Effective descriptions of complex quantum systems: path integrals and operator ordering problems

Abstract

Key words Effective quantum theories, operator ordering, coordinate-dependent mass. PACS 05.30.-d, 05.40.-a, 31.15.Kb Dedicated to Bernhard Mühlschlegel on the occasion of his 80th birthdayWe study certain aspects of the effective, occasionally called collective, description of complex quantum systems within the framework of the path integral formalism, in which the environment is integrated out. Generalising the standard Feynman-Vernon Caldeira-Leggett model to include a non-linear coupling between "particle" and environment, and considering a particular spectral density of the coupling, a coordinatedependent mass (or velocity-dependent potential) is obtained. The related effective quantum theory, which depends on the proper discretisation of the path integral, is derived and discussed. As a result, we find that in general a simple effective low-energy Hamiltonian, in which only the coordinate-dependent mass enters, cannot be formulated. The quantum theory of weakly coupled superconductors and the quantum dynamics of vortices in Josephson junction arrays are physical examples where these considerations, in principle, are of relevance.

Operator Ordering in Two-Dimensional N = 1 Supersymmetry With Curved Manifold

“…Operator ordering problem appears in various context. For example, nonrelativistic Chern-Simons theory [1], path integral in statistical mechanics [2] and so on. There are several ways for this problem such as Weyl ordering [3] or self-adjoint extension of Hermitian operators [4].…”