6.3 Loop states and spin 6 The Formalism6.1 Loop algebra

6.2 Loop quantum gravity 

The kinematics of a quantum theory is defined by an algebra of ``elementary'' operators (such as x and tex2html_wrap_inline2642, or creation and annihilation operators) on a Hilbert space tex2html_wrap_inline2484 . The physical interpretation of the theory is based on the connection between these operators and classical variables, and on the interpretation of tex2html_wrap_inline2484 as the space of the quantum states. The dynamics is governed by a hamiltonian, or, as in general relativity, by a set of quantum constraints, constructed in terms of the elementary operators. To assure that the quantum Heisenberg equations have the correct classical limit, the algebra of the elementary operator has to be isomorphic to the Poisson algebra of the elementary observables. This yields the heuristic quantization rule: ``promote Poisson brackets to commutators''. In other words, define the quantum theory as a linear representation of the Poisson algebra formed by the elementary observables. For the reasons illustrated in section 5, the algebra of elementary observables we choose for the quantization is the loop algebra, defined in section 6.1 . Thus, the kinematic of the quantum theory is defined by a unitary representation of the loop algebra. Here, I construct such representation following a simple path.

We can start ``à la Schrödinger'' by expressing quantum states by means of the amplitude of the connection, namely by means of functionals tex2html_wrap_inline2648 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 tex2html_wrap_inline2650, say with n links, denoted tex2html_wrap_inline2654, immersed in the manifold M . For technical reasons, we require the links to be analytic. Let tex2html_wrap_inline2658 be the parallel transport operator of the connection A along tex2html_wrap_inline2662 . tex2html_wrap_inline2664 is an element of SU (2). Pick a function tex2html_wrap_inline2668 on tex2html_wrap_inline2670 . The graph tex2html_wrap_inline2650 and the function f determine a functional of the connection as follows


(These states are called cylindrical states because they were introduced in [14Jump To The Next Citation Point In The Article, 15Jump To The Next Citation Point In The Article, 16Jump To The Next Citation Point In The Article] as cylindrical functions for the definition of a cylindrical measure.) Notice that we can always ``enlarge the graph'', in the sense that if tex2html_wrap_inline2650 is a subgraph of tex2html_wrap_inline2678, we can always write


by simply choosing f ' independent from the tex2html_wrap_inline2682 's of the links which are in tex2html_wrap_inline2678 but not in tex2html_wrap_inline2650 . 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, 14Jump To The Next Citation Point In The Article, 15Jump To The Next Citation Point In The Article, 16Jump To The Next Citation Point In The Article] 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 (14Popup Equation) 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 tex2html_wrap_inline2484 . This is the (unconstrained) quantum state space of loop gravity. Popup Footnote tex2html_wrap_inline2484 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 (14Popup Equation) is that it is invariant under both these transformations.

tex2html_wrap_inline2484 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 tex2html_wrap_inline2484, and it turns out that using spin networks (discussed below) one can express tex2html_wrap_inline2484 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 tex2html_wrap_inline2722 in ordinary quantum mechanics. As in that case, however, we should remember that tex2html_wrap_inline2724 is not a normalizable state.

6.3 Loop states and spin 6 The Formalism6.1 Loop algebra

image Loop Quantum Gravity
Carlo Rovelli
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de