A Transfer Operator Approach to Relativistic Quantum Wavefunction

TL;DR

This paper introduces a transfer operator framework for relativistic quantum mechanics, deriving a new wavefunction evolution equation that generalizes the Schrödinger and Dirac equations within a geometric setting.

Contribution

It develops a scalar wavefunction formalism on a pseudo-Riemannian manifold, linking transfer operators with relativistic quantum evolution and connecting to string theory considerations.

Findings

Recover the Schrödinger equation in the non-relativistic limit.

Show that Dirac spinor wavefunctions satisfy the new relativistic equation.

Establish a connection between the formalism and string theoretic mass considerations.

Abstract

The original intent of the Koopman-von Neumann formalism was to put classical and quantum mechanics on the same footing by introducing an operator formalism into classical mechanics. Here we pursue their path the opposite way and examine what transfer operators can say about quantum mechanical evolution. To that end, we introduce a physically motivated scalar wavefunction formalism for a velocity field on a 4-dimensional pseudo-Riemannian manifold, and obtain an evolution equation for the associated wavefunction, a generator for an associated weighted transfer operator. The generator of the scalar evolution is of first order in space and time. The probability interpretation of the formalism leads to recovery of the Schrodinger equation in the non-relativistic limit. In the special relativity limit, we show that the scalar wavefunction of Dirac spinors satisfies the new equation. A connection with string theoretic considerations for mass is provided.