We study the canonical thermodynamics of a one-particle model in which a relativistic oscillator in d spatial dimensions is coupled to the finite Laplacian spectrum of a fuzzy two-sphere with angular-momentum cutoff JF; the fuzzy eigenvalues shift the squared mass. In the nonrelativistic window, where the partition function factorises, the internal heat capacity follows the known positive T−2 finite-spectrum law. For the full Klein–Gordon trace at fixed finite JF and fixed model parameters, the fuzzy contribution to the heat capacity is instead negative and decays as T−2, with a mass- and dimension-dependent coefficient; its leading term and, for d ≥ 3, its next-to-leading term are derived and confirmed by exact canonical sums. The two T−2 laws hold in disjoint regimes, because the corresponding limits do not commute. For the three-dimensional Dirac oscillator, the unrestricted canonical trace diverges; spinor results are therefore defined only in a conditional ensemble with fixed conserved total angular momentum, which does not regularise or replace the unrestricted trace. No large-JF or commutative limit is considered.
We investigate the composite system of a 2+1-dimensional electric charge placed in the background field of a Nielsen–Olesen flux tube. The explicit analytic results are obtained in the static probe approximation and in the controlled large-separation regime αx0 ≫ 1, where α=2ea. We show that the combined configuration carries a non-vanishing field angular momentum, closely analogous to the familiar charge–monopole systems in 3+1 dimensions. We also find that the system possesses a finite field momentum, which appears to violate the center-of-energy theorem. A detailed analysis reveals that this field momentum is compensated by a hidden mechanical momentum carried by the flux tube’s currents, thereby preserving the center-of-energy theorem.
