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## Algebraic Preliminaries

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter reviews algebraic identities about gaussian integrals, in particular Wick's theorem, a result also relevant for gaussian probability distributions. It discusses the steepest descent ... More

## Grassmann Calculus

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter sets out basic definitions and features for the differential and integral calculus of c-number and Grassmann functions. Derivatives and integrals of functions of c-number variables are ... 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

## St And Brs Symmetries, Stochastic Field Equations

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0016
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Section 15.3 introduced a transformation depending on anticommuting parameters, to prove the geometric stability of homogeneous spaces under renormalization. There is a set of topics, stochastic ... More

## From Langevin Equation To Supersymmetry

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0017
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

A particular class of Langevin equations is of special interest: equations with gaussian noise, in which the drift force derives from a static action and is thus conservative (the corresponding ... More

## Grassmann numbers: coherent states and the path integral for fermions

*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/9780199699322.003.0029
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter introduces a new type of mathematical object: Grassmann numbers. These anticommute and can be used to describe fermions. It shows how to construct a fermion path integral using Grassmann ... More

## An Introduction to Clifford Algebras and Spinors

*Jayme Vaz, Jr. and Roldão da Rocha, Jr.*

- Published in print:
- 2016
- Published Online:
- August 2016
- ISBN:
- 9780198782926
- eISBN:
- 9780191826146
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198782926.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This book is unique in the literature on spinors and Clifford algebras in that it is accessible to both students and researchers while maintaining a formal approach to these subjects. Besides ... More

## Phase Space Methods for Degenerate Quantum Gases

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

Recent experimental progress has enabled cold atomic gases to be studied at nanokelvin temperatures, creating new states of matter where quantum degeneracy occurs—Bose–Einstein condensates and ... 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

## Complex Numbers and Grassmann Numbers

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter sets out the basic algebraic properties of both complex numbers (c-numbers) and Grassmann numbers (g-numbers). The formal rules of addition and multiplication together with the ... More

## The Origins of Four-Dimensional Geometry

*Tony Robbin*

### in Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought

- Published in print:
- 2006
- Published Online:
- October 2013
- ISBN:
- 9780300110395
- eISBN:
- 9780300129625
- Item type:
- chapter

- Publisher:
- Yale University Press
- DOI:
- 10.12987/yale/9780300110395.003.0001
- Subject:
- Society and Culture, Technology and Society

This chapter takes a look at four-dimensional geometry, which Hermann Grassmann envisioned as a system where space and its geometric elements and descriptions can be assumed to other dimensions. The ... More

## The Emergence of Foundational Study

*Jan von Plato*

### in The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691174174
- eISBN:
- 9781400885039
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174174.003.0003
- Subject:
- History, History of Science, Technology, and Medicine

This chapter looks at how modern foundational study has twofold mathematical roots. One is the discovery of non-Euclidean geometries, especially the proof of independence of the parallel postulate by ... More

## Fermions and Functional Formalism

*Laurent Baulieu, John Iliopoulos, and Roland Sénéor*

### in From Classical to Quantum Fields

- Published in print:
- 2017
- Published Online:
- May 2017
- ISBN:
- 9780198788393
- eISBN:
- 9780191830310
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198788393.003.0011
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

The introduction of spinor fields and the problem of the positivity of the energy. Need for anti-commutators. Calculus in a Grassmannian manifold, the Berezin integral. Clifford algebras. Quantum ... More

## Geometry on my Mind

*David D. Nolte*

### in Galileo Unbound: A Path Across Life, the Universe and Everything

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198805847
- eISBN:
- 9780191843808
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805847.003.0005
- Subject:
- Physics, History of Physics

This chapter reviews the history of modern geometry with a focus on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who ... More

## Introduction

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0001
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter provides an overview of the phase space method for treating the physics of systems of bosons and fermions in regimes where quantum degeneracy occurs. Key concepts such as modes, creation ... More

## Exterior Algebra and Grassmann Algebra

*Jayme Vaz and Roldão da Rocha*

### in An Introduction to Clifford Algebras and Spinors

- Published in print:
- 2016
- Published Online:
- August 2016
- ISBN:
- 9780198782926
- eISBN:
- 9780191826146
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198782926.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In this chapter, exterior algebras and Grassmann algebras are discussed. In particular, their morphisms, inherited from the tensor algebra, are presented in a general framework. In some texts, the ... More

## Clifford, or Geometric, Algebra

*Jayme Vaz and Roldão da Rocha*

### in An Introduction to Clifford Algebras and Spinors

- Published in print:
- 2016
- Published Online:
- August 2016
- ISBN:
- 9780198782926
- eISBN:
- 9780191826146
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198782926.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In this chapter, the so-called geometric algebras, or Clifford algebras, are introduced, with special attention to the universal Clifford algebras. As well as providing the standard definition of a ... More

## Functional Calculus for C-Number and Grassmann Fields

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0011
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter presents the basic mathematical tools required for extending the phase space method involving phase space variables for separate modes to a phase space method involving field functions ... More

## Distribution Functionals in Quantum Atom Optics

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0012
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter formally presents the representation of the quantum density operator by phase space distribution functionals of phase space fields, leading to expressions for quantum correlation ... More

## Noether’s Theorem for Fields

*Peter Mann*

### in Lagrangian and Hamiltonian Dynamics

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0028
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This is a unique chapter that discusses classical path integrals in both configuration space and phase space. It examines both Lagrangian and Hamiltonian formulations before qualitatively discussing ... More

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