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

Russell B. Thompson*

Zarin Tasneem

Yves Caudano

  • Department of Physics & Astronomy and Waterloo Institute for Nanotechnology, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1

  • Department of Physics & Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1

  • *Contact author: thompson@uwaterloo.ca
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Phys. Rev. A 114, 012206 – Published 13 July, 2026

DOI: https://doi.org/10.1103/mpb7-534c

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 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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Information

Phys. Rev. A 114, 012206– Published 13 July, 2026
  • Received 20 November 2025
  • Revised 11 June 2026
  • Accepted 18 June 2026
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