  We present a quantum statistical analysis of a microscopic mean-field model
of structural glasses at low temperatures. The model can be thought of as
arising from a random Born von Karman expansion of the full interaction
potential. The problem is reduced to a single-site theory formulated in terms
of an imaginary-time path integral using replicas to deal with the disorder. We
study the physical properties of the system in thermodynamic equilibrium and
develop both perturbative and non-perturbative methods to solve the model. The
perturbation theory is formulated as a loop expansion in terms of two-particle
irreducible diagrams, and is carried to three-loop order in the effective
action. The non-perturbative description is investigated in two ways, (i) using
a static approximation, and (ii) via Quantum Monte Carlo simulations. Results
for the Matsubara correlations at two-loop order perturbation theory are in
good agreement with those of the Quantum Monte Carlo simulations.
Characteristic low-temperature anomalies of the specific heat are reproduced,
both in the non-perturbative static approximation, and from a three-loop
perturbative evaluation of the free energy. In the latter case the result so
far relies on using Matsubara correlations at two-loop order in the three-loop
expressions for the free energy, as self-consistent Matsubara correlations at
three-loop order are still unavailable. We propose to justify this by the good
agreement of two-loop Matsubara correlations with those obtained
non-perturbatively via Quantum Monte Carlo simulations.
