dorsal/arxiv
View SchemaPauli's Theorem and Quantum Canonical Pairs: The Consistency Of a Bounded, Self-Adjoint Time Operator Canonically Conjugate to a Hamiltonian with Non-empty Point Spectrum
| Authors | Eric A. Galapon |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9908033 |
| URL | https://arxiv.org/abs/quant-ph/9908033 |
| DOI | 10.1098/rspa.2001.0874 |
| Journal | Proc. R. Soc. Lond. A, 458 (2002) 451-472 |
Abstract
In single Hilbert space, Pauli's well-known theorem implies that the existence of a self-adjoint time operator canonically conjugate to a given Hamiltonian signifies that the time operator and the Hamiltonian possess completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self-adjoint time operator conjugate to a semibounded or discrete Hamiltonian despite some well-known illustrative counterexamples. In this paper we evaluate Pauli's theorem against the single Hilbert space formulation of quantum mechanics, and consequently show the consistency of assuming a bounded, self-adjoint time operator canonically conjugate to a Hamiltonian with an unbounded, or semibounded, or finite point spectrum. We point out Pauli's implicit assumptions and show that they are not consistent in a single Hilbert space. We demonstrate our analysis by giving two explicit examples. Moreover, we clarify issues sorrounding the different solutions to the canonical commutation relations, and, consequently, expand the class of acceptable canonical pairs beyond the solutions required by Pauli's theorem.
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"abstract": "In single Hilbert space, Pauli\u0027s well-known theorem implies that the\nexistence of a self-adjoint time operator canonically conjugate to a given\nHamiltonian signifies that the time operator and the Hamiltonian possess\ncompletely continuous spectra spanning the entire real line. Thus the\nconclusion that there exists no self-adjoint time operator conjugate to a\nsemibounded or discrete Hamiltonian despite some well-known illustrative\ncounterexamples. In this paper we evaluate Pauli\u0027s theorem against the single\nHilbert space formulation of quantum mechanics, and consequently show the\nconsistency of assuming a bounded, self-adjoint time operator canonically\nconjugate to a Hamiltonian with an unbounded, or semibounded, or finite point\nspectrum. We point out Pauli\u0027s implicit assumptions and show that they are not\nconsistent in a single Hilbert space. We demonstrate our analysis by giving two\nexplicit examples. Moreover, we clarify issues sorrounding the different\nsolutions to the canonical commutation relations, and, consequently, expand the\nclass of acceptable canonical pairs beyond the solutions required by Pauli\u0027s\ntheorem.",
"arxiv_id": "quant-ph/9908033",
"authors": [
"Eric A. Galapon"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1098/rspa.2001.0874",
"journal_ref": "Proc. R. Soc. Lond. A, 458 (2002) 451-472",
"title": "Pauli\u0027s Theorem and Quantum Canonical Pairs: The Consistency Of a Bounded, Self-Adjoint Time Operator Canonically Conjugate to a Hamiltonian with Non-empty Point Spectrum",
"url": "https://arxiv.org/abs/quant-ph/9908033"
},
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