The connection A plays the rˆole of the configuration variable, and E is the canonically conjugate momentum. The classical configuration space is the space of smooth connections 𝒜. By going over to the quantum regime, the space 𝒜 is extended to 𝒜 ͞͞ , by adding also connections which are “distributional” in the sense that they can have support on lower-dimensional subspaces of . The space 𝒜 ͞͞ can be endowed with a topology that turns it into a compact Hausdorff space, and a regular, normalized Borel measure μ_AL, the Ashtekar-Isham-Lewandowski-measure. The quantum theory is then formulated in the kinematical Hilbert space H_kin = L²(𝒜 ͞͞ , dμ_AL). The space 𝒜 is topologically dense in 𝒜 ͞͞ , but measure-theoretically thin: 𝒜 is contained in a set of measure zero.[4] In the Hamiltonian formulation of GR employed in Loop Quantum Gravity, the dynamics is contained in the constraint functions, which are phase space functions that generate a Hamiltonian flow on the constraint hypersurface. Two of the constraints, the Gauss- and the Diffeomorphism- constraint, encode the invariance of the theory under change of SU(2)-gauge and diffeomorphisms on ∑. Consequently, their Hamiltonian flows generate gauge-transformations and diffeomorphisms on ∑, respectively. The action of the groups ℊ and Diff(∑) can be easily extended to the quantum configuration space 𝒜 ͞͞ . But also the groups themselves can be extended to groups of generalized gauge transformations and generalized diffeomorphisms, ℊ͞ and Diff(∑). The group of generalized gauge transformation ℊ͞ is taken to be the group of all maps from ∑ to SU(2). Although classical GR is not invariant under this group, in the quantum theory this arises as a natural candidate for the extension of the smooth gauge transformations. There are reasons to believe that also the diffeomorphisms have to be extended: First, the Hilbert space H_diff of states invariant under diffeomorphisms, is not separable and contains many degrees of freedom (the so-called moduli) that are believed to be unphysical [4, 5]. Also, there are other physical reasons to believe that the group of diffeomorphisms is too small [6]. For an extension of the diffeomorphisms Diff(∑) to Diff(∑)͞ , several suggestions have been made. In [5], it was shown that already a slight extension of the group of diffeomorphisms gives rise to a separable diff-invariant Hilbert space. In [7], Ashtekar and Lewandowski discussed Cⁿ-diffeomorphisms on ∑, which are analytic except for lower-dimensional subsets of ∑. The proof of the uniqueness of the diffeomorphism invariant state ω_AL on the holonomy-flux-algebra for Loop Quantum Gravity uses these for n ≥ 1 [8]. In [6], the stratified diffeomorphisms, introduced earlier by Fleischhack have been investigated. In [9] the piecewise analytic diffeomorphisms have been introduced, which are bijections on ∑ that leave the set of analytical graphs Γ invariant. In [10], the graphomorphisms extended this concept to the smooth and other categories. In [11, 12], it was displayed how the basic ingredients of Loop Quantum Gravity can be formulated naturally as concepts of category theory, i.e. as morphisms, functors and natural transformations. In this language, the connections arise as functors from the path groupoid P of ∑ to the suspension of the gauge group Susp(SU(2)), and the generalized gauge transformations are in one-to-one correspondence to the natural transformations of these functors. Furthermore, the diffeomorphisms acting on ∑ can be naturally interpreted as elements in the automorphism group Aut(P), i.e. as invertible functors from P to itself. Velhinho pointed out that in the light of category language, Aut(P) arises as a candidate for an extension of the diffeomorphisms Diff(∑), and this extension appears to be natural, at least from the mathematical point of view. In this article, we will investigate the consequences of choosing Diff(∑) ͞ = Aut(P). The automorphisms φ ∈ Aut(P) are invertible functors on the path groupoid P, i.e. they permute points in ∑, and also the paths between them in a consistent way. We will, however, encounter elements in Aut(P) that can not be interpreted as bijections of ∑. By this, the elements in Aut(P) will also be able to map graphs into each other that have the same combinatorics, but lie in different generalized knotting classes. By this, a combinatorial picture emerges, which is a desirable feature for a quantum theory of gravity [4, 13, 14]. The emphasis of this article lies on two topics: First, we will prove that the automorphisms Aut(P) leave the Ashtekar-Isham-Lewandowski measure μ_AL invariant, and hence have a well-defined unitary action on the kinematical Hilbert space H_kin =L²(𝒜 ͞͞ , dμ_AL). Second, we will have a closer look at the automorphisms and the orbits of its action on Hkin, in order to describe the automorphism-invariant Hilbert space. We will start in chapter 2 by reviewing the basic concepts of Loop Quantum Gravity, with emphasis on the categorial formulation, and for general gauge group G. We introduce the concept of a (primitive) metagraph, which will be useful in the investigation of the automorphisms Aut(P). We will continue by presenting two kinds of nontrivial automorphisms in section 3, which will both be not induced by a bijection on ∑, but are most useful in what follows. In particular, with the help of these automorphisms we will prove in chapter 4 that the automorphisms leave the Ashtekar-Isham-Lewandowski measure invariant, but also be able to show that any two graphs (in fact, hyphs) with the same combinatorics can be mapped into each other by an automorphism. It is in particular this fact which suggests that by using the automorphisms, a combinatorial picture emerges. In section 5 we will investigate the orbits of vectors in Hkin under the action of Aut(P). For a certain choice of rigging map, we will define the Hilbert space of vectors invariant under the action of Aut(P). For the case of abelian Loop Quantum Gravity, i.e. G = U(1), an explicit orthonormal basis will be given. for G = SU(2), we will comment on how to obtain such a basis. In the appendix, we will, after briefly presenting notions from category theory and combinatorial group theory, present a way of how to write the exponentiated fluxes in category language, and present a categorial version of the Weyl- and the holonomy-flux algebra. inspirehep.net/record/766542
Posted on: Sun, 17 Aug 2014 20:45:23 +0000
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