Quantization of Fields by Averaging Classical Evolution Equations

TL;DR

This paper introduces a novel method for quantizing fields by averaging classical evolution equations over a fifth dimension, connecting quantum and classical theories and enabling quantization of non-canonical theories.

Contribution

It extends the formalism of quantum field theory quantization to classical evolution equations using a 5-D Wick rotation and averaging, allowing for new approaches to quantum effects.

Findings

Averaging over the fifth dimension reproduces standard quantum diagrams.

The method quantizes theories without canonical quantization.

Expectations are computed as averages over classical PDE solutions.

Abstract

This paper extends the formalism for quantizing field theories via a microcanonical quantum field theory and Hamilton's principle to classical evolution equations. These are based on the well-known correspondence under a Wick rotation between quantum field theories and 4-D statistical mechanical theories. By placing quantum field theories on a 4+1-D under Wick rotation to 5-D, expectations of observables are calculated for a microcanonical field theory averaging Hamiltonian flow over a fifth spacelike dimension, a technique common in lattice gauge simulations but not in perturbation theory. In a novel demonstration, averaging pairs of external lines in the classical Feynman diagrams over the fifth dimension generates diagrams with loops and vacuum fluctuations identical to Standard Model diagrams. Because it is microcanonical, this approach, while equivalent for standard quantum fields theories in the Standard Model, is able to quantize theories that have no canonical quantization. It is also unique in representing expectations as averages over solutions to an ordinary, classical PDE rather than a path integral or operator based approaches. Hence, this approach draws a clear connection between quantum field theory and classical field theory in higher dimensions which has implications towards how quantum effects are interpreted. In particular, it raises questions about how violations of the ergodic hypothesis could influence quantum measurements even in standard, non-statistical quantum field theory.