# Eyvind Hugo Wichmann

## Professor of Physics, Emeritus

Eyvind Hugo Wichmann was born in Stockholm, Sweden, on May 30, 1928, and grew up in Helsinki, Finland. He attended the Finnish Institute of Technology and graduated with a diploma in physics in 1950. Wichmann moved to the United States after receiving a fellowship to study at Columbia University. He received his Ph.D. in physics from Columbia University in 1956. Wichmann then spent a year as a member of the Institute for Advanced Study in Princeton, New Jersey. He joined the faculty of the Department of Physics at the University of California, Berkeley, in 1957, and was promoted to full professor in 1967. Wichmann retired as a professor emeritus in 1993.

Wichmann's early career in theoretical physics began with his 1956 doctoral thesis at Columbia University, written under the supervision of Norman Kroll. During Wichmann's graduate studies, he developed a mastery of quantum electrodynamics, a relatively new field of theoretical physics that combines relativity, quantum mechanics, and electrodynamics. The goal of the theory is to describe the interaction of light and matter. Wichmann's thesis deals with vacuum polarization in a very strong Coulomb field. Vacuum polarization is the phenomenon in which an electron-positron pair is created from the vacuum by an external electric field. It has the effect of altering the Coulomb potential of a point charge. In the thesis, Wichmann developed certain nonperturbative techniques and applied them to this difficult problem in order to accurately compute these modifications.

Wichmann's research focused on technically demanding and physically important calculations such as dispersion relations, scattering theory, and *S *matrix theory in quantum electrodynamics through the end of the fifties. After he arrived at UC Berkeley in 1957, Wichmann became more active in research in mathematical quantum mechanics. This was then a new field that sought to use mathematics to clarify some of the underlying difficulties in theoretical physics, and quantum electrodynamics in particular. Much of Wichmann's research was done in collaboration with his graduate students, some of whom are mentioned in this memorial.

Some of Wichmann's main contributions from the sixties in mathematical nonrelativistic quantum mechanics include a proof of the exponential decay of two- and three-body eigenfunctions for Schrödinger operators, with student L. Slaggie, and the structure of the Green's function for the Schrödinger operator with a Coulomb potential, with another student C.-H. Woo. Wichmann's student J. Finley studied aspects of scattering theory for nonrelativistic quantum systems and characterized short-range interactions using observable properties of wave packets. This project reflected Wichmann's long-standing interest in foundational questions in quantum mechanics such as the measurement problem.

Perturbation theory in nonrelativistic quantum mechanics was another important topic of interest to Wichmann. His student L. Mlodinow developed a mathematical theory of the *1/N*-expansion, for large *N*, a powerful method used by physicists to compute atomic energy levels.

Wichmann continued to work on quantum field theory. The theory of the scattering matrix was studied intensively in the UC Berkeley Physics Department in the sixties, led by colleagues G. Chew and S. Mandelstam. Wichmann actively contributed to this research, emphasizing axiomatic aspects of the theory. With his student J. Crichton, he analyzed the important cluster properties of the *S*-matrix.

One of the challenging problems of quantum field theory is the interaction of light and matter. P. Mohr, one of Wichmann's doctoral students in the late sixties and early seventies, computed the shift in the electronic energy levels of hydrogen-like atoms due to so-called radiative corrections from the electron's interaction with virtual photons. Another characteristic property of the interaction of light with matter is radiative decay. In the late seventies, Wichmann's student M. Hillery analyzed a model of the hydrogen atom coupled to a quantized electromagnetic field. The excited states of the atom are predicted to decay approximately exponentially in time, a characteristic of a quantum resonance. Hillery computed the decay rate of an excited state and its variation from exponential behavior.

Wichmann's interests in axiomatic and algebraic quantum field theory (AQFT) developed in the late sixties and dominated his research for the rest of his career. Axiomatic quantum field theory arose from attempts of theoretical physicists to understand singularities in quantum field theory from a fundamental, mathematical point of view. A. S. Wightman of Princeton University was a leading proponent of this mathematical formulation of quantum field theory, guided by basic physical principles such as locality and Poincaré covariance. Wichmann found this approach very appealing and taught many aspects of it in his graduate quantum field theory course. In the early sixties, R. Haag and D. Kastler reformulated axiomatic quantum field theory as an algebraic theory in which the algebras of local observables play the key role. Wichmann quickly became an expert in AQFT. His landmark contribution, together with his doctoral student J. Bisognano, was the proof of Haag duality for von Neumann algebras of local observables associated with spacelike wedge regions. They proved that the commutant of the local wedge algebra *A*(*W*_{L}) consisting of observables localized to the left wedge region, is the local algebra of observables *A*(*W*_{R}) generated by observables localized in the causal complement of *W*_{L} namely the right wedge region *W*_{R}.

