Commutation relations for time and energy?

The problem of extending Hamiltonian mechanics to include a time operator, and to interpret a time-energy uncertainty relation, first posited (without clear formal discussion) in the early days of quantum mechanics, has a large associated literature; a survey article by

carefully reviews the literature up to the year 2000. There is no natural operator solution in a Hilbert space setting: In his treatise Pauli showed in 1958 by a simple argument that a self-adjoint time operator densely defined in a Hilbert space cannot satisfy a canonical commutation relation (CCR) with the Hamiltonian, as the CCR would imply that the Hamiltonian has as spectrum the whole real line, which is unphysical. Instead, time measurements are in practice treated by so-called POVMs (positive operator valued measures) for the time observable modeling properties of the measuring clock.

On the other hand, there is a need for a covariant interpretation of the 4-momentum in quantum gravity (general relativity), so recently there was renewed activity in the field.

and others look at enlarged Hilbert spaces to shed light on the problem of ''time in quantum physics'' (and especially in quantum gravity).

In this spirit there were a number of papers by Horwitz and Piron, most prominently

They use as single-particle state space the Hilbert space of square integrable functions on space-time - rather than on space, which is the traditional approach that leads to the extremely precise prediction of standard QED.
The problem with the Horwitz-Piron approach is that it has far too many states and far too many observables. The basic fields depend (in momentum space coordinates) on arbitrary 4-vectors p rather than only on time/lightlike vectors on one or several mass shells. As a consequence, their off-shell quantum electrodynamics doesn't reproduce the standard results of QED, and hence is irrelevant for the applications.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