Tribute to Henri Epstein by Jürg Fröhlich

I worked at IHES and lived in Bures-sur-Yvette for four and a half years, from 1978 to 1982. The Institute was truly a paradise back then, and I consider the time I spent there to be one of the best and most delightful periods of my life. One of the inhabitants of that paradise was Henri Epstein.

But let’s go back a little farther into the past. I first met Henri at the famous 1970 Les Houches summer school on Statistical Mechanics and Quantum Field Theory. My PhD advisor, Klaus Hepp, along with other luminaries in mathematical physics, lectured on renormalized perturbation theory in quantum field theory (BPHZ). Henri presented his ongoing work with Jurko Glaser on a new approach to renormalized perturbation theory, that made very clever use of locality and causality and avoided encountering infinities. Their work was not published, yet, but would later become famous. The Les Houches school was also the occasion when I first met further top mathematical physicists, including some of those who were or would become inhabitants of the paradise at Bures-sur-Yvette, in particular Alain Connes, the late Oscar E. Lanford III, and David Ruelle.

In 1970, I was a beginning PhD student, and I must say, I found Henri quite intimidating—a feeling that never entirely left me. He seemed to insist on ultimate mathematical precision and conceptual clarity, both in his thinking and in discussions with colleagues, which made it difficult for me to interact with him.

A year or two later, I met Henri at the University of Zurich, where he gave a great lecture on an application of his work with Glaser to quantum electrodynamics. Most of the theorists in Zurich were hopeful that Henri would be offered a professorship at the university. He shared scientific interests—particularly in axiomatic quantum field theory in the spirit of Arthur S. Wightman—with two of my most admired teachers, Res Jost and Klaus Hepp. Res held Henri in the highest esteem, describing him as “pure gold.”

Unfortunately, for reasons I’d prefer not to go into, the professorship offer never materialized—a significant loss for theoretical physics in Zurich.

In the fall of 1972, I became a postdoctoral researcher at the University of Geneva. Almost every week, Jean-Pierre Eckmann, Jurko Glaser, and Raymond Stora would meet for a working seminar in mathematical physics, and I was one of the regular listeners. This provided yet another opportunity to meet Henri and learn about his work with Glaser on renormalization theory.

In 1974, I encountered Henri again at Princeton, shortly after he had published his new proof of the strong subadditivity property of quantum-mechanical entropy—a result first established by Lieb and Ruskai. Henri’s analysis made use of holomorphic functions, an area of mathematics in which he was an unparalleled master. I found his proof both very illuminating and, relatively easy to follow.

In the summer of 1975, I returned to Europe for a few months. Jean-Pierre Eckmann and Henri were working on a problem that I had also become interested in: proving that perturbation theory is asymptotic to the S-matrix in a family of relativistic scalar quantum field models in two space-time dimensions, known as P(ϕ)₂-models. They showed me a preliminary draft of a paper that was intended to contain a proof of this result. However, there was still a small gap in their argument, which I was able to fill. This is how I became a co-author of a beautiful paper for which Henri deserves ninety-nine percent of the credit.

The paper showcases Henri’s remarkable knowledge of the properties of chronological, retarded, and advanced Green functions in quantum field theory. He had previously applied this expertise in significant work with the late Jacques Bros and others on the properties of scattering amplitudes in relativistic quantum field theory.

During my time at IHES in the late 1970s and early 1980s, I witnessed the beginnings of Henri’s fascination with computers and his growing interest in the theory of dynamical systems. I was deeply impressed by his proof, with Massimo Campanino, and building on partial results by David Ruelle, of the existence of a fixed point in Feigenbaum’s equation—a proof that didn’t rely on any computer assistance. In hindsight, I regret not following Henri more closely in his explorations of dynamical systems and sharing his fascination with computers.

In 2015, I organized a special trimester at IHES on new developments and open problems in quantum physics, entitled “Trimestre sur le Monde Quantique.” I believe this was a successful program, addressing some of the truly deep questions in contemporary physics. My impression was further validated by Henri’s comments—he mentioned that he found the lectures on quantum physics, both theoretical and experimental, to be highly educational and entertaining.

In recent years, Henri and various colleagues, mostly worked on properties of quantum field theory on de Sitter and anti-de Sitter space. The little I learned on these matters from Henri and his friends inspired some results in a paper entitled “KMS, Etc.” I wrote with a student, twenty-three years ago.

Let me conclude with a few remarks on some well-known traits of Henri’s personality, cherished by his friends. Above all, Henri exemplified outstanding intellectual honesty. He possessed a highly critical mind and insisted on accuracy and precision in his professional interactions. He was deeply cultured, and despite being more knowledgeable and intelligent than most of us, he remained modest and kind. Henri appreciated style in all areas of life—whether in conversations, discussions, cultural pursuits, literature, or culinary adventures. He was remarkably loyal and respectful in his relationships with colleagues and friends, and he had a fine sense of humor.

To close this testimonial, I would like to remind readers that Henri’s youth was profoundly shaped by the precarious circumstances in Europe between 1933 and 1945. We all hope such times will never return.

I will miss Henri.

(I would also like to add that I will miss the pleasant encounters and conversations I had with Pierre Cartier, who passed away just a few days after Henri.)

Jürg Fröhlich (ETH Zurich)
September 2024