《量子力学，概念与应用》（英文版）Quantum mechanics of fundamental systems

Quantum Mechanics of Fundamental Systems The Quest for Beauty and Simplicity

Quantum Mechanics of Fundamental Systems: The Quest for Beauty and Simplicity

Marc henneaux Jorge Zanelli Editors Quantum Mechanics of Fundamental Systems The Quest for beauty and Simplicity Claudio bunster festschrift Springer

Marc Henneaux · Jorge Zanelli Editors Quantum Mechanics of Fundamental Systems: The Quest for Beauty and Simplicity Claudio Bunster Festschrift 123

Marc henneaux Universite libre de bruxelles Jorge zanelli Centro de estudios cientificos Service Physique the Valdivia. Chile z @cecs cl 050 Bruxelles Campus de la plaine henneaux@cecscl henneaux@ulb ac be ISBN:978-0-387-87498-2 e-lSBN:978-0-387-874999 DOI:10.1007978-0-387-874999 Library of Congress Control Number: 2008942059 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated ied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, by similar or dissimilar methodology now known or hereafter develop he use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject Printed on acid-free pa

Marc Henneaux Université Libre de Bruxelles Service Physique théorique et mathématique 1050 Bruxelles Campus de la Plaine Belgium henneaux@cecs.cl henneaux@ulb.ac.be Jorge Zanelli Centro de Estudios Cientificos Valdivia, Chile z@cecs.cl ISBN: 978-0-387-87498-2 e-ISBN: 978-0-387-87499-9 DOI: 10.1007/978-0-387-87499-9 Library of Congress Control Number: 2008942059 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

Contents Part 1 OPENING REMARKS Greetings Opening lecture Marc henneaux Part 2 CONTRIBUTED PAPERS On the Symmetries of Classical String Theory Constantin P bacha Eddington-Born-Infeld Action and the dark Side of General relativity Maximo banados Light-Cone Field Theory, Maximal Supersymmetric Theories and Ezn in Light-Cone Superspace Lars brink Strongly Hyperbolic Extensions of the ADM Hamiltonian J. David Brown Black Hole Entropy and the problem of Universalit Sources for chern-Simons theories Jose d. Edelstein and Jorge Zanelli The Emergence of Fermions and the el Content 125 Francois Englert and Laurent Houart

Contents Part 1 OPENING REMARKS Greetings...................................................................... 3 Opening Lecture .............................................................. 11 Marc Henneaux Part 2 CONTRIBUTED PAPERS On the Symmetries of Classical String Theory .............................. 17 Constantin P. Bachas Eddington–Born–Infeld Action and the Dark Side of General Relativity ...................................................................... 27 M´aximo Ba˜nados Light-Cone Field Theory, Maximal Supersymmetric Theories and E7(7) in Light-Cone Superspace ......................................... 33 Lars Brink Strongly Hyperbolic Extensions of the ADM Hamiltonian ................. 71 J. David Brown Black Hole Entropy and the Problem of Universality ....................... 91 S. Carlip Sources for Chern–Simons Theories ......................................... 107 Jos´e D. Edelstein and Jorge Zanelli The Emergence of Fermions and the E11 Content........................... 125 Franc¸ois Englert and Laurent Houart v

ontents Why Does the Universe Inflate? 147 S.w. Hawking Kac-Moody Algebras and the Structure of Cosmology gularities:A N on the belinskii-Khalatnikov-Lifshitz Analysis Marc henneaux Black Holes with a Conformally Coupled Scalar Field 167 Cristian martinez Quantum Mechanics on Some Supermanifolds Luca mezincescu John Wheeler's Quest for Beauty and Simplicity 193 Charles W. misner magnetic Monopoles in Electromagnetism and gravity 197 Ruben portugues The Census taker’sHat 213 Leonard Susskind Static Wormholes in Vacuum and gravity in Diverse Dimensions 267 Ricardo troncoso Part 3 CLOSING Claudio Bunster: A Personal recollection Jorge Zanelli Profile of claudio Bunster Index 303

