Empirical adequacy of the time operator canonically conjugate to a Hamiltonian generating translations

Abstract

Abstract To admit a canonically conjugate time operator, the Hamiltonian has to be a generator of translations (like the momentum operator generates translations in space), so its spectrum must be unbounded. But the Hamiltonian governing our world is thought to be bounded from below. Also, judging by the number of fields and parameters of the Standard Model, the Hamiltonian seems much more complicated. In this article I give examples of worlds governed by Hamiltonians generating translations. They can be expressed as a partial derivative operator just like the momentum operator, but when expressed in function of other observables they can exhibit any level of complexity. The examples include any quantum world realizing a standard ideal measurement, any quantum world containing a clock or a free massless fermion, the quantum representation of any deterministic time-reversible dynamical system without time loops, and any quantum world that cannot return to a past state. Such worlds are as sophisticated as our world, but they admit a time operator. I show that, despite having unbounded Hamiltonian, they do not decay to infinite negative energy any more than any quantum or classical world. Since two such quantum systems of the same Hilbert space dimension are unitarily equivalent even if the physical content of their observables is very different, they are concrete counterexamples to Hilbert Space Fundamentalism (HSF). Taking the observables into account removes the ambiguity of HSF and the clock ambiguity problem attributed to the Page-Wootters formalism, also caused by assuming HSF. These results provide additional motivations to restore the spacetime symmetry in the formulation of Quantum Mechanics and for the Page-Wootters formalism.