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

TL;DR

This paper explores the path integral approach to complex quantum systems, highlighting challenges in deriving simple effective Hamiltonians due to operator ordering and discretisation issues, with applications to superconductors and Josephson junctions.

Contribution

It generalizes the Caldeira-Leggett model to include non-linear couplings and discusses the implications for effective quantum theories and operator ordering problems.

Findings

Coordinate-dependent mass arises in the effective theory.

A simple low-energy Hamiltonian cannot generally be formulated.

Relevance to superconductors and Josephson junctions.

Abstract

We 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 coordinate-dependent 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.