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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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