Henri Epstein and Mathematical Physics

Henri Epstein met Louis Michel in 1955, during his final year as a student at the École polytechnique. This pivotal encounter steered him towards a lifelong career as a theoretical physicist. After completing his thesis, Henri was staff member of CERN’s theoretical physics department from 1967 to 1970. During this time, he continued to make regular visits to IHES, eventually joining the Institute as a CNRS researcher in 1971.

In 1957-58, he attended Arthur Wightman’s course on axiomatic quantum field theory at the Collège de France, a topic that became the focus of his first research endeavors. In the 1960s, Henri achieved remarkable results, such as the proof of the non-positivity of energy density (with Vladimir Jurko Glaser and Arthur Jaffe) and the establishment of analyticity properties, including the crossing symmetry relation, for 2→2 scattering amplitudes (with Jacques Bros and Glaser). The latter, now a classic result, forms one of the foundations of the S-matrix bootstrap program, which has seen a significant revival in recent years.

In the 1970s, Henri’s work on quantum field theory led to the famous Epstein-Glaser construction of causal perturbation theory, which offers a novel approach to handling UV divergences in Feynman diagrams. The full potential of this method, particularly for perturbations of conformal field theories, still remains an active area of research.

In the 1980s, Henri’s focus shifted to discrete dynamical systems, and particularly Feigenbaum’s universality properties. He notably came up with an ingenious new proof for the existence of fixed points for the Feigenbaum-Cvitanović equation. Unlike the original argument by Oscar Lanford III, his proof, based on the Schauder-Tikhonov fixed point theorem, does not require computer assistance.

In the latter part of his life and scientific career, Henri devoted himself primarily to the study of quantum fields in de Sitter and anti-de Sitter universes, collaborating with Jacques Bros, Ugo Moschella, and occasionally with Michel Gaudin and Vincent Pasquier. De Sitter universes, which are solutions to Einstein’s cosmological equations without matter, play a central role in contemporary theoretical physics. Mathematically, they are analytic Lorentzian manifolds, structures particularly well suited to the application of methods from the theory of analytic functions of several complex variables—a field in which Henri was the last surviving master of a glorious era.

The results Henri and his colleagues achieved over 25 years range from general structural properties of de Sitterian quantum field theories to the derivation of concrete formulas such as the Källén–Lehmann spectral representations. The latter are non-trivial, exact formulas with surprising implications for particle stability and the existence of bound states.

Henri continued his research as long as he could, up until the last days of his life. His final paper was published on the very day of his passing. He loved beauty and elegance in his research and in all aspects of life. He had a deep appreciation for literature and music. His sense of humor and clarity of mind made working with him a true pleasure and joy. Until the end, he remained youthful in spirit, enthusiastic, sincere, and generous. Mozart and Schubert, whom he loved so much, accompanied him on his last journey.

Ugo Moschella and Slava Rychkov