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7.2 Basic operators

In the classical reduction, symmetry conditions are imposed on both connections and triads, but in our description so far, at the level of states, only connections have been taken into account. Configuration and momentum variables play different roles in any quantum theory since a polarization is necessary. As we based the construction on the connection representation, symmetric triads have to be implemented at the operator level. (There cannot be additional reduction steps at the state level since, as we already observed, states just implement the right number of reduced degrees of freedom.)

Classically, the reduction of phase-space functions is simply done by pull back to the reduced phase space. The flow generated by the reduced functions then necessarily stays in the reduced phase space and defines canonical transformations for the model. An analogous statement in the corresponding quantum theory would mean that the reduced state space would be fixed by full operators such that their action (or dual action on distributions) could directly be used in the model without further work. This, however, is not the case with the reduction performed so far. We have considered only connections in the reduction of states; and also classically a reduction to a subspace 𝒜inv × ℰ, where connections are invariant but not triads, would be incomplete. First, this would not define a phase space of its own with a non-degenerate symplectic structure. More important in this context is the fact that this subspace would not be preserved by the flow of reduced functions.

As an example (see also [60Jump To The Next Citation Point] for a different discussion in the spherically-symmetric model) we consider a diagonal homogeneous model, such as Bianchi I for simplicity, with connections of the form Ai dxa = &tidle;c ΛiωI a (I) I and look at the flow generated by the full volume ------- V = ∫ d3x ∘ |detE |. It is straightforward to evaluate the Poisson bracket

i ijk b c ∘ ------- {A a(x),V } = 2πγG εabcε E jE k∕ |det E|

already used in Equation (13View Equation). A point on 𝒜inv × ℰ characterized by i &tidle;c(I)Λ I and an arbitrary triad thus changes infinitesimally by

∘ ------- δ(&tidle;c(I)ΛiI) = 2π γG εIbcεijkEbjEck∕ |det E |,

which does not preserve the invariant form. First, on the right-hand side we have arbitrary fields E such that i δ(&tidle;c(I)Λ I) is not homogeneous. Second, even if we would restrict ourselves to homogeneous E, δ(&tidle;c(I)ΛiI) would not be of the original diagonal form. This is the case only if δ(&tidle;c(I)Λi ) = Λiδ(&tidle;c(I)) I I, since only the &tidle;cI are canonical variables. The latter condition is satisfied only if

ijk j i j b c ∘ ------- ε Λ Iδ(&tidle;c(I)Λ I) = 4 πγG εIbcΛ (I)E iE j∕ |det E|

vanishes, which is not the case in general. This condition is true only if Eai ∝ Λai, i.e., if we restrict the triads to be of diagonal homogeneous form, just as the connections.

A reduction of only one part of the canonical variables is thus incomplete and leads to a situation in which most phase-space functions generate a flow that does not stay in the reduced space. Analogously, the dual action of full operators on symmetric distributional states does not in general map this space to itself. Thus, an arbitrary full operator maps a symmetric state to a non-symmetric one and cannot be used to define the reduced operator. In general, one needs a second reduction step that implements invariant triads at the level of operators by an appropriate projection of its action back to the symmetric space. This can be quite complicated, and fortunately there are special full operators adapted to the symmetry for which this step is not necessary.

From the above example it is clear that those operators must be linear in the momenta a Ei, for otherwise, one would have a triad remaining after evaluating the Poisson bracket, which on 𝒜inv × ℰ would not be symmetric everywhere. Fluxes are linear in the momenta, so we can try pK (z ) := ∫ d2y Λk Ea ωK 0 Sz0 (K) k a where S z0 is a surface in the IJ-plane at position z = z 0 in the K-direction. By choosing a surface along symmetry generators XI and XJ this expression is adapted to the symmetry, even though it is not fully symmetric yet since the position z0 has to be chosen. Again, we compute the Poisson bracket

∫ i K i K 2 {A a(x),p (z0)} = 8πγG Λ (K ) δ(x,y )ω a (y) d y Sz0

resulting in

δ(&tidle;c Λi ) = 8 πγG Λiδ(z,z ) . (I) I I 0

Here, also, the right-hand side is not homogeneous, but we have ijk j k ε Λ Iδ(&tidle;c(I)ΛI) = 0 such that the diagonal form is preserved. The violation of homogeneity is expected since the flux is not homogeneous. This can easily be remedied by “averaging” the flux in the K-direction to

∑N pK := lim N − 1 pK(αN −1L0) , N →∞ α=1

where L0 is the coordinate length of the K-direction if it is compact. For any finite N the expression is well defined and can directly be quantized, and the limit can be performed in a well-defined manner at the quantum level of the full theory.

Most importantly, the resulting operator preserves the form of symmetric states for the diagonal homogeneous model in its dual action, corresponding to the flux operator of the reduced model as used before. In averaging the full operator the partial background provided by the group action has been used, which is responsible for the degeneracy between edge length and spin in one reduced flux label. Similarly, one can obtain holonomy operators along the I-direction, which preserve the form of symmetric states after averaging them along the J and K directions (in such a way that the edge length is variable in the averaging limit). Thus, the dual action of full operators is sufficient to derive all basic operators of the model from the full theory [66Jump To The Next Citation Point]. (See [88Jump To The Next Citation Point] for a simpler illustration of the reduction from anisotropic to isotropic models.) The representation of states and basic operators, which was seen to be responsible for most effects in loop quantum cosmology, is thus directly linked to the full theory. An elaboration of this algebraic version of the symmetry reduction can be found in [207208], which also shows promise in extending the reduction to non-basic operators such as the Hamiltonian constraint. This, then, defines the cosmological sector of loop quantum gravity.


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