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Time in Quantum Mechanics - Vol. 2 pp 25–63Cite as

Post Pauli’s Theorem Emerging Perspective on Time in Quantum Mechanics

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Part of the book series: Lecture Notes in Physics ((LNP,volume 789))

Abstract

In a Hilbert space setting, Pauli’s well-known theorem Pauli’s theorem asserts that no self-adjoint operator exists that is conjugate to a semibounded or discrete Hamiltonian [58]. Pauli’s argument goes as follows. Assume that there exists a self-adjoint operator T conjugate to a given Hamiltonian H, that is, [T,H]=iћ I such an operator conjugate to the Hamiltonian is known as a time operator. Since T is self-adjoint, the operator U ε=exp(–iεT) is unitary for all real number ε. Now if φE is an eigenvector of H with the eigenvalue E, then, according to Pauli, the conjugacy relation [T,H]=iћ I implies that T is a generator of energy shifts so that (E+ε)φE+e; this means that H has a continuous spectrum spanning the entire real line because ε is an arbitrary real number. Hence, the ‘inevitable’ conclusion that if the Hamiltonian is semibounded or discrete no self-adjoint time operator T will exist

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Acknowledgments

The works reported here were funded by the National Research Council of the Philippines through Grant No. I-81-NRCP, the UP Office of the Vice Chancellor for Research andDevelopment through UP-OVCRD Outright Research Grants PNSE-050509 and PNSE-070703,and by the Office of the Vice Chancellor for Academic Affairs through UP System Grants 2004 and 2007. The theory of confined time of arrival operators was developed with the help of R. Caballar, R. Bahague, R. Vitancol, and A. Villanueva. The idea of taking the limit of the confining length to infinity arose out of a collaboration with J.G. Muga, I. Egusquiza, and F. Delgado. The author especially acknowledges J.G. Muga for his encouragements that have become a major motivation for the author's personal crusade to understand time in quantum mechanics.

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Authors and Affiliations

  1. Theoretical Physics Group, National Institute of Physics, University of the Philippines, Diliman, Quezon City, Philippines

    Eric A. Galapon

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  1. Eric A. Galapon
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Correspondence to Eric A. Galapon .

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Editors and Affiliations

  1. Depto. Quimica Fisica EHU, Universidad Pais Vasco, Apartado 644, Bilbao, 48080, Spain

    Gonzalo Muga

  2. Inst. Mathematische Physik, TU Braunschweig, Mendelssohnstr. 3, Braunschweig, 38106, Germany

    Andreas Ruschhaupt

  3. Inst. Mathematical Sciences, Imperial College London, Prince's Gate 53, London, SW7 2PG, UK

    Adolfo del Campo

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Galapon, E.A. (2009). Post Pauli’s Theorem Emerging Perspective on Time in Quantum Mechanics. In: Muga, G., Ruschhaupt, A., del Campo, A. (eds) Time in Quantum Mechanics - Vol. 2. Lecture Notes in Physics, vol 789. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03174-8_3

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  • DOI: https://doi.org/10.1007/978-3-642-03174-8_3

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