Pauli's seven decade myth: The consistency of a bounded, self-adjoint time operator canonically conjugate to a Hamiltonian with non-empty point spectrum

Abstract

Despite the immense success of the standard single Hilbert space formulation of quantum mechanics, there still remains a gaping dissatisfaction due to its difficulty in accommodating some nontrivial problems such as the role of time in the quantum domain. In standard quantum mechanics, time stands as a mere parameter, an external variable independent from the dynamics of any given system. But time undoubtedly acquires dynamical significance in questions involving the occurrence of an event, e.g. when does a nucleon decay, or when does a particle arrive at a given spatial point, or how long a particle tunnels through a barrier. Furthermore, the time-energy uncertainty principle requires more than a parametric treatment of time. Naïve in keeping with standard quantum mechanics requires a self-adjoint time operator conjugate to the Hamiltonian for each of the above mentioned examples. However with the Hamiltonian generally possessing a semi-bounded, pure point spectrum, finding self-adjoint time operators has been tagged impossible to achieve. The culprit is the theorem due to Pauli, which asserts that there exists no self-adjoint time operator canonically conjugate to a semi-bounded Hamiltonian: The existence of a self-adjoint time operator implies that the time operator and the Hamiltonian have completely continuous spectra spanning the entire real line. The embargo imposed by Pauli's theorem has led to diverse treatments of time within and beyond the usual formulation and interpretation of quantum mechanics. It is the objective of this paper to critically evaluate the validity of the objections raised by Pauli. Specifically we will show the consistency of assuming the existence of a self-adjoint, bounded time operator canonically conjugate to a Hamiltonian with a non-empty unbounded, semibounded, or finitely countable point spectrum, which is in direct opposition to the claims of Pauli.