# Arthur Jaffe

Arthur Jaffe is known for his research in both mathematics and physics, as well as for training many of leading persons in these fields. In early work Jaffe and Glimm established the field known as *constructive quantum field theo*ry, by showing the mathematical compatibility of relativity with quantum theory and interaction. They constructed the first family of non-linear quantum field theories in 2D (partially in collaboration with Tom Spencer and others). Later Jaffe and Glimm also established the only complete example of a field theory in dimention 3D (with contributions from Osterwalder and Feldman). All these field theories satisfy the standard properties proposed by Wightman, Haag, Kastler, Ruelle, Osterwalder, and Schrader. Constructing a complete theory in dimension four remains one of the most important open questions.

Jaffe (with Glimm and Spencer) gave the first mathematical proof of the existence of symmetry breaking and phase transitions in quantum field theory.

Solving these problems involved developing a non-perturbative theory of renormalization they called "phase-cell localization." It also relied on using the "reflection positivity property" to establish results in quantum theory using properties of statistical mechanics and probability theory.

Jaffe contributed to a number of other subjects in over 200 papers and books; see his personal web site. These subjects include classical gauge theories, reflection positivity, non-commutative geometry, and supersymmetry. His collaborations with F. Pedrocchi starting in 2012, and with B. Janssens starting in 2014 led to his collaboration with Zhengwei Liu starting in 2015.

The picture language project at Harvard with Zhengwei Liu began formally in September 2017.