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 Γ on energy shell for n = 2, it is not possible to completely disentangle the coupled oscillators (x2 y2 model) from the Higgs sector. The situation is different for n = 3 for which Γ 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.