vi Contents Why Does the Universe Inflate? .............................................. 147 S.W. Hawking Kac–Moody Algebras and the Structure of Cosmological Singularities: A New Light on the Belinskii–Khalatnikov–Lifshitz Analysis............................. 155 Marc Henneaux Black Holes with a Conformally Coupled Scalar Field...................... 167 Cristi´an Mart´ınez Quantum Mechanics on Some Supermanifolds ............................. 181 Luca Mezincescu John Wheeler’s Quest for Beauty and Simplicity ........................... 193 Charles W. Misner Magnetic Monopoles in Electromagnetism and Gravity .................... 197 Rub´en Portugue´s The Census Taker’s Hat ...................................................... 213 Leonard Susskind Static Wormholes in Vacuum and Gravity in Diverse Dimensions.................................................................... 267 Ricardo Troncoso Part 3 CLOSING Claudio Bunster: A Personal Recollection ................................... 289 Jorge Zanelli Profile of Claudio Bunster .................................................... 295 Index ................................................................................ 303

Greetings Recorded in santa barbara David gross Hi Claudio Good to see you. I am sorry I am not able to be there for your birthday. Jackie and I really loved our visit last year, and I am sure we would have enormously enjoyed what I expect will be a great party, but at the moment we have a previous engagement somewhere in Thailand. I have known Claudio since he was a mere lad of 22, a young graduate student of ohnnie Wheeler when I first came to Princeton. He then stayed on as an Assistant Professor. so i have known him for more than half of his life. After princeton he went to Texas and a few years later, much to my surprise, returned to Chile, first on a part time basis, and then full time. I must say that at the time I was surprised and amazed that he did this. I really admired him for his courage and dedication in going back to Chile to help build science at such a very difficult and dangerous time I was delighted over the years to visit Claudio at the institute that he established in Santiago and later in Valdivia. I tried to do the little I could do to help him in his remarkable leadership in developing science in Chile and in healing the wounds of hard times Claudio Teitelboim is truly a great scientist and a great statesman of science He has helped to transform Chilean science. I have enormous respect for him and all that he has achieved and I just wish that I was there in person to offer him my congratulations. I cannot be in Chile, so from afar, congratulations Claudio! To paraphrase a well known Hebrew saying that is often said on such occasions until 120. You are already half way there, so enjoy the rest, the second half Best of wishes David gross I Kavli Institute of Theoretical Physics, University of California, Santa Barbara, CA 93106, USA, e-mail: gross@kitp. ucsb. edu M. Henneaux, J Zanelli(eds ) Quantum Mechanics of Fundamental Systems: The Quest for Beauty and Simplicity. DOI 10.1007/978-0-387-87499-9-1 C Springer Science+Business Media LLC 2009

Greetings Recorded in Santa Barbara David Gross1 Hi Claudio, Good to see you. I am sorry I am not able to be there for your birthday. Jackie and I really loved our visit last year, and I am sure we would have enormously enjoyed what I expect will be a great party, but at the moment we have a previous engagement somewhere in Thailand. I have known Claudio since he was a mere lad of 22, a young graduate student of Johnnie Wheeler when I first came to Princeton. He then stayed on as an Assistant Professor, so I have known him for more than half of his life. After Princeton he went to Texas and a few years later, much to my surprise, returned to Chile, first on a part time basis, and then full time. I must say that at the time I was surprised and amazed that he did this. I really admired him for his courage and dedication in going back to Chile to help build science at such a very difficult and dangerous time. I was delighted over the years to visit Claudio at the institute that he established in Santiago and later in Valdivia. I tried to do the little I could do to help him in his remarkable leadership in developing science in Chile and in healing the wounds of previous hard times. Claudio Teitelboim is truly a great scientist and a great statesman of science. He has helped to transform Chilean science. I have enormous respect for him and all that he has achieved and I just wish that I was there in person to offer him my congratulations. I cannot be in Chile, so from afar, congratulations Claudio! To paraphrase a well known Hebrew saying that is often said on such occasions “until 120”. You are already half way there, so enjoy the rest, the second half. Best of wishes, Bye David Gross 1 Kavli Institute of Theoretical Physics, University of California, Santa Barbara, CA 93106, USA, e-mail: gross@kitp.ucsb.edu M. Henneaux, J. Zanelli (eds.), Quantum Mechanics of Fundamental Systems: The Quest 3 for Beauty and Simplicity, DOI 10.1007/978-0-387-87499-9 1, c Springer Science+Business Media LLC 2009

