Positive mass gap of quantum Yang-Mills Fields
Adrian P. C. Lim
TL;DR
The paper constructs a rigorous 4D quantum Yang–Mills framework by organizing the Hilbert space as $\mathbb{H}_{YM}(\frak g)=\{1\}\oplus\bigoplus_{n\ge1}\mathscr{H}(\rho_n)$, with each $\mathscr{H}(\rho_n)$ encoding states smeared on space-like surfaces and carrying a representation $\rho_n$ of a compact simple Lie algebra $\frak g$. Momentum and energy spectra are derived from a Yang–Mills path integral via the quadratic Casimir, yielding eigenvalues $\hat{H}(\rho_n)$ and $\hat{P}(\rho_n)$ satisfying $\hat{H}(\rho_n)^2-\hat{P}(\rho_n)^2=m_n^2>0$, and the infimum mass gap $m_0>0$ is shown to exist through a Callan–Symanzik–driven renormalization analysis with asymptotic freedom. The work proves local (anti)commutativity and CPT-type symmetries, demonstrates the area-law behavior underpinning the non-Abelian structure, and establishes a clustering theorem by connecting the mass gap to exponential decay of vacuum correlations. It also clarifies that the abelian case (e.g., $U(1)$) does not admit a mass gap in this construction, highlighting the essential role of non-Abelian interactions. Overall, the paper provides a constructive, axiomatic route to a 4D Yang–Mills theory with a positive mass gap, tying spectral data from Casimir operators to physically meaningful mass and clustering phenomena within a Wightman-framework setting.
Abstract
We construct a 4-dimensional quantum field theory on a Hilbert space, dependent on a simple Lie Algebra of a compact Lie group, that satisfies Wightman's axioms. This Hilbert space can be written as a countable sum of non-separable Hilbert spaces, each indexed by a non-trivial, inequivalent irreducible representation of the Lie Algebra. In each component Hilbert space, a state is given by a triple, a space-like rectangular surface $S$ in $\mathbb{R}^4$, a measurable section of the Lie Algebra bundle over this surface $S$, represented irreducibly as a matrix, and a Minkowski frame. The inner product is associated with the area of the surface $S$. In our previous work, we constructed a Yang-Mills measure for a compact semi-simple gauge group. We will use a Yang-Mills path integral to quantize the momentum and energy in this theory. During the quantization process, renormalization techniques and asymptotic freedom will be used. Each component Hilbert space is the eigenspace for the momentum operator and Hamiltonian, and the corresponding Hamiltonian eigenvalue is given by the quadratic Casimir operator. The eigenvalue of the corresponding momentum operator will be shown to be strictly less than the eigenvalue of the Hamiltonian, hence showing the existence of a positive mass gap in each component Hilbert space. We will further show that the infimum of the set containing positive mass gaps, each indexed by an irreducible representation, is strictly positive. In the last section, we will show how the positive mass gap will imply the Clustering Theorem.