Quantum Mechanics and Spacetime

I was rather harsh with James Hartle about the colloqium he gave the other day. As a rule, i'm not picky but i'm more demanding of celebrities. Last week, James also gave a theory seminar that i didn't like either. However, this time i'm going to hold my temper and just report the story told.

The title was Generalizing quantum mechanics for quantum spacetime. The usual formulation of quantum mechanics, with the Schrödinger equation and the unitary evolution of states, is based on predefined time and space. James was discussing a framework that is suitable for a system with no fixed spacetime geometry. One obvious motivation for such a generalization is the application to the early universe.

The framework discussed in the seminar derives from the quantum mechanics of closed systems. This formulation requires the Hamiltonian and the initial quantum state as an input, but does not rely on the notions of classical regime, external observers or measurement. The key object is a set of coarse-grained alternative histories of the particles in a system. These are bundles of fine-grained histories - the Feynman paths of particles. For example, a coarse-grained history could be a position of the center-of-mass of the Earth at a certain accuracy. One defines branch state vectors and the decoherence functional to quantify the quantum interference between alternative histories. Probabilities can be asssigned to sets of histories for which the interference between its members is negligible.

In the presence of a fixed spacetime geometry, in which we can define the timelike direction and the foliation into spacelike surfaces, this formulation can be proven equivalent to the standard one. Basically, this is just the equivalence between the Feynman path integral and the usual formulation of quantum mechanics. James argued that this formulation can accommodate quantum spacetime too. In this case, the alternative histories do not represent evolution in spacetime, but evolution of spacetime. The key object would be a set of coarse-grained histories of a metric and matter fields on a fixed manifold.

Although all this sounds plausible, not much more than words words words were presented during the seminar. There was not the slightest mention of a possible experimental verification of these ideas. Another worry is that this formulation, by itself, does not address the divergences problem of quantum gravity. In all known cases where this is taken care of, e.g. in string theory, the spacetime description breaks down at some energy scale. It is not clear how the alternative histories formulation could fit in such a picture.

The transparencies, of course, are not available (why oh why?). Luckily enough, James has this article on the archive that pretty well covers the material presented in the seminar.