Recorded in Cambridge, MA Frank wilczek Greetings Claudio, greetings friends in Chile Claudio, as you will know, time is an illusion. The Hamiltonian that evolves systems in time is just a constraint, and it is zero, so your 60th birthday should not be a cause for alarm and there are other branches of the wave function in which you are celebrating your 20th birthday or 30th, or 40th or whichever one your prefer. Even on this branch of the wave function the 60th birthday is really a cause for celebration. It is a chance to celebrate what has been achieved by this time that ite an impressive thing to contemplate Your contributions to fundamental physics are in retrospect even more remark able than they seemed at the time. Thinking about abstract problems on how you quantize constraint systems or how you deal with extended objects and generalize the Dirac quantization conditions or how you understand black holes as quantum mechanical objects. These things to which you contributed so much and focused interest on, have proved to be some of the most unlikely, yet rich and fertile fields of theoretical physics in recent decades. You have also of course founded the institute in Chile. which has been an extraor- dinary place for intellectual adventure, not only in theoretical physics but in topics that have proved to be, again, amazing choices of things to focus on Glaciology now is at the forefront of interest in the worlds problems of climate change and what we are going to do about it and understanding it and of course, understanding how the mind works is going to be the great occupation, I,m sure, of science in the later parts of the twenty-first century. Besides intellectual achievements and up other people's intellectual achievements, you have had direct effect on people's lives, family, friends, coworkers, and in future years you'll have not only of extending your own ventures, but of watching their adventures as the solution of time marches on. So you can look back with satisfaction and look ahead with anticipation. Happy Birthday! Frank wilczek 2 Department of Physics, Center for Theoretical Physics, Massachusetts Institute of Technology Cambridge MA 02139. USA. e-mail: wilczek@mitedu

4 Greetings Recorded in Cambridge, MA Frank Wilczek2 Greetings Claudio, greetings friends in Chile. Claudio, as you will know, time is an illusion. The Hamiltonian that evolves systems in time is just a constraint, and it is zero, so your 60th birthday should not be a cause for alarm and there are other branches of the wave function in which you are celebrating your 20th birthday or 30th, or 40th or whichever one your prefer. Even on this branch of the wave function the 60th birthday is really a cause for celebration. It is a chance to celebrate what has been achieved by this time that is quite an impressive thing to contemplate. Your contributions to fundamental physics are in retrospect even more remark￾able than they seemed at the time. Thinking about abstract problems on how you quantize constraint systems or how you deal with extended objects and generalize the Dirac quantization conditions or how you understand black holes as quantum mechanical objects. These things to which you contributed so much and focused interest on, have proved to be some of the most unlikely, yet rich and fertile fields of theoretical physics in recent decades. You have also of course founded the institute in Chile, which has been an extraor￾dinary place for intellectual adventure, not only in theoretical physics but in topics that have proved to be, again, amazing choices of things to focus on. Glaciology now is at the forefront of interest in the world’s problems of climate change and what we are going to do about it and understanding it and of course, understanding how the mind works is going to be the great occupation, I’m sure, of science in the later parts of the twenty-first century. Besides intellectual achievements and setting up other people’s intellectual achievements, you have had direct effect on people’s lives, family, friends, coworkers, and in future years you’ll have the joy not only of extending your own adventures, but of watching their adventures as the solution of time marches on. So you can look back with satisfaction and look ahead with anticipation. Happy Birthday! Frank Wilczek 2 Department of Physics, Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA, e-mail: wilczek@mit.edu

Greetings Greeting from Utrecht Gerard 't Hooft Dear Marc, Jorge and other Organisers of the Claudio Fest Although, regretfully, I will not be able to be physically present at this gather ing, I do wish to send Claudio my very warmest wishes and congratulations for his 60th birthday. As the title of the meeting shows, 60 is the respectable age when one is beginning to be more reflective, pondering about the real essentials of our research topics: what is quantum mechanics, what is beauty and simplicity? In what directions should future searches go, and what is it that we can expect? I hope the meeting, by its informal nature, will be thought provoking. My warmest greetings also to all my friends and colleagues who did manage to be present and I hope they will bring claudio the hommage he deserves In case it is still opportune: Merry Christmas and a Happy 2008 Gerard t Hooft 3 Institute for Theoretical Physics, Utrecht University and Spinoza Institute, Postbus 8000, 3508 TA Utrecht, the Netherlands, e-mail: GtHooft@ phys. uu. n

