Quantum spin dynamics (QSD): II. The kernel of the Wheeler - DeWitt constraint operator

The Higgs constraint can be analysed more easily in the reduced model because there is only one Higgs vertex.

“In particular, the Higgs constraint ( [20] and section 2) can be analysed more easily because there is just one Higgs vertex in the reduced model.”

For LRS models, the possible conjugacy classes of homomorphisms λ: F→G have representatives λ_k: U(1)→SU(2), exp(tτ3) → exp(ktτ3), for k ∈ N0.

“For LRS models their representatives are given by λ k : U(1) → SU (2), exp(tτ 3 ) → exp(ktτ 3 ) for k ∈ N 0 = {0, 1, . ..} (for a derivation see the example of spherical symmetry in [20] which is in many respects similar to LRS models).”

The reconstruction of invariant connections from the Higgs field pulls back a spin network function to a function on A_B × U_B; these functions are distributional states of the unreduced theory symmetric under [λ], and all symmetric states arise this way.

“The basic idea of [20] is to use the reconstruction of invariant connections from the Higgs field to pull back a spin network function, which is a function on the space A of generalized connections on modulo gauge transformations, to a function on the space A B × U B of generalized connections and Higgs fields on the reduced manifold B.”

The paper uses symmetric distributional states of loop quantum gravity defined in [20], which are exactly symmetric rather than only approximately symmetric at large scales.

“In the present paper we make use of symmetric, distributional states of loop quantum gravity which have been defined and investigated in [20].Contrary to weaves, they are exactly symmetric, not only approximately at large scales.”

For a non-compact space manifold, the symmetries can be interpreted as global asymptotic transformations at spatial infinity that are not part of the gauge group.

“For a non-compact space manifold one can interpret the symmetries as global asymptotic transformations at spatial infinity which are not regarded as elements of the gauge group [23].”

For the non-symmetric operator, the complete kernel of the Wheeler-DeWitt constraint operator is found in the companion paper.

“For the non-symmetric operator we are able to find the complete kernel of the Wheeler-DeWitt constraint operator in [1].”

This does not guarantee that Ĥ^E(N) itself is symmetric.

“Notice that this is far from guaranteeing that Ĥ E (N ) itself is symmetric (see [1]).”

For the symmetric operator, anomaly freedom is obtained only if all loops α_ij are chosen to be kinks with vertex at v(T), i.e. the arc joins endpoints in at least C^1 fashion; arbitrary attachment is insufficient.

“(ii) for the symmetric operator only if all the loops α ij ( ) are chosen to be kinks with vertex at v( )!That is, the arc a ij ( ) joins the endpoints of e i ( ), e j ( ) in at least a C 1 fashion (see [1] for the details of the attachment).An arbitrary attachment of a ij ( ) is insufficient to guarantee anomaly-freeness.”

The commutator in equation (3.16) vanishes identically because the volume operator part at one vertex does not act on holonomies along edges incident at the other vertex.

“Now we just need to use corollary 3.1 of [1] (the non-symmetric operator is treated in a comment after this corollary) to see that the commutator in (3.16) vanishes identically because the part Vv of the volume operator involved in Ĥ E (γ ) does not act on the holonomies along edges incident at v involved in Ĥ E (γ ) and vice versa since the vertices v, v are different.”

In the diffeomorphism-invariant context, the constraint algebra becomes Abelian.

“Expectedly, in the diffeomorphism-invariant context it is true that the constraint algebra becomes Abelian which makes group-averaging especially attractive (see [1]).”