I count it as one of the great gifts of my professional life to have had V. S. Varadarajan as my de facto undergraduate advisor in the late 1970s — only de facto because at UCLA in those days undergraduate advising was not formalized at all; but Professor Varadarajan had so profound an effect on my academic life, then and throughout my later career, that I cannot count anyone as having had, or having, a greater role. I am excited to be in a position to review his new book, Reflections on Quanta, Symmetries, and Supersymmetries: so many things he said to me long ago, and also during a sabbatical I spent with him in the middle 1990s, not only resonate loudly, but, encountered in their context here, reveal themselves as having taken on increased meaning over the years.
Varadarajan said to me in my junior year that mathematics should be seen as an organic unity, with its various parts deeply (and sometimes) mysteriously linked. His entire professional trajectory illustrates his commitment to this vision and his unrelenting labor to reveal this beautiful inner harmony that is bewitching to so many of us. When I first heard him say this I registered it, to be sure, but I had no idea at the time of how such a view of things, if pursued, opens up incomparable research venues.
Just consider the fact that Varadarajan began as a young probabilist in India, became enchanted with quantum mechanics due to Dirac’s influence, went on to become a leading expert in Lie groups and Lie algebras as a consequence of his acquaintance and then deep friendship with Harish-Chandra, and has most recently been involved with such avant garde themes as “super geometry” (cf. pp. 100ff in the book under review). When I studied with him in the ’70s all I knew about him was that he had written the book on Lie Groups, Lie Algebras, and Their Representations, surely a model of classical exposition at a very deep level (Varadarajan is not for beginners!): every i dotted, every t crossed, and pure mathematics at its very best. I had no idea in my myopic youth that his scholarship was rooted in a desire to understand symmetry at a level well beyond what even his beautiful book conveyed.
The areas in which he taught me, however, were predominantly number theoretic. When I approached Professor Varadarajan with the complaint that I knew too little analysis and would like him to give me a tutorial he agreed and had me study André Weil’s gorgeous monograph Elliptic Functions According to Eisenstein and Kronecker. Analysis indeed… But I was hooked: Professor Varadarajan (VSV from now on) was teaching a research seminar on class field theory which I attended (I still have his superb class notes — three full terms worth, covering both local and global Klassenkörpertheorie in his usual encyclopædic and meticulous manner, replete with historical motivation, and doing scrupulous justice to the powerful modern machinery of cohomology. idèles, and adèles). VSV admonished me for neglecting to sign up for Robert Steinberg’s graduate course in number theory, referring to Steinberg as “a master,” and I moved like lightning to correct this and learned just how right he was. And when it came time for me to head to graduate school VSV said that, given what I was doing (largely under his influence), there were only three to choose from, one of which was indeed where I ended up attending, my admission guaranteed by VSV’s letter of recommendation.
Accordingly, in my twenties I thought of VSV as a combination Lie theorist/algebraic number theorist, his recent foci being class field theory, oriented to Kronecker’s Jugendtraum, and automorphic forms. (Oops. I forgot: after finishing my reading course with him on Weil’s book, he directed me to his next seminar based on Gôro Shimura’s Introduction to the Arithmetic Theory of Automorphic Functions: never let it be said that VSV’s taste in books is anything short of superb.) Little did I know that his scholarship was much broader even than that. I should have gotten a clue when he said to me at some point that number theory and physics are two sides of the same coin, but it did not resonate with me at that time.
This resonance came with time. Over the years I did learn about VSV’s expertise in quantum mechanics (viz. his early book, Geometry of Quantum Theory), and, indeed, his prevailing and recently burgeoning interest in questions on the very frontier of modern physics: for example, his website contains his two recent talks titled, “Has God made the (quantum) world p-adic?,” as well as his talk on “Symmetry and Supersymmetry.” And I am beginning to understand what he meant back in the late 1970s by the above remark.
