Quantum Processes and Energy-Momentum Flow

TL;DR

This paper explores energy flows in simple quantum systems through the quantum Hamilton-Jacobi equation, linking it to standard formalisms and proposing empirical methods to analyze energy-momentum components.

Contribution

It introduces a novel approach to analyze energy-momentum flow in quantum systems via the quantum Hamilton-Jacobi equation and its relation to weak values and non-commutative geometry.

Findings

Energy-momentum tensor components relate to weak values.

The quantum Hamilton-Jacobi equation appears in multiple formalisms.

New empirical avenues for studying quantum energy flow.

Abstract

In this paper we focus on energy flows in simple quantum systems. This is achieved by concentrating on the quantum Hamilton-Jacobi equation. We show how this equation appears in the standard quantum formalism in essentially three different but related ways, from the standard Schr\"{o}dingier equation, from Lagrangian field theory and from the von Neumann-Moyal algebra. This equation allows us to track the energy flow using the energy-momentum tensor, the components of which are related to weak values of the four-momentum operator. This opens up a new way to explore these components empirically. The algebraic approach enables us to discuss the physical significance of the underlying non-commutative symplectic geometry, raising questions as to the structure of particles in quantum systems.