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Title: Canonical quantization of Galilean covariant field theories

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:
 [1];  [2];  [3]
  1. Theoretical Physics Institute, University of Alberta, Edmonton, Alta., T6G 2J1 (Canada)
  2. Theoretical Physics Institute, University of Alberta, Edmonton, Alta., T6G 2J1 (Canada) and Faculte Saint-Jean, University of Alberta, Edmonton, Alta., T6C 4G9 (Canada)
  3. Theoretical Physics Institute, University of Alberta, Edmonton, Alta., T6G 2J1 (Canada) and TRIUMF, 4004, Wesbrook Mall, Vancouver, BC, V6T 2A3 (Canada)
Publication Date:
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.
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
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.
@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}
}