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

  • Eric A. GalaponEmail author
Chapter
Part of the Lecture Notes in Physics book series (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]=iћI such an operator conjugate to the Hamiltonian is known as a time operator. Since T is self-adjoint, the operator Uε=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]=iћI 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

Keywords

Hilbert Space Arrival Distribution Canonical Commutation Relation Standard Quantum Mechanic Canonical Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Theoretical Physics Group, National Institute of PhysicsUniversity of the PhilippinesQuezon CityPhilippines

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