To explain this in more detail, one of the major discoveries in the theory of operator algebras in the late sixties was Tomita's theory of modular operators for von Neumann algebras. The impact on both operator algebra theory and on algebraic quantum field theory was profound. Briefly, the theory discovered by Tomita, and expounded on by Takesaki, now called Tomita-Takesaki theory of modular operators, identified a pair of operators, the modular involution and the modular generator, associated with a von Neumann algebra. This pair of operators has the properties that the modular involution maps the von Neumann algebra to its commutant and the modular operator generates a one-parameter semigroup of automorphism of the von Neumann algebra.

In AQFT, the relationship between a von Neumann algebra and its commutant is intimately tied to the notion of locality, an important physical property believed to be fundamental to any physical theory of quantum particles. Locality is a key characteristic of field theories as conceptualized by R. Haag in the late fifties. In AQFT, Wichmann and Bisogano proved that the commutant of the von Neumann algebra of local observables associated with the left spacelike wedge region *W*_{L} in Minkowski space is the von Neumann algebra associated with the right wedge region *W*_{R}. They did this by constructing the modular operators for the von Neumann algebra associated with the left wedge. The modular involution is related to the fundamental TCP operator of local quantum field theory. Furthermore, the group associated with the modular generator was identified as a one-parameter group, implementing a specific Lorentz boost transformation preserving the wedge region. Haag duality for double-cone regions in massless free quantum field theory was proved by P. Hislop, another of Wichmann's students. The algebraic approach to scattering theory was investigated by Wichmann’s student H. Warchall for models in which a massless quantum field interacts linearly with a quantum oscillator.

A related problem that held Wichmann's interests for many decades was the association between local quantum fields and local von Neumann algebras of observables. On a technical level, this is related to the self-adjointness of the local fields smoothed with test functions. In that case, the local algebras would be generated by field operators applied to local test functions. In absence of a self-adjointness result, Wichmann explored local von Neumann algebras affiliated with closed extensions of the field operators. He and colleagues, including D. Buchholz and S. Summers, discovered that the requirement that the net of local algebras satisfy a condition of Bisognano-Wichmann duality is essential for the locality of the theory. Local and internal symmetries in AQFT were thoroughly explored by Wichmann with his students L. Landau and I. Herbst. Wichmann's last doctoral student, L. Thomas, also studied aspects of local algebras in AQFT.

On the basis of his research in theoretical physics, Wichmann was elected a Fellow of the American Physical Society and a member of the Finnish Academy of Science and Letters. From 1961 to 1963, Wichmann was a Sloan Research Fellow.

A dedicated and exacting teacher, Wichmann cared deeply about his many undergraduate and graduate students, including more than fifteen successful doctoral students. In 1988, Wichmann received UC Berkeley’s Distinguished Teaching Award, honoring his lifelong commitment to physics education.

Wichmann’s devotion to his students was manifest in many ways. In the decades before the internet and the personal computer revolution, Wichmann meticulously typed notes for his undergraduate and graduate classes, always improving upon them using the ‘cut and paste’ method. Students would pay a few dollars for the notes that were distributed by a punch card system. Many of his doctoral students valued these notes decades after the classes.

Based on his undergraduate quantum physics course, Wichmann authored the widely-used textbook *Quantum Physics*, *volume 4 of the Berkeley Physics *Course, in 1971. This demanding text is structured to reinforce basic physical principles and computational proficiency.

The international conference "Theoretical Physics and Quantum Field Theory" was held at UC Berkeley, June 4-6, 1999, to celebrate Wichmann’s contributions to theoretical physics. The proceedings are published in the *Electronic Journal of Differential Equations*, Conference 04, 2000.

On a personal note, all of Wichmann’s students knew he had a fondness for cigars and pipes. They could determine whether he was in his office or not by just walking down the corridor. Wichmann also enjoyed telling people about Finnish culture and unusual food items such as Mykyrokka (blood dumpling soup), smoked reindeer meat, and a liqueur made from berries that only grow above the Arctic Circle. One of his favorite books was* The Good Soldier Schweik*, by Jaroslav Hašek, a darkly satirical novel and a commentary on the absurdity of many bureaucratic organizations.

Eyvind Wichmann passed away on February 16, 2019, at the age of 90, in Berkeley, California. He was preceded in death by his wife of 52 years, Marianne, an accomplished artist. He is survived by his children Mats (Beth) and Sonia, and grandchildren Mora, Elina, and Ville.

Peter D. Hislop

2020