TowardsNonperbativeQuantizationOfYangMillsFieldsAndTheMassGap - crowlogic/arb4j GitHub Wiki

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

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.