Temperature-time duality exemplified by Ising magnets and quantum-chemical many electron theory

Abstract

In this work, we first present a detailed analysis of temperature-time duality in the 3D Ising model, by inspecting the resemblance between the density operator in quantum statistical mechanics and the evolution operator in quantum field theory, with the mapping β = (kBT)−1 → it. We point out that in systems like the 3D Ising model, for the nontrivial topological contributions, the time necessary for the time averaging must be infinite, being comparable with or even much larger than the time of measurement of the physical quantity of interest. The time averaging is equivalent to the temperature averaging. The phase transitions in the parametric plane (β, it) are discussed, and a singularity (a second-order phase transition) is found to occur at the critical time tc, corresponding to the critical point βc (i.e, Tc). It is necessary to use the 4-fold integral form for the partition function for the 3D Ising model. The time is needed to construct the (3 + 1)D framework for the quaternionic sequence of Jordan algebras, in order to employ the Jordan-von Neumann-Wigner procedure. We then turn to discuss quite briefly temperature-time duality in quantum-chemical many-electron theory. We find that one can use the known one-dimensional differential equation for the Slater sum S(x, β) to write a corresponding form for the diagonal element of the Feynman propagator, again with the mapping β → it.