Greetings 5 Greeting from Utrecht Gerard ’t Hooft3 Dear Marc, Jorge and other Organisers of the Claudio Fest, Although, regretfully, I will not be able to be physically present at this gather￾ing, I do wish to send Claudio my very warmest wishes and congratulations for his 60th birthday. As the title of the meeting shows, 60 is the respectable age when one is beginning to be more reflective, pondering about the real essentials of our research topics: what is quantum mechanics, what is beauty and simplicity? In what directions should future searches go, and what is it that we can expect? I hope the meeting, by its informal nature, will be thought provoking. My warmest greetings also to all my friends and colleagues who did manage to be present and I hope they will bring Claudio the hommage he deserves. In case it is still opportune: Merry Christmas and a Happy 2008, Gerard ’t Hooft 3 Institute for Theoretical Physics, Utrecht University and Spinoza Institute, Postbus 8000, 3508 TA Utrecht, the Netherlands, e-mail: G.tHooft@phys.uu.nl

For Claudio: A 60th Birthday greeting Stanley Deser+ Dear claudio Let me preface this, at times indiscreet, 60th birthday message with the consol tion that, were it not for the impossible 18h(each way)plane journeys, you would have had to suffer the insult of hearing it live! The much appreciated invitation to your celebration made me both count and re member: we have known each other for over half your life so far, and it led me back to our first encounter, of early 1975, in Princeton. You were then already famous the successor in the lineage of Feynman and Misner-as the latest of Wheeler's discoveries, " having reached New Jersey by the route closest to Johns heart, crazy electrodynamics. Since were also traveling backward in time, that takes care of leining Dicks way, while Charlies(more recent)exploit was to destroy the beauty of Maxwell-Einstein theory by(re-discovering its-horribly complicated, already metrical version. Your road was the radiation reaction prob- lem, which you clarified so much that it forced Sidney Coleman to(sort of) do it right later, also using exotic Feynman propagators, come to think of it. What most pressed me however, in that initial meeting, was of a non-physics nature: you looked like a Jeune Premier(someone will translate)and you(claimed you)were equipping an ancient and highly unlikely-looking wreck of a yellow Land Rover for the journey home(merely to Santiago, rather than Valdivia, but still) down the ( then highly unfinished)Pan-American highway. In drab, conventional, Princeton one can imagine that these characteristics stood out most vividly! I had been invited for an informal visit by the Physics Department. I had a lot of fun -except when the whole Physics Department-from Wigner down, would close for grading Freshman hysics exams(a Princeton custom, I was told! ) Despite this quaint custom, you and I were able start what would become a long-standing research collaboration. whose first result was our Phys Rev electromagnetic duality paper(surviving those unsettling local mores). It was to become famous, long before duality became fash- ionable, but not before going through the usual "it's wrong and trivial"scoffing. It still gets quoted, and as you know, served as a basis of three of our further collab- orations, with Henneaux and gomberoff, at your Institute some two decades later. as well as of more recent papers by you and Marc, that in turn generated ones by Domenico Seminara and me. Duality has indeed evolved from breakthrough to tru- ism, as(legend has it)good physics ideas always do! The second big thing for us came less than 2 years after, early in 1977, when we met on a freezing day in Harvard Yard (they all are) and realized that we had both been dreaming that the then brand-new Supergravity might be the key to one of the Department of Physics, Brandeis University, Waltham, MA, USA and Caltech, Pasadena CA, USA. e-mail: deser@ brandeis. edu

