Canonical quantization of Galilean covariant field theories

doi:https://doi.org/10.1016/j.aop.2005.04.013

Title: Canonical quantization of Galilean covariant field theories

Full Record
Other Related Research

Abstract

The Galilean-invariant field theories are quantized by using the canonical method and the five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. This method is motivated by the fact that the extended Galilei group in 3 + 1 dimensions is a subgroup of the inhomogeneous Lorentz group in 4 + 1 dimensions. First, we consider complex scalar fields, where the Schroedinger field follows from a reduction of the Klein-Gordon equation in the extended space. The underlying discrete symmetries are discussed, and we calculate the scattering cross-sections for the Coulomb interaction and for the self-interacting term {lambda}{phi} {sup 4}. Then, we turn to the Dirac equation, which, upon dimensional reduction, leads to the Levy-Leblond equations. Like its relativistic analogue, the model allows for the existence of antiparticles. Scattering amplitudes and cross-sections are calculated for the Coulomb interaction, the electron-electron and the electron-positron scattering. These examples show that the so-called 'non-relativistic' approximations, obtained in low-velocity limits, must be treated with great care to be Galilei-invariant. The non-relativistic Proca field is discussed briefly.

Authors:

Santos, E S ^[1]; Montigny, M de ^[2]; Khanna, F C ^[3]

Theoretical Physics Institute, University of Alberta, Edmonton, Alta., T6G 2J1 (Canada)
Theoretical Physics Institute, University of Alberta, Edmonton, Alta., T6G 2J1 (Canada) and Faculte Saint-Jean, University of Alberta, Edmonton, Alta., T6C 4G9 (Canada)
Theoretical Physics Institute, University of Alberta, Edmonton, Alta., T6G 2J1 (Canada) and TRIUMF, 4004, Wesbrook Mall, Vancouver, BC, V6T 2A3 (Canada)

Publication Date:: 2005-11-01

OSTI Identifier:: 20690201

Resource Type:: Journal Article

Journal Name:: Annals of Physics (New York)

Additional Journal Information:: Journal Volume: 320; Journal Issue: 1; Other Information: DOI: 10.1016/j.aop.2005.04.013; PII: S0003-4916(05)00068-0; Copyright (c) 2005 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANTIPARTICLES; CROSS SECTIONS; DIRAC EQUATION; ELECTRON-POSITRON COLLISIONS; ELECTRON-POSITRON INTERACTIONS; ELECTRONS; FIELD THEORIES; KLEIN-GORDON EQUATION; LORENTZ GROUPS; MATHEMATICAL SPACE; QUANTIZATION; RELATIVISTIC RANGE; SCALAR FIELDS; SCATTERING; SCATTERING AMPLITUDES; SYMMETRY

Citation Formats


                    Santos, E S, Montigny, M de, and Khanna, F C. Canonical quantization of Galilean covariant field theories.  United States: N. p., 2005. 
        Web.  doi:10.1016/j.aop.2005.04.013.

Copy to clipboard


                    Santos, E S, Montigny, M de, & Khanna, F C. Canonical quantization of Galilean covariant field theories.  United States.  https://doi.org/10.1016/j.aop.2005.04.013

Copy to clipboard


                    Santos, E S, Montigny, M de, and Khanna, F C. Tue .  
        "Canonical quantization of Galilean covariant field theories".  United States.  https://doi.org/10.1016/j.aop.2005.04.013.

Copy to clipboard


                    
@article{osti_20690201,

  title        = {Canonical quantization of Galilean covariant field theories},

  author       = {Santos, E S and Montigny, M de and Khanna, F C},

  abstractNote = {The Galilean-invariant field theories are quantized by using the canonical method and the five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. This method is motivated by the fact that the extended Galilei group in 3 + 1 dimensions is a subgroup of the inhomogeneous Lorentz group in 4 + 1 dimensions. First, we consider complex scalar fields, where the Schroedinger field follows from a reduction of the Klein-Gordon equation in the extended space. The underlying discrete symmetries are discussed, and we calculate the scattering cross-sections for the Coulomb interaction and for the self-interacting term {lambda}{phi} {sup 4}. Then, we turn to the Dirac equation, which, upon dimensional reduction, leads to the Levy-Leblond equations. Like its relativistic analogue, the model allows for the existence of antiparticles. Scattering amplitudes and cross-sections are calculated for the Coulomb interaction, the electron-electron and the electron-positron scattering. These examples show that the so-called 'non-relativistic' approximations, obtained in low-velocity limits, must be treated with great care to be Galilei-invariant. The non-relativistic Proca field is discussed briefly.},

  doi          = {10.1016/j.aop.2005.04.013},

  url          = {https://www.osti.gov/biblio/20690201},
  journal      = {Annals of Physics (New York)},

  issn         = {0003-4916},

  number       = 1,

  volume       = 320,

  place        = {United States},

  year         = {2005},

  month        = {11}

}

Copy to clipboard

Journal Article:

https://doi.org/10.1016/j.aop.2005.04.013

Other availability

Find in Google Scholar

Search WorldCat to find libraries that may hold this journal

Save / Share:

