Towards non-perturbative quantization and the mass gap problem for the Yang-Mills Field

TL;DR

This paper proposes a new approach to quantize the Yang-Mills field by defining a probability measure on gauge equivalence classes, demonstrating a non-perturbative quantization with a mass gap for the Abelian case.

Contribution

It introduces a formal self-adjoint quantized Hamiltonian for Yang-Mills fields and constructs a Gaussian measure for the Abelian case, revealing a mass gap.

Findings

Defined a probability measure for the Abelian U(1) case

Constructed a self-adjoint quantized Hamiltonian with a spectral gap

Provided a non-perturbative quantization framework

Abstract

We reduce the problem of quantization of the Yang-Mills field Hamiltonian to a problem for defining a probability measure on an infinite-dimensional space of gauge equivalence classes of connections on $\mathbb{R}^3$. We suggest a formally self-adjoint expression for the quantized Yang-Mills Hamiltonian as an operator on the corresponding Lebesgue $L^2$-space. In the case when the Yang-Mills field is associated to the Abelian group $U(1)$ we define the probability measure which depends on two real parameters $m>0$ and $c\neq 0$. This yields a non-standard quantization of the Hamiltonian of the electromagnetic field, and the associated probability measure is Gaussian. The corresponding quantized Hamiltonian is a self-adjoint operator in a Fock space the spectrum of which is $\{0\}\cup[\frac12m, \infty)$, i.e. it has a gap.