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Title: Quantum mechanics in fractional and other anomalous spacetimes

Abstract

We formulate quantum mechanics in spacetimes with real-order fractional geometry and more general factorizable measures. In spacetimes where coordinates and momenta span the whole real line, Heisenberg's principle is proven and the wave-functions minimizing the uncertainty are found. In spite of the fact that ordinary time and spatial translations are broken and the dynamics is not unitary, the theory is in one-to-one correspondence with a unitary one, thus allowing us to employ standard tools of analysis. These features are illustrated in the examples of the free particle and the harmonic oscillator. While fractional (and the more general anomalous-spacetime) free models are formally indistinguishable from ordinary ones at the classical level, at the quantum level they differ both in the Hilbert space and for a topological term fixing the classical action in the path integral formulation. Thus, all non-unitarity in fractional quantum dynamics is encoded in a contribution depending only on the initial and final states.

Authors:
 [1];  [2];  [1]
  1. Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Muehlenberg 1, D-14476 Golm (Germany)
  2. Dipartimento di Matematica e Fisica, Universita Cattolica, via Musei 41, 25121 Brescia (Italy)
Publication Date:
OSTI Identifier:
22093755
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 53; Journal Issue: 10; Other Information: (c) 2012 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONFIGURATION; HARMONIC OSCILLATORS; HILBERT SPACE; QUANTUM MECHANICS; SPACE-TIME; UNITARITY; WAVE FUNCTIONS

Citation Formats

Calcagni, Gianluca, Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Nardelli, Giuseppe, INFN Gruppo Collegato di Trento, Universita di Trento, 38100 Povo, Scalisi, Marco, and Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen. Quantum mechanics in fractional and other anomalous spacetimes. United States: N. p., 2012. Web. doi:10.1063/1.4757647.
Calcagni, Gianluca, Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Nardelli, Giuseppe, INFN Gruppo Collegato di Trento, Universita di Trento, 38100 Povo, Scalisi, Marco, & Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen. Quantum mechanics in fractional and other anomalous spacetimes. United States. https://doi.org/10.1063/1.4757647
Calcagni, Gianluca, Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Nardelli, Giuseppe, INFN Gruppo Collegato di Trento, Universita di Trento, 38100 Povo, Scalisi, Marco, and Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen. 2012. "Quantum mechanics in fractional and other anomalous spacetimes". United States. https://doi.org/10.1063/1.4757647.
@article{osti_22093755,
title = {Quantum mechanics in fractional and other anomalous spacetimes},
author = {Calcagni, Gianluca and Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid and Nardelli, Giuseppe and INFN Gruppo Collegato di Trento, Universita di Trento, 38100 Povo and Scalisi, Marco and Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen},
abstractNote = {We formulate quantum mechanics in spacetimes with real-order fractional geometry and more general factorizable measures. In spacetimes where coordinates and momenta span the whole real line, Heisenberg's principle is proven and the wave-functions minimizing the uncertainty are found. In spite of the fact that ordinary time and spatial translations are broken and the dynamics is not unitary, the theory is in one-to-one correspondence with a unitary one, thus allowing us to employ standard tools of analysis. These features are illustrated in the examples of the free particle and the harmonic oscillator. While fractional (and the more general anomalous-spacetime) free models are formally indistinguishable from ordinary ones at the classical level, at the quantum level they differ both in the Hilbert space and for a topological term fixing the classical action in the path integral formulation. Thus, all non-unitarity in fractional quantum dynamics is encoded in a contribution depending only on the initial and final states.},
doi = {10.1063/1.4757647},
url = {https://www.osti.gov/biblio/22093755}, journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 10,
volume = 53,
place = {United States},
year = {Mon Oct 15 00:00:00 EDT 2012},
month = {Mon Oct 15 00:00:00 EDT 2012}
}