Paul C. Martin, many-body theorist

I met Paul C. Martin when he lectured on many-body theory and fluids at a Spring School of physics at Naples, 1964. He has done work of permanent importance on equilibrium and non-equilibrium quantum statistical physics. Of great importance also for the theory of C*-algebras is his formulation, with J. Schwinger of the KMS condition, also used by R. Kubo. This is a generalisation of the theorem concerning Laplace transforms, which says that the Laplace transform of a function of energy E, which is zero for E < 0, is analytic as a function of the conjugate variable (here, time t) in the upper half-t-plane. The KMS generalisation of this says that, if the energy of a theory is positive (one of the axioms!) then the equilibrium state gives a two-point two-time correlation function that is analytic in time-difference in a strip in the upper-half plane of width 1/T, where T is the temperature, and is periodic with imaginary period 1/T. The periodicity follows from the cyclicity of the trace. This idea was exploited by Haag and Hugenholtz to obtain results in C*algebraic statistical mechanics, and was then incorporated into the general theory of von Neumann algebras by Takesaki and Connes.