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
Die allgemeinen Prinzipien der Wellenmechanik,
Handbuch der Physik (S. Fluegge, ed.), Vol V/1, p. 60,
Springer, Berlin 1958.
The general principles of quantum mechanics, p. 63,
Springer, Berlin 1980.
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
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
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
Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