Covariant canonical quantization and the problem of time

Abstract

In the covariant canonical approach to classical physics, each point in phase space represents an entire classical trajectory. Initial data at a fixed time serve as coordinates for this ``timeless'' phase space, and time evolution can be viewed as a coordinate change. We argue for a similar view in quantum theory. As in the Heisenberg picture, the wave function is fundamentally time-independent. On any given time slice, however, we can diagonalize a complete set of position operators to form a basis, in which the projected wave function depends on the choice of time. In this picture, time evolution can be viewed as a basis change in what is otherwise a block universe. We argue that this may help solve the ``problem of time'' in quantum gravity, and illustrate the idea with an example from three-dimensional quantum gravity.

contribution to the book "Time and timelessness in fundamental physics and cosmology," edited by Silvia De Bianchi, Marco Forgione and Laura Marongiu , to be published by Springer; v2: eqn (3.1) corrected (commutators in covariant phase space), no change in conclusions