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Path Integrals in Quantum Mechanics$
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Jean Zinn-Justin

Print publication date: 2004

Print ISBN-13: 9780198566748

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198566748.001.0001

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CLASSICAL AND QUANTUM STATISTICAL PHYSICS

CLASSICAL AND QUANTUM STATISTICAL PHYSICS

Chapter:
(p.91) 4 CLASSICAL AND QUANTUM STATISTICAL PHYSICS
Source:
Path Integrals in Quantum Mechanics
Author(s):

Jean Zinn-Justin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566748.003.0004

This chapter provides a simple physical interpretation to the formal continuum limit that has led, from an integral over position variables corresponding to discrete times, to a path integral. It shows that the integral corresponding to discrete times can be considered as the partition function of a classical statistical system in one space dimension. The continuum limit, then, corresponds to a limit where the correlation length, which characterizes the decay of correlations at large distance, diverges. This limit has some universality properties in the sense that different discretized forms lead to the same path integral. In this statistical framework, the correlation functions that have been introduced earlier appear as continuum limits of the correlation functions of classical statistical models on a one-dimensional lattice. Thus, the path integral can be used to exhibit a mathematical relation between classical statistical physics on a line and quantum statistical physics of a point-like particle at thermal equilibrium.

Keywords:   continuum limit, classical statistical physics, quantum statistical physics, path integral, correlation functions, partition function, discrete times, one-dimensional lattice

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