A foundational derivation of quantum weak values and time-dependent density functional theory

The equations of time-dependent density functional theory are derived, via the expression for the quantum weak value, from ring polymer quantum theory using a symmetry 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 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 the formula for the quantum weak value. As a special case, this in turn reduces 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 derived using the limits of the equilibrium equations and the ground state dynamics. This finite temperature expression shows that a ring polymer model for quantum particles holds for time-dependent systems as well as static 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. Connections with a variety of quantum phenomena is reviewed, and a possible link with de Broglie-Bohm theory is mentioned.