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7.1 Symmetric states

One can imagine states that are invariant under a given action of a symmetry group on space by starting with a general state and naively summing over all its possible translates by elements of the symmetry group. For instance on spin network states, the symmetry group acts by moving the graph underlying the spin network, keeping the labels fixed. Since states with different graphs are orthogonal to each other, the sum over uncountably many different translates cannot be normalizable. In simple cases, such as for graphs with a single edge along a symmetry generator, one can easily make sense of the sum as a distribution. But this is not clear for arbitrary states, in particular for states whose graphs have vertices, which on the other hand would be needed for sufficient generality. A further problem is that any such action of a symmetry group is a subgroup of the diffeomorphism group. At least on compact space manifolds where there are no asymptotic conditions for diffeomorphisms in the gauge group, it then seems that any group-averaged diffeomorphism-invariant state would already be symmetric with respect to arbitrary symmetries, which is obviously not sensible.

In fact, symmetries and (gauge) diffeomorphisms are conceptually very different, even though mathematically they are both expressed by group actions on a space manifold. Gauge diffeomorphisms are generated by first class constraints of the theory, which in canonical quantum gravity are imposed in the Dirac manner [153] or following refined algebraic quantization [31], conveniently done by group averaging [223]. Symmetries, however, are additional conditions imposed on a given theory to extract a particular sector of special interest. They can also be formulated as constraints added to the theory, but these constraints must be second class for a well-defined framework: one obtains a consistent reduced theory, e.g., with a non-degenerate symplectic structure, only if configuration and momentum variables are required to be symmetric in the same (or dual) way.

In the case of gravity in Ashtekar variables, the symmetry type determines, along the lines of Appendix A, the form of invariant connections and densitized triads defining the phase space of the reduced model. At the quantum level, however, one cannot keep connections and triads on the same footing since a polarization is required. One usually uses the connection representation in loop quantum gravity such that states are functionals on the space of connections. In a minisuperspace quantization of the classically reduced model, states would then be functionals only of invariant connections for the given symmetry type. This suggests that one should define symmetric states in the full theory to be those states, whose support contains invariant connections as a dense subset [96Jump To The Next Citation Point46] (one requires only a dense subset because possible generalized connections must be allowed for). As such, they must necessarily be distributional, as already expected from the naive attempt at a construction. Symmetric states thus form a subset of the distributional space Cyl ∗. In this manner, only the reduced degrees of freedom are relevant, i.e., the reduction is complete, and all of them are indeed realized, i.e., the reduction is not too strong. Moreover, an “averaging” map from a non-symmetric state to a symmetric one can easily be defined by restricting the non-symmetric state to the space of invariant connections and requiring it to vanish everywhere else.

This procedure defines states as functionals, but since there is no inner product on the full Cyl∗ this does not automatically result in a Hilbert space. Appropriately defined subspaces of Cyl∗, nevertheless, often carry natural inner products, which is also the case here. In fact, since the reduced space of invariant connections can be treated by the same mathematical techniques as the full space, it carries an analog of the full Ashtekar–Lewandowski measure and this is indeed induced from the unique representation of the full theory. The only difference is that in general an invariant connection is not only determined by a reduced connection but also by scalar fields (see Appendix A). As in the full theory, this space π’œinv of reduced connections and scalars is compactified to the space ¯ π’œinv of generalized invariant connections on which the reduced Hilbert space is defined. One thus arrives at the same Hilbert space for the subset of symmetric states in Cyl ∗ as used before for reduced models, e.g., using the Bohr compactification in isotropic models. The new ingredient now is that these states have meaning in the full theory as distributions, whose evaluation on normalizable states depends on the symmetry type and partial background structure used.

Whether or not the symmetric Hilbert space obtained in this manner is identical to the reduced loop quantization of Section 5 is not a matter of definition, but would be a result of the procedure. The support of a distribution is by definition a closed subset of the configuration space. If the set of generalized invariant connections of π’œ¯inv is not a closed subset of π’œ¯, the support of symmetric distributional states would be larger than the space of invariant connections. In such a case, the reduction at the quantum level would give rise to more degrees of freedom than loop quantization of the classically reduced model. Whether or not this is the case and what such degrees of freedom could be is still open; the situation is rather subtle. While the classical space of invariant connections is embedded in the space of all connections, this is not the case for generalized connections [117]. It is thus not obvious, whether the space of invariant connections is closed or what its closure is.

Alternative reduction procedures that are not based on states, which require the connection to be symmetric, but on coherent states, are being studied in [163164]. This has been shown to work well for free quantum-field theories, where also the Hamiltonian operator of the reduced model can be derived from the full one. However, since the existence of dynamical coherent states for the free theory is exploited in the construction it remains unclear how general Hamiltonian operators can be reduced.

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