SU(2) Yang-Mills quantum mechanics of spatially constant fields

TL;DR

This paper investigates the SU(2) Yang-Mills quantum mechanics of spatially constant fields using a symmetric gauge and variational methods, achieving high-precision results for eigenstates and exploring excitation effects on eigenvalues.

Contribution

It introduces an analytical approach to solve the eigenstates of SU(2) Yang-Mills quantum mechanics with a variational basis, providing detailed insights into excitation energies and eigenvalue behavior.

Findings

High-precision eigenstates for spin-0 and higher spins.

Excitation energies increase the largest eigenvalue expectation value.

Other eigenvalues and magnetic field components remain at vacuum values.

Abstract

As a first step towards a strong coupling expansion of Yang-Mills theory, the SU(2) Yang-Mills quantum mechanics of spatially constant gauge fields is investigated in the symmetric gauge, with the six physical fields represented in terms of a positive definite symmetric (3 x 3) matrix S. Representing the eigenvalues of S in terms of elementary symmetric polynomials, the eigenstates of the corresponding harmonic oscillator problem can be calculated analytically and used as orthonormal basis of trial states for a variational calculation of the Yang-Mills quantum mechanics. In this way high precision results are obtained in a very effective way for the lowest eigenstates in the spin-0 sector as well as for higher spin. Furthermore I find, that practically all excitation energy of the eigenstates, independently of whether it is a vibrational or a rotational excitation, leads to an increase of the expectation value of the largest eigenvalue <\phi_3>, whereas the expectation values of the other two eigenvalues, <\phi_1> and <\phi_2>, and also the component <B_3> = g<\phi_1\phi_2> of the magnetic field, remain at their vacuum values.