Pauli'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

摘要

In single Hilbert space, Pauli's well-known theorem implies that theexistence of a self-adjoint time operator canonically conjugate to a givenHamiltonian signifies that the time operator and the Hamiltonian possesscompletely continuous spectra spanning the entire real line. Thus theconclusion that there exists no self-adjoint time operator conjugate to asemibounded or discrete Hamiltonian despite some well-known illustrativecounterexamples. In this paper we evaluate Pauli's theorem against the singleHilbert space formulation of quantum mechanics, and consequently show theconsistency of assuming a bounded, self-adjoint time operator canonicallyconjugate to a Hamiltonian with an unbounded, or semibounded, or finite pointspectrum. We point out Pauli's implicit assumptions and show that they are notconsistent in a single Hilbert space. We demonstrate our analysis by giving twoexplicit examples. Moreover, we clarify issues sorrounding the differentsolutions to the canonical commutation relations, and, consequently, expand theclass of acceptable canonical pairs beyond the solutions required by Pauli'stheorem.