Quantum mechanics of a spin particle in a curved spacetime with absolute time
type: Item
instance of: scholarly article
title: Quantum mechanics of a spin particle in a curved spacetime with absolute time
zbMATH Open document ID: 0888.53052
DOI: 10.1016/0034-4877(96)82487-6
author: Daniel Canarutto
author: Marco Modugno
author: Arkadiusz Jadczyk
published in: Reports on Mathematical Physics
publication date: 1998-04-02T00:00:00Z
review text: The authors extend the geometric formulation of nonrelativistic charged scalar particles given by two of them [\textit{A. Jadczyk} and \textit{M. Modugno}, Proc. XXI Internat. Conf. Differ. Geom. Methods in Theoretical Physics, Tianjin 1992, 543-556, World Scientific, Singapore (1992; per bibl.)] to particles with spin. The spacetime is a four-dimensional manifold with a Galileian structure and Newtonian connection. It is described here as a fibre bundle \(E\) over the one-dimensional time manifold, the fibres being equipped with a Riemannian metric. The connection is metric compatible and given on the first jet bundle \(J^1E\). It is determined by the metric on the fibres and a cosymplectic (or Poisson) form \(\Omega\) on the spacetime \(E\). For a scalar particle a quantum bundle \(Q\) is then a Hermitian line bundle over spacetime (rather than over phase space as in geometric quantization) with a connection whose curvature is the cosymplectic form on spacetime. For particles with spin the quantum bundle is replaced by a tensor product of \(Q\) and a spinor bundle (of complex dimension 2). Analogues of the Schrödinger and Schrödinger-Pauli equations are then derived. Since these quantum bundles are over spacetime, a polarization is not needed in this approach, but the class of quantizable functions is nevertheless restricted, essentially to quadratic functions in the velocity. While most constructions involve an observer field explicitly, it is verified that the main physical content of the equations is invariant under changes of the observer. For the flat Galilein spacetime, the standard nonrelativistic quantum mechanics is recovered.
Mathematics Subject Classification ID: 53Z05
Mathematics Subject Classification ID: 81R25
Mathematics Subject Classification ID: 81T20
Mathematics Subject Classification ID: 83C60
zbMATH DE Number: 1087268
zbMATH Keywords: quantization
zbMATH Keywords: spin
zbMATH Keywords: Galileian space-time
zbMATH Keywords: jet bundles
zbMATH Keywords: spin connections
MaRDI profile type: Publication
full work available at URL: 0034-4877(96)82487-6
OpenAlex ID: W2150487474
cites work: Q3805391
cites work: Q4721276
cites work: Q3963734
cites work: Connections and 1-jet fiber bundles
cites work: Q4048996
cites work: Q5562497
cites work: Multilinear algebra. 2nd ed
cites work: Four-Dimensional Formulations of Newtonian Mechanics and Their Relation to the Special and the General Theory of Relativity
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Q4256628
cites work: Q4856853
cites work: Q4486549
cites work: Q3706139
cites work: Covariant Newtonian limit of Lorentz space-times
cites work: Q4043261
cites work: Q5521405
cites work: Q3326383
cites work: Q3668376
cites work: Q3968481
cites work: Torsion and Ricci tensor for nonlinear connections
cites work: Q3284485
cites work: Q3324593
cites work: Quantum geometry. A framework for quantum general relativity
cites work: Q3231026
cites work: Q5724561
cites work: An intrinsic formulation of nonrelativistic analytical mechanics and wave mechanics
cites work: Q4003058
Recommended article: Quantum mechanics in curved space-time
Recommended article: The phase of a quantum mechanical particle in curved spacetime
Recommended article: Quantum kinematics of a test particle in a curved spacetime
Recommended article: Relativistic wave mechanics of spinless particles in a curved space-time
Recommended article: Q4895885
Recommended article: Spin decoherence by spacetime curvature
Recommended article: Covariant Hamiltonian spin dynamics in curved space-time
Recommended article: Q4394719
Recommended article: Q5315682
Recommended article: Some remarks on quantum mechanics in a curved spacetime, especially for a Dirac particle
description: scientific article; zbMATH DE number 1087268