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Gaussian expectation values. Steepest descent method

Jean Zinn-Justin

in Phase Transitions and Renormalization Group

Published in print:
2007
Published Online:
January 2010
ISBN:
9780199227198
eISBN:
9780191711107
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199227198.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter presents a number of technical results concerning generating functions, Gaussian measures, and the steepest descent method. It begins by introducing the notion of generating function of ... More


Formal means

Helmut Hofmann

in The Physics of Warm Nuclei: with Analogies to Mesoscopic Systems

Published in print:
2008
Published Online:
September 2008
ISBN:
9780198504016
eISBN:
9780191708480
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198504016.003.0026
Subject:
Physics, Nuclear and Plasma Physics

This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, ... More


Continuum limit and path integrals

Jean Zinn-Justin

in Phase Transitions and Renormalization Group

Published in print:
2007
Published Online:
January 2010
ISBN:
9780199227198
eISBN:
9780191711107
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199227198.003.0005
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter shows that, as in the case of the random walk, one can associate to the continuum limit a path integral, which generalizes the path integral of the Brownian motion. It first studies the ... More


GAUSSIAN INTEGRALS

Jean Zinn-Justin

in Path Integrals in Quantum Mechanics

Published in print:
2004
Published Online:
January 2010
ISBN:
9780198566748
eISBN:
9780191717994
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566748.003.0001
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

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 ... More


PATH INTEGRALS: FERMIONS

Jean Zinn-Justin

in Path Integrals in Quantum Mechanics

Published in print:
2004
Published Online:
January 2010
ISBN:
9780198566748
eISBN:
9780191717994
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566748.003.0007
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

The methods that have been described in previous chapters can be used to study general quantum Bose systems. They are based in a direct way on the introduction of a generating function of symmetric ... More


A formula for semisimple orbital integrals

Jean-Michel Bismut

in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691151298
eISBN:
9781400840571
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691151298.003.0007
Subject:
Mathematics, Geometry / Topology

This chapter states without proof the main result of this volume, which expresses the elliptic orbital integrals associated with the heat kernel as a Gaussian integral on t(γ‎). In addition, the ... More


The case where [k(γ),p0]=0

Jean-Michel Bismut

in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691151298
eISBN:
9781400840571
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691151298.003.0009
Subject:
Mathematics, Geometry / Topology

This chapter explicitly evaluates the Gaussian integral that appears in the right-hand side of a formula in Chapter 6 for the orbital integrals of the heat kernel, when γ‎ is nonelliptic and ... More


Fundamental Ideas

Nicholas Manton and Nicholas Mee

in The Physical World: An Inspirational Tour of Fundamental Physics

Published in print:
2017
Published Online:
July 2017
ISBN:
9780198795933
eISBN:
9780191837111
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198795933.003.0002
Subject:
Physics, Condensed Matter Physics / Materials

Chapter 1 offers a simple introduction to the use of variational principles in physics. This approach to physics plays a key role in the book. The chapter starts with a look at how we might minimize ... More


Path integrals: I said to him, ‘You’re crazy’

Tom Lancaster and Stephen J. Blundell

in Quantum Field Theory for the Gifted Amateur

Published in print:
2014
Published Online:
June 2014
ISBN:
9780199699322
eISBN:
9780191779435
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199699322.003.0024
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

The chapter describes Feynman"s path integral approach and demonstrate how to evaluate certain Gaussian integrals, using the simple harmonic oscillator as an example.


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