6 Greetings For Claudio: A 60th Birthday Greeting Stanley Deser4 Dear Claudio, Let me preface this, at times indiscreet, 60th birthday message with the consola￾tion that, were it not for the impossible 18 h (each way) plane journeys, you would have had to suffer the insult of hearing it live! The much appreciated invitation to your celebration made me both count and re￾member: we have known each other for over half your life so far, and it led me back to our first encounter, of early 1975, in Princeton. You were then already famous – the successor in the lineage of Feynman and Misner – as the latest of Wheeler’s “discoveries,” having reached New Jersey by the route closest to John’s heart, crazy electrodynamics. Since we’re also traveling backward in time, that takes care of defining Dick’s way, while Charlie’s (more recent) exploit was to destroy the beauty of Maxwell–Einstein theory by (re-) discovering its – horribly complicated, “already unified” – purely geometrical version. Your road was the radiation reaction prob￾lem, which you clarified so much that it forced Sidney Coleman to (sort of) do it right later, also using exotic Feynman propagators, come to think of it. What most impressed me however, in that initial meeting, was of a non-physics nature: you looked like a Jeune Premier (someone will translate) and you (claimed you) were equipping an ancient and highly unlikely-looking wreck of a yellow Land Rover for the journey home (merely to Santiago, rather than Valdivia, but still) down the (then highly unfinished) Pan-American highway. In drab, conventional, Princeton, one can imagine that these characteristics stood out most vividly! I had been invited for an informal visit by the Physics Department. I had a lot of fun – except when the whole Physics Department – from Wigner down, would close for grading Freshman physics exams (a Princeton custom, I was told!). Despite this quaint custom, you and I were able start what would become a long-standing research collaboration, whose first result was our Phys Rev electromagnetic duality paper (surviving those unsettling local mores). It was to become famous, long before duality became fash￾ionable, but not before going through the usual “it’s wrong and trivial” scoffing. It still gets quoted, and as you know, served as a basis of three of our further collab￾orations, with Henneaux and Gomberoff, at your Institute some two decades later, as well as of more recent papers by you and Marc, that in turn generated ones by Domenico Seminara and me. Duality has indeed evolved from breakthrough to tru￾ism, as (legend has it) good physics ideas always do! The second big thing for us came less than 2 years after, early in 1977, when we met on a freezing day in Harvard Yard (they all are) and realized that we had both been dreaming that the then brand-new Supergravity might be the key to one of the 4 Department of Physics, Brandeis University, Waltham, MA, USA and Caltech, Pasadena CA, USA, e-mail: deser@brandeis.edu

Greetings really big problems in ordinary General Relativity: positive energy. For background (in case there are younger members of the audience, unaware that there ever might have been such a problem), it had long been suspected- I had wasted countless years on it myself -that the Gr Hamiltonian was nonnegative, and only vanished for vacuum= flat space. A proof had actually quite recently appeared, by Schoen and Yau, but it was very pure mathematics of the sort that no one whom I understood understood, if you see what I mean. For you, the flash came via the notion that SUGRA was some sort of Dirac square root of gravity, for me it was the same fact, but stated as the SUGRa algebras relation that the Hamiltonian was the(hermitian) square of the supercharge. We compared notes, calculated some more-and it held up! I was eager to publish this final validation of SUGRA as also the savior of gr stability and to finally shake the problem, while you wisely hesitated because our proof was seemingly for the purely QUANTUM(because of the fermions)SUGRA, ather than its classical GR counterpart, and you hoped we could soon overcome that hurdle. I"won, so we had to later cede a bit of the glory to witten, who extracted the classical content via Killing spinors, and to Grisaru who simply noted that classical GR was the h=0 limit. restricted to the no external fermion sector of sugra t least formally. Nevertheless, ours represented (if I may say so)a most rewardin accomplishment of SUGRA (and of its devoted servants), providing a clear physical basis for a deep necessity. I cannot speak as authoritatively about your many other non-research feats, such as the enormous service rendered to Chile and to science by the Center you have so tirelessly served, starting from very lean and difficult times; many of your Northern Hemisphere colleagues have witnessed this first-hand. Certainly, the now-legendary South Pole theoretical physics conference of a few years back will eternally resonate in the hearts of all its survivors, at many levels, including its superb organization- even unto mobilizing the entire air force as well as commanding the winter south Pole's weather to obey! On a personal note, Gary Gibbons and I are indebted to this meeting for having gotten us started on perhaps the world's lowest paper, if only (we hope) latitude-wise! I can, however, say a bit more about the remarkable evolution of some of the physics ideas that you have produced; this requires making a severe election, but the four surviving my triage should give the overall flavor: In sheer SLAC citation density, of course it is BTZ that leads the list at 10(and counting) That a non-dynamical theory like 2+ 1 GR could be tortured into exhibiting a black hole solution is already amazing: that it then became the first entropy-explanation laboratory(triggered by Brown and Henneaux's work), and still keeps on giving, tes- tifies to its depth. It should also be a special source of all-Chilean pride Next, there is Regge-Teitelboim, as it is simply known, a clear and detailed exposition of the ins and outs of the dynamics of GR, that has weaned whole generations of relativists, despite its mysterious samizdat-like Italian publication. Then there is another, if less famous, Regge-Teitelboim, one that is especially close to my heart, though for im- pure reasons. Your work was a most original attempt to give a description of gR in terms of higher-dimensional embeddings, and it prompted a followup by Pirani Robinson and me. We submitted ours to the Physical Review, in the days when or did not toy with that Journals majesty: our title, "Embedding the G-String, " was not

