5.2 One additional assumption 

In choosing the loop algebra as the basis for the quantization, we are essentially assuming that Wilson loop operators are well defined in the Hilbert space of the theory; in other words, that certain states concentrated on one dimensional structures (loops and graphs) have finite norm. This is a subtle non trivial assumption entering the theory. It is the key assumption that characterizes loop gravity. If the approach turned out to be wrong, it will likely be because this assumption is wrong. The Hilbert space resulting from adopting this assumption is not a Fock space. Physically, the assumption corresponds to the idea that quantum states can be decomposed on a basis of ``Faraday lines'' excitations (as Minkowski QFT states can be decomposed on a particle basis).

Furthermore, this is an assumption that fails in conventional quantum field theory, because in that context well defined operators and finite norm states need to be smeared in at least three dimensions, and one-dimensional objects are too singular. The fact that at the basis of loop gravity there is a mathematical assumption that fails for conventional Yang-Mills quantum field theory is probably at the origin of some of the resistance that loop quantum gravity encounters among some high energy theorists. What distinguishes gravity from Yang-Mills theories, however, and makes this assumption viable in gravity, even if it fails for Yang-Mills theory, is diffeomorphism invariance. The loop states are singular states that span a ``huge'' non-separable state space. (Non-perturbative) diffeomorphism invariance plays two roles. First, it wipes away the infinite redundancy. Second, it ``smears'' a loop state into a knot state, so that the physical states are not really concentrated in one dimension, but are, in a sense, smeared all over the entire manifold by the nonperturbative diffeomorphisms. This will be more clear in the next section.

Loop Quantum Gravity Carlo Rovelli http://www.livingreviews.org/lrr-1998-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de