This brings me to the book under review. As then title clearly indicates, it is, roughly speaking, about physics. But it is the kind of physics every mathematician should find irresistible, featuring such themes as quantum algebra, probability in what VSV calls the quantum world, super geometry and unitary representations of super Lie groups, and non-archimedean physics (where we encounter p-adic analysis, of course:
The simplest way to enforce nonarchimedean geometry is to assume that spacetime is a manifold over Qp (…) Since no single prime can be given distinguished status, it is perhaps even more natural to see if one could really work with an adèlic geometry as the basis for space-time… (p.157)
The latter lecture (for that is the format in which this book’s essays were first presented) even goes on to discuss the shape of a putative nonarchimedean quantum field theory; à propos, Feynman path integrals are covered earlier, in the lecture on “Probability in the quantum world,” where VSV gives, among other things, a superb sketch of Kac’s critical manouevre, which also illustrates the crystal clarity of VSV’s expository style:
In the 1950s (…) Mark Kac noticed that if we change in the Schrödinger equation the time variable t to it where i = (-1)1/2 then it becomes a diffusion equation which governs the transition probabilities of a Markoff process. To any stochastic process one can always associate a probability measure on the space of paths of the process and so one has a genuine path integral for computing the propagator of this process. This path integral can then be analytically continued in the time variable and the Feynman propagator can be obtained by this an analytic continuation…
The lectures themselves are truly vintage Varadarajan. They are individually complete entities, launched by a compact introduction, containing a wealth of insights, solid mathematics including treatments of how things are proved (viz. VSV’s occasional “proof sketches”), and personal reminiscences. The latter are particularly evocative; consider, for example, the following passage (pp. 54–55) regarding the late Moshe Flato, who died in 1998:
As one gets older, one finds invariably that one cannot do all the things one did when one was young, and certainly not at the depth that was reached in youth. The natural tendency to compensate for this is to reduce one’s ambitions and do what one can to advance science. Moshe scorned this way of living. He pushed himself and those around him without any slackening and lived on the creative edge all the time, right up to the moment of his death. According to the Hindu view of life this is one of the noblest ways to live and die (…) When I learned of Moshe’s death I was reminded of Isaac Stern’s words on the occasion of David Oistrakh’s death, when he referred to Oistrakh as a golden man. Moshe was a golden man…
Along these lines I also want to mention the book’s last chapter, “Mackey, Harish-Chandra, and representation theory,” given VSV’s opening remark, in the book’s Preface, to the effect that his “own mathematical education evolved out of interactions with Mackey and Harish-Chandra.” He goes on to characterize the last chapter as “brief portraits of their work, embedded in the context of personal reminiscences.” Two quotes are in order here. First, regarding George Mackey (pp. 207–208):
It was a great opportunity for me to see a great master at work, and that encounter shaped my entire mathematical career. I became a great admirer of his way of thinking that encompasses a broad picture of mathematics and physics, and emphasized concepts above brute calculation and ideas above technique. I was deeply impressed by the curiosity as well as humility with which he viewed the role of mathematics and the mathematician in the understanding, description, and interpretation of the world of phenomena around us.
Second, regarding Harish-Chandra (p. 220):
[On] one occasion in 1968 … [Harish-Chandra], who was a creature of regular habits, was suddenly forced to eat lunch at the Institute [of Advanced Study] one day, and a bunch of us accompanied him to the Institute cafeteria which was housed at the top of the building then. He shared the table with us and the discussion naturally turned to one of his favorite topics — how an artist gives shape to his dreams. He said that in his experience one has a dream that is crystal-clear but in the process of bringing it down to earth the dream gets mangled so much that at the end one does not know whether to feel relief or elation. I have pondered about this remark many times and marveled at his imagination, which allowed him to dare to dream those great dreams of his, and his power and persistence that made them a reality no matter how long it took.
There is a good reason (obvious to any one who knows him) why this remark by Harish-Chandra should resonate so well with Varadarajan.
Varadarajan’s Reflections on Quanta, Symmetries, and Supersymmetries is a wonderful collection of essays by a master of the genre (of the interphase of physics and mathematics, from the standpoint of a fine mathematician, with unitary representations of Lie groups hiding behind almost every corner — and rightly so), presented in a compact fashion, richly peppered with pithy personal remarks, both biographical and autobiographical. I dearly love Professor Varadarajan, and I am ecstatic at having a copy of this book all my own. Get a copy, too.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.