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 to 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.

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• Classical statistical mechanics
• Quantum foundations
• Weak values & weak measurements
• Density functional theory
• Self-consistent field theory
• Time-dependent DFT