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From Random Walks to Random Matrices$
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Jean Zinn-Justin

Print publication date: 2019

Print ISBN-13: 9780198787754

Published to Oxford Scholarship Online: August 2019

DOI: 10.1093/oso/9780198787754.001.0001

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Functional integration: From path to field integrals

Functional integration: From path to field integrals

(p.13) 2 Functional integration: From path to field integrals
From Random Walks to Random Matrices

Jean Zinn-Justin

Oxford University Press

Chapter 2 is rather descriptive and introduces the notion of functional (path and field) integrals, for boson (the holomorphic formalism) as well as fermion systems (this necessitates the introduction of Grassmann integration) as they are used in physics. Prior to the second half of the twentieth century, the technical tools of theoretical physics were mainly differential or partial differential equations. However, when systematic investigations of large scale systems with quantum of statistical fluctuations began, new tools were required. This led to the development of functional integration. In this chapter, the role of Gaussian measures and Gaussian expectation values is emphasized, leading to Wick’s theorem, a tool for perturbative calculations. Functional integrals provide remarkable tools to study the classical limit and barrier penetration. They provide a simple bridge between non–relativistic quantum mechanics and quantum field theory.

Keywords:   functional integral, Gaussian measure, Gaussian expectation value, Grassmann integration, holomorphic formalism, barrier penetration, perturbative calculation, quantum field theory, Wick’s theorem

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