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## 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

## What Is a Quantum State?

*Gary E. Bowman*

### in Essential Quantum Mechanics

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199228928
- eISBN:
- 9780191711206
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199228928.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter first discusses statistical techniques and concepts — including probability, probability distribution, average and expectation value, and uncertainty — that are relevant to quantum ... 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

## Time Evolution

*Gary E. Bowman*

### in Essential Quantum Mechanics

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199228928
- eISBN:
- 9780191711206
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199228928.003.0011
- Subject:
- Physics, Condensed Matter Physics / Materials

The time-dependent Schrödinger equation forms the chapter's starting point. For a time-independent Hamiltonian, a quantum state may be time evolved with the standard, unitary time evolution operator. ... More

## PATH INTEGRALS IN QUANTUM MECHANICS

*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/9780198566748.003.0002
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

This chapter constructs the path integral associated with the statistical operator e -βH in the case of Hamiltonians of the simple form p2/2m + V (q). The path ... More

## Probability Distributions on Hilbert Space

*Heinz-Peter Breuer and Francesco Petruccione*

### in The Theory of Open Quantum Systems

- Published in print:
- 2007
- Published Online:
- February 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.05
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter characterizes the statistical properties of a quantum mechanical ensemble in terms of a density matrix. However, if selective measurements of one or several observables are carried out ... More

## Statistical field theory: Perturbative expansion

*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/9780199227198.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses the perturbative calculation of correlation or vertex functions expressed in terms of field (functional) integrals. The successive contributions to the perturbative expansion ... More

## Functional integration: From path to field integrals

*Jean Zinn-Justin*

### in From Random Walks to Random Matrices

- Published in print:
- 2019
- Published Online:
- August 2019
- ISBN:
- 9780198787754
- eISBN:
- 9780191829840
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198787754.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

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 ... More

## The “Higgs” Boson, and its Field in the “Vacuum”

*Alvaro De Rújula*

### in Enjoy Our Universe: You Have No Other Choice

- Published in print:
- 2018
- Published Online:
- February 2020
- ISBN:
- 9780198817802
- eISBN:
- 9780191859366
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198817802.003.0022
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Spontaneous symmetry breaking. The symmetric Higgs bottle and its asymmetric state of lowest energy. The “expectation value” of the Higgs field in the vacuum. Why and how the vacuum is not void, it ... More

## The double scaling limit

*Razvan Gurau*

### in Random Tensors

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198787938
- eISBN:
- 9780191829918
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198787938.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Particle Physics / Astrophysics / Cosmology

This chapter presents the double scaling limit of quartic tensor models. In the first part of this chapter it is shown show that the melonic family in the quartic melonic tensor model can be ... More

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