Path Integrals from Spacetime Quantum Actions

TL;DR

This paper develops a novel formalism linking quantum path integrals with a spacetime symmetric quantum action, enabling covariant treatment of quantum mechanics and potential quantum computation applications.

Contribution

It introduces a new trace-based formalism connecting the Quantum Action with Feynman's Path Integral, facilitating covariant quantum descriptions and computational approaches.

Findings

Identifies the path integral as a quantum trace in an extended Hilbert space.

Reveals different path integral representations depend on basis choices.

Proposes a new continuum limit approach for path integrals.

Abstract

The possibility of extending the canonical formulation of quantum mechanics (QM) to a space-time symmetric form has recently attracted wide interest. In this context, a recent proposal has shown that a spacetime symmetric many-body extension of the Page and Wootters mechanism naturally leads to the so-called Quantum Action (QA) operator, a quantum version of the action of classical mechanics. In this work, we focus on connecting the QA with the well-established Feynman's Path Integral (PI). In particular, we present a novel formalism which allows one to identify the "sum over histories" with a quantum trace, where the role of the classical action is replaced by the corresponding QA. The trace is defined in the extended Hilbert space resulting from assigning a conventional Hilbert space to each time slice and then taking their tensor product. The formalism opens the way to the application of quantum computation protocols to the evaluation of PIs and general correlation functions, and reveals that different representations of the PI arise from distinct choices of basis in the evaluation of the same trace expression. The Hilbert space embedding of the PIs also discloses a new approach to their continuum time limit. Finally, we discuss how the ensuing canonical-like version of QM inherits many properties from the PI formulation, thus allowing an explicitly covariant treatment of spacetime symmetries.