The Partition Function and Level Density for Yang-Mills-Higgs Quantum Mechanics

We calculate the partition function $Z(t)$ and the asymptotic integrated level density $N(E)$ for Yang-Mills-Higgs Quantum Mechanics for two and three dimensions ($n = 2, 3$). Due to the infinite volume of the phase space $\Gamma$ on energy shell for $n= 2$, it is not possible to disentangle completely the coupled oscillators ($x^2 y^2$-model) from the Higgs sector. The situation is different for $n = 3$ for which $\Gamma$ is finite. The transition from order to chaos in these systems is expressed by the corresponding transitions in $Z(t)$ and $N(E)$, analogous to the transitions in adjacent level spacing distribution from Poisson distribution to Wigner-Dyson distribution. We also discuss a related system with quartic coupled oscillators and two dimensional quartic free oscillators for which, contrary to YMHQM, both coupling constants are dimensionless.