The second reason is that turns out to have a natural basis labeled by knots. More precisely by ``s-knots''. An s-knot s is an equivalence class of spin networks S under diffeomorphisms. An s-knot is characterized by its ``abstract'' graph (defined only by the adjacency relations between links and nodes), by the coloring, and by its knotting and linking properties, as in knot-theory. Thus, the physical quantum states of the gravitational field turn out to be essentially classified by knot theory.
There are various equivalent ways of obtaining from . One can use regularization techniques for defining the quantum operator corresponding to the classical diffeomorphism constraint in terms of elementary loop operators, and then find the kernel of such operator. Equivalently, one can factor by the natural action of the diffeomorphism group that it carries. Namely
There are several rigorous ways for defining the quotient of a Hilbert space by the unitary action of a group. See in particular the construction in , which follows the ideas of Marolf and Higuchi [145, 147, 148, 108].
In the quantum gravity literature, a big deal has been made of the problem that a scalar product is not defined on the space of solutions of a constraint , defined on a Hilbert space . This, however, is a false problem. It is true that if zero is in the continuum spectrum of , then the corresponding eigenstates are generalized states and the scalar product is not defined between them. But the generalized eigenspaces of , including the kernel, inherit nevertheless a scalar product from . This can be seen in a variety of equivalent ways. For instance, it can be seen from the following theorem. If is self adjoint, then there exist a measure on its spectrum and a family of Hilbert spaces such that
where the integral is the continuous sum of Hilbert spaces described, for instance, in . Clearly is the kernel of equipped with a scalar product. This is discussed, for instance, in .
There are two distinct possibilities for factoring away the diffeomorphisms in the quantum theory, yielding two distinct version of the theory. The first possibility is to factor away smooth transformations of the manifold. In doing so, finite dimensional moduli spaces associated with high valence nodes appear , so that the resulting Hilbert space is still non-separable. The physical relevance of these moduli parameters is unclear at this stage, since they do not seem to play any role in the quantum theory. Alternatively, one can consistently factor away continuous transformations of the manifold. This possibility has been explored by Zapata in [215, 216], and seems to lead to a consistent theory free of the residual non separability.
|Loop Quantum Gravity
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