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Time in Quantum Physics: From an External Parameter to an Intrinsic Observable

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Abstract

In the Schrödinger equation, time plays a special role as an external parameter. We show that in an enlarged system where the time variable denotes an additional degree of freedom, solutions of the Schrödinger equation give rise to weights on the enlarged algebra of observables. States in the associated GNS representation correspond to states on the original algebra composed with a completely positive unit preserving map. Application of this map to the functions of the time operator on the large system delivers the positive operator valued maps which were previously proposed by two of us as time observables. As an example we discuss the application of this formalism to the Wheeler-DeWitt theory of a scalar field on a Robertson-Walker spacetime.

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References

  1. Doplicher, S., Fredenhagen, K., Roberts, J.E.: The quantum structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. 172, 187 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Grosse, H., Wulkenhaar, R.: Renormalisation of φ 4-theory on noncommutative R 4 to all orders. arXiv:hep-th/0403232v1 (2004)

  3. Rovelli, C.: Partial observables. Phys. Rev. D 65, 124013 (2002)

    MathSciNet  ADS  Google Scholar 

  4. Dittrich, B.: Partial and complete observables for canonical general relativity. Class. Quantum Gravity 23, 6155 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Thiemann, T.: Reduced phase space quantization and Dirac observables. Class. Quantum Gravity 23, 1163 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Ashtekar, A., Pawlowski, T., Singh, P.: Quantum nature of the big bang: an analytical and numerical investigation. Phys. Rev. D 73, 124038 (2006)

    MathSciNet  ADS  Google Scholar 

  7. Brunetti, R., Fredenhagen, K.: Time of occurrence observable in quantum mechanics. Phys. Rev. A 66, 044101 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  8. Buchholz, D., Porrmann, M., Stein, U.: Dirac versus Wigner. Towards a universal particle concept in local quantum field theory. Phys. Lett. B 267, 377 (1991)

    MathSciNet  ADS  Google Scholar 

  9. Porrmann, M.: Particle weights and their disintegration. I. Commun. Math. Phys. 248, 269 (2004)

    MATH  MathSciNet  ADS  Google Scholar 

  10. Porrmann, M.: Particle weights and their disintegration. II. Phys. Lett. B 248, 305 (2004)

    MATH  MathSciNet  Google Scholar 

  11. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics 1: C*- and W*-algebras. Symmetry groups. Decomposition of states. In: Texts and Monographs in Physics, 2nd edn. Springer, Berlin (2002)

  12. Gambini, R., Porto, R.A., Pullin, J., Torterolo, S.: Conditional probabilities with Dirac observables and the problem of time in quantum gravity. Phys. Rev. D 79, 041501(R) (2009)

    MathSciNet  ADS  Google Scholar 

  13. Aharonov, Y., Bohm, D.: Time in the quantum theory and the uncertainty relation for time and energy. Phys. Rev. 122, 1649 (1961)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Busch, P.: The time-energy uncertainty relation. In: Muga, J.G., Sala Mayato, R., Egusquiza, I.L. (eds.) Time in Quantum Mechanics, pp. 69–98, 2nd rev. edn. Springer, Berlin (2002). 2007, and related papers therein

    Chapter  Google Scholar 

  15. Ashtekar, A., Kaminski, W., Lewandowski, J.: Quantum field theory on a cosmological, quantum space-time. arXiv:0901.0933 [gr-qc] (2009)

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123, Povo, TN, Italy

    Romeo Brunetti

  2. II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761, Hamburg, Germany

    Klaus Fredenhagen & Marc Hoge

Authors
  1. Romeo Brunetti
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  2. Klaus Fredenhagen
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  3. Marc Hoge
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Correspondence to Klaus Fredenhagen.

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Brunetti, R., Fredenhagen, K. & Hoge, M. Time in Quantum Physics: From an External Parameter to an Intrinsic Observable. Found Phys 40, 1368–1378 (2010). https://doi.org/10.1007/s10701-009-9400-z

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  • DOI: https://doi.org/10.1007/s10701-009-9400-z

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