Weak Values: Approach through the Clifford and Moyal Algebras

TL;DR

This paper explores weak values in quantum mechanics for Schrödinger and Pauli particles using Clifford and Moyal algebras, revealing their relation to Bohmian quantities and providing multiple derivations.

Contribution

It introduces a unified algebraic framework combining Clifford and Moyal algebras to analyze weak values for different quantum particles.

Findings

Weak values relate to Bohm momentum, energy, and quantum potential.

Multiple methods (standard, Clifford, Moyal) yield consistent results for Schrödinger particles.

Unified algebraic approach extends to Pauli particles, enriching the theoretical understanding.

Abstract

In this paper we calculate various transition probability amplitudes, TPAs, known as `weak values' for the Schrodinger and Pauli particles. It is shown that these values are related to the Bohm momentum, the Bohm energy and the quantum potential in each case. The results for the Schrodinger particle are obtained in three ways, the standard approach, the Clifford algebra approach of Hiley and Callaghan, and the Moyal approach. To obtain the results for the Pauli particle, we combine the Clifford and Moyal algebras into one structure. The consequences of these results are discussed.