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
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