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Title: Quantum mechanics in non-inertial reference frames: Time-dependent rotations and loop prolongations

Abstract

This is the fourth in a series of papers on developing a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group to include transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. In previous work, we have shown that there exist representations of the Galilean line group that uphold the non-relativistic equivalence principle as well as representations that violate the equivalence principle. In these previous studies, the focus was on linear accelerations. In this paper, we undertake an extension of the formulation to include rotational accelerations. We show that the incorporation of rotational accelerations requires a class of loop prolongations of the Galilean line group and their unitary cocycle representations. We recover the centrifugal and Coriolis force effects from these loop representations. Loops are more general than groups in that their multiplication law need not be associative. Hence, our broad theoretical claim is that a Galilean quantum theory that holds in arbitrary non-inertial referencemore » frames requires going beyond groups and group representations, the well-established framework for implementing symmetry transformations in quantum mechanics. -- Highlights: •A formulation of Galilean quantum mechanics in non-inertial reference frames is presented. •The Galilei group is generalized to infinite dimensional Galilean line group. •Loop prolongations of Galilean line group contain central extensions of Galilei group. •Unitary representations of the loops are constructed. •These representations lead to terms in the Hamiltonian corresponding to fictitious forces, including centrifugal and Coriolis forces.« less

Authors:
 [1];  [1];  [2]
  1. Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242 (United States)
  2. (United States)
Publication Date:
OSTI Identifier:
22220781
Resource Type:
Journal Article
Journal Name:
Annals of Physics (New York)
Additional Journal Information:
Journal Volume: 336; Other Information: Copyright (c) 2013 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; ACCELERATION; CORIOLIS FORCE; EQUIVALENCE PRINCIPLE; HAMILTONIANS; LIE GROUPS; MAGNETIC FIELDS; QUANTUM MECHANICS; RELATIVISTIC RANGE; ROTATION; TIME DEPENDENCE; TRANSFORMATIONS

Citation Formats

Klink, W.H., E-mail: william-klink@uiowa.edu, Wickramasekara, S., E-mail: wickrama@grinnell.edu, and Department of Physics, Grinnell College, Grinnell, IA 50112. Quantum mechanics in non-inertial reference frames: Time-dependent rotations and loop prolongations. United States: N. p., 2013. Web. doi:10.1016/J.AOP.2013.06.004.
Klink, W.H., E-mail: william-klink@uiowa.edu, Wickramasekara, S., E-mail: wickrama@grinnell.edu, & Department of Physics, Grinnell College, Grinnell, IA 50112. Quantum mechanics in non-inertial reference frames: Time-dependent rotations and loop prolongations. United States. https://doi.org/10.1016/J.AOP.2013.06.004
Klink, W.H., E-mail: william-klink@uiowa.edu, Wickramasekara, S., E-mail: wickrama@grinnell.edu, and Department of Physics, Grinnell College, Grinnell, IA 50112. 2013. "Quantum mechanics in non-inertial reference frames: Time-dependent rotations and loop prolongations". United States. https://doi.org/10.1016/J.AOP.2013.06.004.
@article{osti_22220781,
title = {Quantum mechanics in non-inertial reference frames: Time-dependent rotations and loop prolongations},
author = {Klink, W.H., E-mail: william-klink@uiowa.edu and Wickramasekara, S., E-mail: wickrama@grinnell.edu and Department of Physics, Grinnell College, Grinnell, IA 50112},
abstractNote = {This is the fourth in a series of papers on developing a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group to include transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. In previous work, we have shown that there exist representations of the Galilean line group that uphold the non-relativistic equivalence principle as well as representations that violate the equivalence principle. In these previous studies, the focus was on linear accelerations. In this paper, we undertake an extension of the formulation to include rotational accelerations. We show that the incorporation of rotational accelerations requires a class of loop prolongations of the Galilean line group and their unitary cocycle representations. We recover the centrifugal and Coriolis force effects from these loop representations. Loops are more general than groups in that their multiplication law need not be associative. Hence, our broad theoretical claim is that a Galilean quantum theory that holds in arbitrary non-inertial reference frames requires going beyond groups and group representations, the well-established framework for implementing symmetry transformations in quantum mechanics. -- Highlights: •A formulation of Galilean quantum mechanics in non-inertial reference frames is presented. •The Galilei group is generalized to infinite dimensional Galilean line group. •Loop prolongations of Galilean line group contain central extensions of Galilei group. •Unitary representations of the loops are constructed. •These representations lead to terms in the Hamiltonian corresponding to fictitious forces, including centrifugal and Coriolis forces.},
doi = {10.1016/J.AOP.2013.06.004},
url = {https://www.osti.gov/biblio/22220781}, journal = {Annals of Physics (New York)},
issn = {0003-4916},
number = ,
volume = 336,
place = {United States},
year = {Sun Sep 15 00:00:00 EDT 2013},
month = {Sun Sep 15 00:00:00 EDT 2013}
}