The Consistency Of a Bounded, Self-Adjoint Time Operator Canonically Conjugate to a Hamiltonian with Non-empty Point Spectrum

Pauli’s well known theorem (W. Pauli, Hanbuch der Physik vol. 5/1, ed. S. Flugge, (1926) p.60) asserts that the existence of a self-adjoint time operator canonically conjugate to a given Hamiltonian implies that the time operator and the Hamiltonian posses completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self-adjoint time operator conjugate to a Hamiltonian with a spectrum which is a proper subspace of the real line. But we challenge this conclusion. We show rigorously the consistency of assuming a bounded, self-adjoint time operator conjugate to a Hamiltonian with an unbounded, or semibounded, or finite point spectrum. Pauli implicitly assumed unconditionally that the domain of the Hamiltonian is invariant under the action of Uβ = exp(−iβT ) where T is the time operator for arbitrary real number β. But we show that the β’s are at most the differences of the eigenvalues of the Hamiltonian. And this happens, under some other conditions, when the Hamiltonian has a non-empty point spectrum extending from negative to positive infinity. For a Hamiltonian with a semibounded or countable finite point spectrum, we show that no β exists such that the domain of the Hamiltonian is invariant under Uβ . We demonstrate our claim by giving an explicit example.