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## 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

## 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

## 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

## Gaussian integrals. Algebraic preliminaries

*Jean Zinn-Justin*

### in Quantum Field Theory and Critical Phenomena: Fifth Edition

- Published in print:
- 2021
- Published Online:
- June 2021
- ISBN:
- 9780198834625
- eISBN:
- 9780191872723
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198834625.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In this work, the perturbative aspects of quantum mechanics (QM) and quantum field theory (QFT), to a large extent, are studied with functional (path or field) integrals and functional techniques. ... 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

## Quantum mechanics (QM): Path integrals in phase space

*Jean Zinn-Justin*

### in Quantum Field Theory and Critical Phenomena: Fifth Edition

- Published in print:
- 2021
- Published Online:
- June 2021
- ISBN:
- 9780198834625
- eISBN:
- 9780191872723
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198834625.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to ... 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.

## Relativistic fermions: Introduction

*Jean Zinn-Justin*

### in Quantum Field Theory and Critical Phenomena: Fifth Edition

- Published in print:
- 2021
- Published Online:
- June 2021
- ISBN:
- 9780198834625
- eISBN:
- 9780191872723
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198834625.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Some basic concepts needed for the discussion of Fermi fields have been introduced earlier, as in quantum mechanics (QM) with Grassmann variables, a representation by field integrals of the ... More

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