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

Recapitulation of the Introduction

The introduction succinctly sets up the framework for reducing the problem of non-perturbative quantization of Yang--Mills fields into a more manageable question involving probability measures on infinite-dimensional spaces. The core goal is to simplify the Yang--Mills field Hamiltonian's quantization into defining a probability measure on the space of gauge equivalence classes of connections in $\mathbb{R}^3$. The Hamiltonian, linked to a compact Lie group and its Lie algebra, centers on the quadratic momentum and the square of the curvature tensor, utilizing a natural metric. It is pointed out that using the Hodge star operator transforms the curvature tensor into a potential vector field, leading to a potential term in the Hamiltonian represented by the square of this vector field. This insight suggests that the problem can be addressed from the perspective of classical Hamiltonian mechanics.

The discussion details how any Hamiltonian on a symplectic manifold can admit canonical quantizations, exemplified by a Hamiltonian formula involving both momentum and a potential vector derived from a gradient of a functional. This leads to a specific operator form that incorporates a measure affected by a function $\psi$, illustrating the significant role this function plays in the quantization process. The function $\psi$ introduces flexibility in quantization, permitting various properties of the quantized Hamiltonian depending on its choice, thus highlighting the potential for experimental and mathematical influences in selecting $\psi$.

Further, the implications are explored by considering the quantum mechanics of a free particle and a harmonic oscillator, showing how different choices of $\psi$ impact the quantization outcome. This analogy helps clarify the potential gap in the spectrum of the quantum Hamiltonian, emphasizing the practical and theoretical flexibility in quantum field theories.

Lastly, the quantization of the Yang--Mills field is directly addressed, showing how this approach helps solve the normal ordering problem in quantization and reduces the problem to defining a suitable measure on the configuration space. The chosen function $\psi$ ensures the measure behaves appropriately at infinity, providing a consistent approach even in complex cases like the abelian $U(1)$ gauge field.

Overall, the introduction comprehensively outlines the mathematical structure and theoretical considerations necessary for approaching the quantization of Yang--Mills fields, setting a robust foundation for detailed exploration in subsequent sections.

Theorem 13: Spectral Properties

Theorem 13 addresses the spectral properties of a specific Hamiltonian operator $H$. It states:

The spectrum of $H$ consists of the point eigenvalue 0 (which is one-dimensional and generated by the constant function 1, considered the ground state) and a continuous spectrum from $\frac{m}{2}$ to infinity $[\frac{m}{2}, \infty)$.
The continuous spectrum is of Lebesgue type, indicating that it fills intervals of the real line densely.
There is a gap in the spectrum of $H$, specifically between 0 and $\frac{m}{2}$, which characterizes the difference between the lowest eigenvalue and the rest of the spectrum.

Remark 14: Quantization of Yang-Mills Fields

Remark 14 delves into the quantization of Yang-Mills fields:

It emphasizes that the mass parameter $m > 0$ is crucial for the spectral gap in the Abelian case of Yang-Mills field quantization.
The standard quantum electrodynamics (QED) approach results in a massless theory, contrasting with the approach discussed, which suggests a mass $m$ leading to a self-adjoint realization in an $L^2$-space.
This quantization method for the Abelian case, as presented, differs fundamentally from non-Abelian theories, where such straightforward quantization and realization in $L^2$-space may not be directly applicable.
A proper quantization of the non-Abelian Yang-Mills Hamiltonian would require a "density" function and renormalization for defining a corresponding measure, which once established, defines the quantized Hamiltonian.

These points are crucial as they not only discuss technical aspects of quantum field theory but also touch upon fundamental differences in handling Abelian versus non-Abelian gauge theories. The spectral gap, essential for the stability and physical interpretation of the quantized theories, has critical implications for the theories' mathematical structure and the types of quantum states they admit.