We can start ``à la Schrödinger'' by expressing quantum states by means of the amplitude of the connection, namely by means of functionals of the (smooth) connection. These functionals form a linear space, which we promote to a Hilbert space by defining a inner product. To define the inner product, we choose a particular set of states, which we denote ``cylindrical states'' and begin by defining the scalar product between these.
Pick a graph , say with n links, denoted , immersed in the manifold M . For technical reasons, we require the links to be analytic. Let be the parallel transport operator of the connection A along . is an element of SU (2). Pick a function on . The graph and the function f determine a functional of the connection as follows
(These states are called cylindrical states because they were introduced in [14, 15, 16] as cylindrical functions for the definition of a cylindrical measure.) Notice that we can always ``enlarge the graph'', in the sense that if is a subgraph of , we can always write
by simply choosing f ' independent from the 's of the links which are in but not in . Thus, given any two cylindrical functions, we can always view them as having the same graph (formed by the union of the two graphs). Given this observation, we define the scalar product between any two cylindrical functions [137, 14, 15, 16] by
where dg is the Haar measure on SU (2). This scalar product extends by linearity to finite linear combinations of cylindrical functions. It is not difficult to show that (14) defines a well defined scalar product on the space of these linear combinations. Completing the space of these linear combinations in the Hilbert norm, we obtain a Hilbert space . This is the (unconstrained) quantum state space of loop gravity. carries a natural unitary representation of the diffeomorphism group and of the group of the local SU (2) transformations, obtained transforming the argument of the functionals. An important property of the scalar product (14) is that it is invariant under both these transformations.
is non-separable. At first sight, this may seem a serious obstacle to its physical interpretation. But we will see below that, after factoring away diffeomorphism invariance, we may obtain a separable Hilbert space (see section 6.8). Also, standard spectral theory holds on , and it turns out that using spin networks (discussed below) one can express as a direct sum over finite dimensional subspaces which have the structure of Hilbert spaces of spin systems; this makes practical calculations very manageable.
Finally, we will use a Dirac notation and write
in the same manner in which one may write in ordinary quantum mechanics. As in that case, however, we should remember that is not a normalizable state.
|Loop Quantum Gravity
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