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arXiv:1807.10462

Rigorous construction of coherent state path integrals through dualization

Falk Bruckmann A, Juan Diego Urbina

Published Jul 27, 2018• 6 citations
arXivHTMLPDF

Abstract

Two long-standing problems in the construction of coherent state path integrals, the unwarranted assumption of path continuity and the ambiguous definition of the Hamiltonian symbol, are rigorously solved. To this end the fully controlled dual representation familiar from lattice quantum field theories is introduced. Dualization allows for both the step-by-step check in the construction of discrete path integrals and for the identification of the Hamiltonian and Berry phase part of the action, thus making both discrete and continuous path integrals consistently defined. Once the correct action is constructed, we are able to follow the transition to the continuum for the polar form of general Bose-Hubbard models and to provide an exact form of the path integral for general spin systems, where previous works showed its failure for all standard choices of operator ordering.

DOI
10.48550/arxiv.1807.10462

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Recently, doubts have been cast on the validity of the continuous-time coherent state path integral. This has led to controversies regarding the correct way of performing calculations with path integrals, and to several alternative definitions of what should be their continuous limit. Furthermore, the issue of a supposedly proper ordering of the Hamiltonian operator, entangled with the continuous-time limit, has led to considerable confusion in the literature. Since coherent state path integrals are at the basis of the modern formulation of many-body quantum theory, it should be laid on solid foundations. Here, we show that the standard coherent state path integral is fine as it is, and does not need any modification. We show that the issues raised above are coming from illegitimate use of the (standard) rules of calculus, which are not necessarily valid in path integrals. This is well known in the context of stochastic equations, in particular in their path integral formulation. This insight allows for solving these issues, and addressing the correspondence between the various orderings at the level of the path integral. We also use this opportunity to address the proper calculation of a functional determinant in the presence of a Hubbard-Stratonovich field, which shares in the controversies.
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Abstract Recently, doubts have been cast on the validity of the continuous-time coherent state path integral. This has led to controversies regarding the correct way of performing calculations with path integrals, and to several alternative definitions of what should be their continuous limit. Furthermore, the issue of a supposedly proper ordering of the Hamiltonian operator, entangled with the continuous-time limit, has led to considerable confusion in the literature. Since coherent state path integrals are at the basis of the modern formulation of many-body quantum theory, it should be laid on solid foundations. Here, we show that the issues raised above are coming from the illegitimate use of the (standard) rules of calculus, which are not necessarily valid in path integrals. This is well known in the context of stochastic equations, in particular in their path integral formulation. This insight allows for solving these issues and addressing the correspondence between the various orderings at the level of the path integral. We also use this opportunity to address the proper calculation of a functional determinant in the presence of a Hubbard–Stratonovich field, which shares in the controversies.