Quantization due to the time evolution, with applications to Quantum Yang-Mills theory, Quantum Gravity and Classical Statistical Field Theory

TL;DR

This paper introduces a rigorous mathematical framework for statistical field theories with non-deterministic time evolution, successfully extending to gauge theories and non-polynomial Hamiltonians, with applications to Quantum Yang-Mills, Quantum Gravity, and classical statistical mechanics.

Contribution

It develops a mathematically rigorous class of statistical field theories that incorporate non-deterministic time evolution and gauge symmetries, unifying various quantum and classical field theories.

Findings

Verified canonical commutation relations under non-deterministic evolution

Extended Gaussian measures to non-polynomial Hamiltonians

Tested formalism with free electromagnetic field quantization

Abstract

Quantum Yang-Mills theory, Classical Statistical Field Theory (for Hamiltonians which are non-polynomial in the fields, e.g. General relativistic statistical mechanics) and Quantum Gravity all suffer from severe mathematical inconsistencies and produce unreliable predictions at best. We define with mathematical rigor, a class of statistical field theories in Minkowski space-time where the (classical) canonical coordinates when modified by a non-deterministic time evolution, verify the canonical commutation relations. We then extend these statistical field theories to include non-trivial gauge symmetries and show that these theories have all the features of a Quantum Yang-Mills theory in four-dimensional space-time. We generalize the Gaussian measure to allow for the definition of Hamiltonians which are non-polynomial in the fields, such as in Classical Statistical Field Theory and Quantum Gravity. Finally, we test the consistency of our formalism with the quantization of the free Electromagnetic field.