Multipleevent probability in generalrelativistic quantum mechanics
Abstract
We discuss the definition of quantum probability in the context of 'timeless' generalrelativistic quantum mechanics. In particular, we study the probability of sequences of events, or multievent probability. In conventional quantum mechanics this can be obtained by means of the 'wave function collapse' algorithm. We first point out certain difficulties of some natural definitions of multievent probability, including the conditional probability widely considered in the literature. We then observe that multievent probability can be reduced to singleevent probability, by taking into account the quantum nature of the measuring apparatus. In fact, by exploiting the vonNeumann freedom of moving the quantum/classical boundary, one can always trade a sequence of noncommuting quantum measurements at different times, with an ensemble of simultaneous commuting measurements on the joint system+apparatus system. This observation permits a formulation of quantum theory based only on singleevent probability, where the results of the wave function collapse algorithm can nevertheless be recovered. The discussion also bears on the nature of the quantum collapse.
 Authors:
 Fakultaet fuer Physik, LudwigMaximiliansUniversitaet, D80799 Munich (Germany)
 (France)
 Centre de Physique Theorique de Luminy, Universite de la Mediterranee, F13288 Marseille (France)
 Publication Date:
 OSTI Identifier:
 21020402
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 8; Other Information: DOI: 10.1103/PhysRevD.75.084033; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ALGORITHMS; COSMOLOGY; GENERAL RELATIVITY THEORY; PROBABILITY; QUANTUM FIELD THEORY; QUANTUM MECHANICS; RELATIVISTIC RANGE; WAVE FUNCTIONS
Citation Formats
Hellmann, Frank, Centre de Physique Theorique de Luminy, Universite de la Mediterranee, F13288 Marseille, Mondragon, Mauricio, Perez, Alejandro, and Rovelli, Carlo. Multipleevent probability in generalrelativistic quantum mechanics. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.084033.
Hellmann, Frank, Centre de Physique Theorique de Luminy, Universite de la Mediterranee, F13288 Marseille, Mondragon, Mauricio, Perez, Alejandro, & Rovelli, Carlo. Multipleevent probability in generalrelativistic quantum mechanics. United States. doi:10.1103/PHYSREVD.75.084033.
Hellmann, Frank, Centre de Physique Theorique de Luminy, Universite de la Mediterranee, F13288 Marseille, Mondragon, Mauricio, Perez, Alejandro, and Rovelli, Carlo. Sun .
"Multipleevent probability in generalrelativistic quantum mechanics". United States.
doi:10.1103/PHYSREVD.75.084033.
@article{osti_21020402,
title = {Multipleevent probability in generalrelativistic quantum mechanics},
author = {Hellmann, Frank and Centre de Physique Theorique de Luminy, Universite de la Mediterranee, F13288 Marseille and Mondragon, Mauricio and Perez, Alejandro and Rovelli, Carlo},
abstractNote = {We discuss the definition of quantum probability in the context of 'timeless' generalrelativistic quantum mechanics. In particular, we study the probability of sequences of events, or multievent probability. In conventional quantum mechanics this can be obtained by means of the 'wave function collapse' algorithm. We first point out certain difficulties of some natural definitions of multievent probability, including the conditional probability widely considered in the literature. We then observe that multievent probability can be reduced to singleevent probability, by taking into account the quantum nature of the measuring apparatus. In fact, by exploiting the vonNeumann freedom of moving the quantum/classical boundary, one can always trade a sequence of noncommuting quantum measurements at different times, with an ensemble of simultaneous commuting measurements on the joint system+apparatus system. This observation permits a formulation of quantum theory based only on singleevent probability, where the results of the wave function collapse algorithm can nevertheless be recovered. The discussion also bears on the nature of the quantum collapse.},
doi = {10.1103/PHYSREVD.75.084033},
journal = {Physical Review. D, Particles Fields},
number = 8,
volume = 75,
place = {United States},
year = {Sun Apr 15 00:00:00 EDT 2007},
month = {Sun Apr 15 00:00:00 EDT 2007}
}

We introduce a simple quantum mechanical model in which time and space are discrete and periodic. These features avoid the complications related to continuousspectrum operators and infinitenorm states. The model provides a tool for discussing the probabilistic interpretation of generally covariant quantum systems, without the confusion generated by spurious infinities. We use the model to illustrate the formalism of generalrelativistic quantum mechanics, and to test the definition of multipleevent probability introduced in a companion paper [Phys. Rev. D 75, 084033 (2007)]. We consider a version of the model with unitary time evolution and a version without unitary time evolution.

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