Export Metadata

Save to My Library

Similar records in OSTI.GOV collections:

Path-integral quantization of Galilean Fermi fields

Journal Article Montigny, M de ; Campus Saint-Jean, University of Alberta, Edmonton, Alta., T6C 4G9 ; Khanna, F C ; ... - Annals of Physics (New York)

The Galilei-covariant fermionic field theories are quantized by using the path-integral method and five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. First, we review the five-dimensional approach to the Galilean Dirac equation, which leads to the Levy-Leblond equations, and define the Galilean generating functional and Green's functions for positive- and negative-energy/mass solutions. Then, as an example of interactions, we consider the quartic self-interacting potential {lambda}({psi}-bar {psi}){sup 2}, and we derive expressions for the 2- and 4-point Green's functions. Our results are compatible with those found in the literature on non-relativistic many-body systems. The extended manifold allows for compact expressions ofmore »« less
https://doi.org/10.1016/j.aop.2007.08.002
From Galilean-invariant to relativistic wave equations

Journal Article Elizalde, E ; Lobo, J A - Phys. Rev., D; (United States)

Through an imaginary change of coordinates in the Galilei algebra in 4 space dimensions and making use of an original idea of Dirac and Levy-Leblond, we are able to obtain the relativistic equations of Dirac and of Bargmann and Wigner starting with the (Galilean-invariant) Schroedinger equation.
https://doi.org/10.1103/PhysRevD.22.884
Bosons, fermions and anyons in the plane, and supersymmetry

Journal Article Horvathy, Peter A., E-mail: horvathy@univ-tours.f ; Plyushchay, Mikhail S., E-mail: mplyushc@lauca.usach.c ; Valenzuela, Mauricio - Annals of Physics (New York)

Universal vector wave equations allowing for a unified description of anyons, and also of usual bosons and fermions in the plane are proposed. The existence of two essentially different types of anyons, based on unitary and also on non-unitary infinite-dimensional half-bounded representations of the (2 + 1)D Lorentz algebra is revealed. Those associated with non-unitary representations interpolate between bosons and fermions. The extended formulation of the theory includes the previously known Jackiw-Nair (JN) and Majorana-Dirac (MD) descriptions of anyons as particular cases, and allows us to compose bosons and fermions from entangled anyons. The theory admits a simple supersymmetric generalization,more »« less
https://doi.org/10.1016/j.aop.2010.02.007
Physical theories in Galilean space-time and the origin of Schroedinger-like equations

Journal Article Musielak, Z.E. ; Fry, J L - Annals of Physics (New York)

A method to develop physical theories of free particles in space-time with the Galilean metric is presented. The method is based on a Principle of Analyticity and a Principle of Relativity, and uses the Galilei group of the metric. The first principle requires that state functions describing the particles are analytic and the second principle demands that dynamical equations for these functions are Galilean invariant. It is shown that the method can be used to formally derive Schroedinger-like equations and to determine modifications of the Galilei group of the metric that are necessary to fullfil the requirements of analyticity andmore »« less
https://doi.org/10.1016/j.aop.2008.06.006
Construction of Non-Perturbative, Unitary Particle-Antiparticle Amplitudes for Finite Particle Number Scattering Formalisms

Technical Report Lindesay, James V

Starting from a unitary, Lorentz invariant two-particle scattering amplitude, we show how to use an identification and replacement process to construct a unique, unitary particle-antiparticle amplitude. This process differs from conventional on-shell Mandelstam s,t,u crossing in that the input and constructed amplitudes can be off-diagonal and off-energy shell. Further, amplitudes are constructed using the invariant parameters which are appropriate to use as driving terms in the multi-particle, multichannel nonperturbative, cluster decomposable, relativistic scattering equations of the Faddeev-type integral equations recently presented by Alfred, Kwizera, Lindesay and Noyes. It is therefore anticipated that when so employed, the resulting multi-channel solutions willmore »« less
https://doi.org/10.2172/799052

Full Text Available

Similar Records