Wick rotation derivation for weak values of the density and time-dependent density functional theory

Abstract

The equations of time-dependent density functional theory are derived, via the expression for a quantum weak value, from ring polymer self-consistent field theory using a mathematical correspondence between time and imaginary time. The imaginary time path integral formalism of Feynman, in which inverse temperature is seen to be a Wick rotation of time, allows one to write the equilibrium partition function of a quantum system in a form mathematically isomorphic with the path integral expression for the dynamics. Therefore the self-consistent field theory equations which are solutions to the equilibrium partition function are Wick rotated back into a set of dynamic equations, which are shown to give an expression for a quantum weak value of the one-particle density. Remarkably, weak values emerge naturally here without being postulated, as an intermediate step before recovering the standard expression for the density. The weak value expression in turn leads to the equations of time-dependent density functional theory. This first-principles derivation does not use the theorems of density functional theory, which are instead applied to guarantee equivalence with standard quantum mechanics. An expression for finite-temperature dynamics is also given, which shows that a ring polymer model for quantum particles holds for time-dependent systems as well as equilibrium situations. Issues arising in time-dependent density functional theory, such as causality, initial state dependence, and $v$-representability, are discussed in the context of the ring polymer derivation.