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3.3 Representation

The key new aspect now is that we can choose the space of Ashtekar connections as our configuration space, whose structure is much better understood than that of a space of metrics. Moreover, the formulation lends itself easily to background-independent quantization. To see this we need to remember that quantizing field theories requires one to smear fields, i.e., to integrate them over regions in order to obtain a well-defined algebra without δ-functions, as in Equation (3View Equation). This is usually done by integrating both configuration and momentum variables over three-dimensional regions, which requires an integration measure. This is no problem in ordinary field theories, which are formulated on a background such as Minkowski or a curved space. However, doing this here for gravity in terms of Ashtekar variables would immediately spoil any possible background independence since a background would already occur at this very basic step.

There is now a different smearing available that does not require a background metric. Instead of using three-dimensional regions we integrate the connection along one-dimensional curves e and exponentiate in a path-ordered manner, resulting in holonomies

∫ he(A ) = 𝒫 exp τiAie˙a dt (8 ) e a
with tangent vector a ˙e to the curve e and 1 τj = − 2iσj in terms of Pauli matrices. The path-ordered exponentiation needs to be done in order to obtain a covariant object from the non-Abelian connection. The prevalence of holonomies or, in their most simple gauge-invariant form as Wilson loops trhe(A ) for closed e, is the origin of loop quantum gravity and its name [258]. Similarly, densitized vector fields can naturally be integrated over two-dimensional surfaces, resulting in fluxes
∫ i a 2 FS(E ) = τ Ei na d y (9 ) S
with the co-normal na to the surface.

The Poisson algebra of holonomies and fluxes is now well defined and one can look for representations on a Hilbert space. We also require diffeomorphism invariance, i.e., there must be a unitary action of the diffeomorphism group on the representation by moving edges and surfaces in space. This is required since the diffeomorphism constraint has to be imposed later. Under this condition, together with the assumption of cyclicity of the representation, there is even a unique representation that defines the kinematical Hilbert space [263264245265266166213]. (See [301], however, for examples of other, non-cyclic representations.)

We can construct the Hilbert space in the representation where states are functionals of connections. This can easily be done by using holonomies as “creation operators” starting with a “ground state”, which does not depend on connections at all. Multiplying with holonomies then generates states that do depend on connections but only along the edges used in the process. These edges can be collected in a graph appearing as a label of the state. An independent set of states is given by spin-network states [261] associated with graphs, whose edges are labeled by irreducible representations of the gauge group SU(2), in which to evaluate the edge holonomies, and whose vertices are labeled by matrices specifying how holonomies leaving or entering the vertex are multiplied together. The inner product on this state space is such that these states, with an appropriate definition of independent contraction matrices in vertices, are orthonormal.

Spatial geometry can be obtained from fluxes representing the densitized triad. Since these are now momenta, they are represented by derivative operators with respect to values of connections on the flux surface. States, as constructed above, depend on the connection only along edges of graphs, such that the flux operator is non-zero only if there are intersection points between its surface and the graph in the state it acts on [212]. Moreover, the contribution from each intersection point can be seen to be analogous to an angular momentum operator in quantum mechanics, which has a discrete spectrum [21Jump To The Next Citation Point]. Thus, when acting on a given state, we obtain a finite sum of discrete contributions and thus a discrete spectrum of flux operators. The spectrum depends on the value of the Barbero–Immirzi parameter, which can accordingly be fixed using implications of the spectrum such as black-hole entropy, which gives a value on the order of, but smaller than, one [11Jump To The Next Citation Point12Jump To The Next Citation Point158Jump To The Next Citation Point225Jump To The Next Citation Point]. Moreover, since angular-momentum operators do not commute, flux operators do not commute in general [18]. There is, thus, no triad representation, which is another reason that using a metric formulation and trying to build its quantization with functionals on a metric space is difficult.

There are important basic properties of this representation, which we will use later on. First, as already noted, flux operators have discrete spectra and, secondly, holonomies of connections are well-defined operators. It is, however, not possible to obtain operators for connection components or their integrations directly but only in the exponentiated form. These are the direct consequence of background-independent quantization and translate to particular properties of more complicated operators.


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