Greetings 7 really big problems in ordinary General Relativity: positive energy. For background (in case there are younger members of the audience, unaware that there ever might have been such a problem), it had long been suspected – I had wasted countless years on it myself – that the GR Hamiltonian was nonnegative, and only vanished for vacuum = flat space. A proof had actually quite recently appeared, by Schoen and Yau, but it was very pure mathematics of the sort that no one whom I understood understood, if you see what I mean. For you, the flash came via the notion that SUGRA was some sort of Dirac square root of gravity, for me it was the same fact, but stated as the SUGRA algebra’s relation that the Hamiltonian was the (hermitian) square of the supercharge. We compared notes, calculated some more-and it held up! I was eager to publish this final validation of SUGRA as also the savior of GR stability and to finally shake the problem, while you wisely hesitated because our proof was seemingly for the purely QUANTUM (because of the fermions) SUGRA, rather than its classical GR counterpart, and you hoped we could soon overcome that hurdle. I “won,” so we had to later cede a bit of the glory to Witten, who extracted the classical content via Killing spinors, and to Grisaru who simply noted that classical GR was the h = 0 limit, restricted to the no external fermion sector, of SUGRA, at least formally. Nevertheless, ours represented (if I may say so) a most rewarding accomplishment of SUGRA (and of its devoted servants), providing a clear physical basis for a deep necessity. I cannot speak as authoritatively about your many other non-research feats, such as the enormous service rendered to Chile and to Science by the Center you have so tirelessly served, starting from very lean and difficult times; many of your Northern Hemisphere colleagues have witnessed this first-hand. Certainly, the now-legendary South Pole theoretical physics conference of a few years back will eternally resonate in the hearts of all its survivors, at many levels, including its superb organization￾even unto mobilizing the entire Air Force as well as commanding the Winter South Pole’s weather to obey! On a personal note, Gary Gibbons and I are indebted to this meeting for having gotten us started on perhaps the world’s lowest paper, if only (we hope) latitude-wise! I can, however, say a bit more about the remarkable evolution of some of the physics ideas that you have produced; this requires making a severe selection, but the four surviving my triage should give the overall flavor: In sheer SLAC citation density, of course it is BTZ that leads the list at 103 (and counting). That a non-dynamical theory like 2 + 1 GR could be tortured into exhibiting a black hole solution is already amazing; that it then became the first entropy-explanation laboratory (triggered by Brown and Henneaux’s work), and still keeps on giving, tes￾tifies to its depth. It should also be a special source of all-Chilean pride. Next, there is Regge–Teitelboim, as it is simply known, a clear and detailed exposition of the ins and outs of the dynamics of GR, that has weaned whole generations of relativists, despite its mysterious samizdat-like Italian publication. Then there is another, if less famous, Regge–Teitelboim, one that is especially close to my heart, though for im￾pure reasons. Your work was a most original attempt to give a description of GR in terms of higher-dimensional embeddings, and it prompted a followup by Pirani, Robinson and me. We submitted ours to the Physical Review, in the days when one did not toy with that Journal’s majesty; our title, “Embedding the G-String,” was not