and made valuable remarks. Special thanks go to Yuri Korjevsky, Alexandra Rozenman and January 29, 2007 11:38 WSPC/Trim Size: 9in x 6in for Proceedings Foreword

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Leigh Simmons who worked hard to make this book visually appeal- ing. They did a great job, creating several successive versions of the cover design and illustrations. I am grateful to Roman Kovalev and Pavel Moshin who carried the main burden of translation. Regarding the translation of numerous poetic passages scattered in Karpel’s memoir I consulted Professor Anatoly Liberman, to whom I would like to say thank you. I am grateful to Stan Lipovetsky for connecting me with Yuri Korjevsky in Moscow and assisting in subsequent communications with him. As usual, my World Scientific contact, Lakshmi Narayanan — “my Editor” — was instrumental in the speedy completion of this project. This publication, which I hope will have a lasting impact in the community of mathematical physicists, was made possible by the enthusiastic participation of friends, colleagues and students of Fe- lix Berezin from whom I received much advice and encouragement. Above all, I want to say thank you to the contributors who are listed below.

Additional acknowledgments Paintings on the cover and pages 1 and 53 are by Alexandra Rozen- man. The painting on the cover (Where is Everybody?) is from the collection of Caitlin Skinner and Robert Marlott. The painting on page 1 (Again) is courtesy of the artist. The painting on page 53 (One-Way Ticket) is from the M.&M. Shifman collection. Black-and- white sketches are by Yuri Korjevsky (M.&M. Shifman collection). All photographs are from the collection of Elena Karpel. All rights reserved. I thank the American Mathematical Society for their kind per- mission to reprint the articles of V. Maslov, R. Minlos, M. Shubin, N. Vvedenskaya and the first article of A. Vershik from the Amer- ican Mathematical Society Translations, Series 2, Advances in the Mathematical Sciences, Volume 175, 1996. January 16, 2007, Minneapolis January 29, 2007 11:38 WSPC/Trim Size: 9in x 6in for Proceedings Foreword

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List of contributors Natalie Berezin, Dmitri Gitman, Elena Karpel, Dimitry Leites, An- drei Losev, Victor Maslov, Robert Minlos, Victor Palamodov, Misha Shubin, Anatoly Vershik, Nikita Vvedenskaya

Disclaimer: The opinions expressed by the writers herein are not necessarily those of the Editor. January 29, 2007 11:38 WSPC/Trim Size: 9in x 6in for Proceedings Foreword

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CONTENTS

Foreword v M. Shifman

Part I

From Berezin Integral to Batalin–Vilkovisky Formalism: A Mathematical Physicist’s Point of View 3 Andrei Losev

Felix Aleksandrovich Berezin: A Brief Scientific Biography 31 Robert Minlos

Part II

On a envie de parler ... 54 Natalie Berezin

The Last Journey. Remembering F.A. Berezin 55 Elena Karpel

Remembering Alik Berezin 149 Victor Maslov

On Berezin 152 Misha Shubin

My Recollections on Berezin 156 Anatoly Vershik

About Alik Berezin and Some of His Works. Ten Years Later 161 Anatoly Vershik January 29, 2007 11:38 WSPC/Trim Size: 9in x 6in for Proceedings Foreword

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Reminiscences of a Close Friend 171 Nikita Vvedenskaya

My Encounters with Felix Aleksandrovich Berezin: Snapshots of Our Life in the 1960s, ’70s and Beyond 181 Dmitri Gitman

About Alik Berezin and His Time 206 Victor Palamodov

With and Without Berezin 218 Dimitri Leites

Part III From Felix Berezin’s Archives

Letter to Academician R.V. Khokhlov, the Rector of the Moscow State University 231

Letter to the Governing Board of the Moscow Mathematical Society 242

From Wikipedia 244