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7.3 Quantization before reduction

When quantizing a model after a classical reduction, there is much freedom even in choosing the basic representation. For instance, in homogeneous models one can use the Wheeler–DeWitt formulation based on the Schrödinger representation of quantum mechanics. In other models one could choose different smearings, e.g., treating triad components by holonomies and connection components by fluxes, since transformation properties can change from the reduced point of view (see, e.g., [60]). There is, thus, no analog to the uniqueness theorem of the full theory, and models constructed in this manner would have much inherent freedom even at a basic level. With the link to the full theory, however, properties of the unique representation there are transferred directly to models, resulting in analogous properties such as discrete fluxes and an action only of exponentiated connection components. This is sufficient for a construction by analogy of composite operators, such as the Hamiltonian constraint according to the general scheme.

If the basic representation is taken from the full quantization, one makes sure that many consistency conditions of quantum gravity are already observed. This can never be guaranteed when classically reduced models are quantized since then many consistency conditions trivialize as a consequence of simplifications in the model. In particular, background independence requires special properties, as emphasized before. A symmetric model, however, always incorporates a partial background and, within a model alone, one cannot determine which structures are required for background independence. In loop quantum cosmology, on the other hand, this is realized thanks to the link to the full theory. Even though a model in loop quantum cosmology can also be seen as obtained by a particular minisuperspace quantization, it is distinguished by the fact that its representation is derived by quantizing before performing the reduction.

In general, symmetry conditions take the form of second-class constraints since they are imposed for both connections and triads. It is often said that second-class constraints always have to be solved classically before the quantization because of quantum uncertainty relations. This seems to make impossible the above statement that symmetry conditions can be imposed after quantizing. It is certainly true that there is no state in a quantum system satisfying all second class constraints of a given reduction. In addition, using distributional states, as required for first-class constraints with zero in the continuous spectrum, does not help. The reduction described above does not simply proceed in this way by finding states, normalizable or distributional, in the full quantization. Instead, the reduction is done at the operator algebra level, or, alternatively, the selection of symmetric states is accompanied by a reduction of operators which, at least for basic ones, can be performed explicitly. In general terms, one does not look for a sub-representation of the full quantum representation, but for a representation of a suitable sub-algebra of operators related to the symmetry. This gives a well-defined map from the full basic representation to a new basic representation of the model. In this map, non-symmetric degrees of freedom are removed irrespective of the uncertainty relations from the full point of view.

Since the basic representations of the full theory and the model are related, it is clear that similar ambiguities arise in the construction of composite operators. Some of them are inherited directly, such as the representation label j one can choose when connection components are represented through holonomies [170]. Other ambiguities are reduced in models since many choices can result in the same form or are restricted by adaptations to the symmetry. This is, for instance, the case for positions of new vertices created by the Hamiltonian constraint. However, new ambiguities can also arise from degeneracies, such as that between spin labels and edge lengths resulting in the parameter δ in Section 5.4. Factor ordering can also appear more ambiguously in a model and lead to less unique operators than in the full theory. As a simple example we can consider a system with two degrees of freedom (q1,p1;q2,p2) constrained to be equal to each other: C1 = q1 − q2, C2 = p1 − p2. In the unconstrained plane (q1,q2), angular momentum is given by J = q1p2 − q2p1 with an unambiguous quantization. Classically, J vanishes on the constraint surface C1 = 0 = C2, but in the quantum system ambiguities arise: q1 and p 2 commute before but not after reduction. There is thus a factor-ordering ambiguity in the reduction, which is absent in the unconstrained system. Since angular momentum operators formally appear in the volume operator of loop quantum gravity, it is not surprising that models have additional factor-ordering ambiguities in their volume operators. Fortunately, they are harmless and result, e.g., in differences as an isotropic volume spectrum |μ |3∕2 compared to ∘ (|μ|-−-1)|μ-|(|μ| +-1)-, where the second form [45] is closer to SU(2) as compared to U(1) expressions.


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