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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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GAUSSIAN INTEGRALS

GAUSSIAN INTEGRALS

Chapter:
(p.1) 1 GAUSSIAN INTEGRALS
Source:
Path Integrals in Quantum Mechanics
Author(s):

Jean Zinn-Justin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566748.003.0001

This chapter introduces, in the case of ordinary integrals, concepts and methods that can be generalized to path integrals. The first part is devoted to the calculation of ordinary Gaussian integrals, Gaussian expectation values, and the proof of the corresponding Wick's theorem. The notion of connected contributions is discussed, and it is shown that expectation values of monomials can conveniently be associated with graphs called Feynman diagrams. Expectation values corresponding to measures that deviate slightly from Gaussian measures can be reduced to sums of infinite series of Gaussian expectation values, a method known as perturbation theory. This chapter also contains a short presentation of the steepest descent method, a method that allows evaluating a class of integrals by reducing them, in some limit, to Gaussian integrals. To discuss properties of expectation values with respect to some measure or probability distribution, it is always convenient to introduce a generating function of the moments of the distribution.

Keywords:   path integrals, Gaussian integrals, Wick's theorem, expectation values, connected contributions, steepest descent method, perturbation theory, generating function

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