Quantum thermalization and ℏ-expansion via quantization deformation

Using the phase-space formalism of quantum mechanics and results from deformation quantization, we establish a systematic framework for obtaining the expectation value of a quantum observable as an ℏ-expansion, whose terms involve phase-space integrals of a function of the Hamiltonian and its derivatives. The convergence of this expansion is examined for several few-body quantum systems with nonlinear couplings where the density operator corresponds to a Gaussian window of variable width σ in the energy eigenspectrum. The choice of σ strongly influences the validity of the expansion, which can be examined by considering tractable higher-order terms. The formalism extends the Bohr–Sommerfeld quantization condition when considering the limit σ → 0, and also connects to the Wigner–Kirkwood expansion via a Laplace transform. For finite widths, the coarse-grained density of states and general observables derived from the Gaussian ensemble can probe different spectral characteristics, including level clustering, shell structure, and deviations from quantum ergodicity. When applied to many-body quantum systems, the formalism yields corrections to the eigenstate thermalization hypothesis, thereby providing insights into thermalization that extend beyond the predictions of random matrix theory.