On the support of the Ashtekar-Lewandowski measure

The new functional calculus does not use any background fields and is therefore well-suited to a fully nonperturbative treatment.

“This calculus does not use any background fields (such as a metric) and is therefore well-suited to a fully nonperturbative treatment.”

Over the past three years, the non-perturbative canonical approach to quantum gravity based on connections has been put on a firm mathematical footing through the development of a new functional calculus on the space of gauge equivalent connections.

“Over the past three years, this approach has been put on a firm mathematical footing through the development of a new functional calculus on the space of gauge equivalent connections [4-11].”

Previous work [9,10,27] showed spin network states provide a natural orthogonal decomposition of H̃_o into finite dimensional subspaces.

“The previous work showed that the spin network states provide us with a natural orthogonal decomposition of the Hilbert space ˜ H o of gauge invariant states in to fi nite dimensional sub-spaces.”

On H_{γ,π}, the action of gauge transformations group G is trivial and there is an orthogonal decomposition (4.23) of H̃_o; this recovers result on spin network states obtained in [9,10].

“On H γ,⃗ π , the action of the gauge transformations group G is trivial and we have the following orthogonal decomposition of the Hilbert space of gauge invariant cylindrical functions,”

Every π_I is necessarily non-trivial whereas ρ_α may be trivial; in gauge invariant context [9,10], ρ_α are all trivial whence items iii and v are unnecessary.

“It should be emphasized that every π I is necessarily non-trivial whereas ρ α may be trivial (i.e., 1-dimensional). In the gauge invariant context [9,10], ρ α are all trivial whence the items iii) and v) are unnecessary.”

The spin network states provide a decomposition of the full Hilbert space H_o into finite dimensional orthogonal subspaces.

“It turns out that the spin network states provide a decomposition of the full Hilbert space H o into fi nite dimensional orthogonal sub-spaces (compare with [9,10]).”

If one does not work with half densities, the gravitational connection becomes complex valued, so the gravitational gauge group would become SL(2,C), which is non-compact and would make the techniques developed in [4,5,6,7,8,3] inapplicable.

“If one does not work with half densities then it turns out that a) The gravitational connection becomes complex valued. Thus, the gravitational gauge group would become become SL (2 , C) which is non-compact and would make the arsenal of techniques developed in [4, 5, 6, 7, 8, 3] inapplicable.”

The Higgs field framework in the diffeomorphism invariant context is developed by analogy between path holonomy and point holonomy, translating the gauge-field framework to arrive at the Higgs field Hilbert space.

“We develop the framework for the Higgs field, in the diffeomorphism invariant context, by pushing the analogy between a usual path holonomy and a 'point holonomy' constructed from a Higgs Field (see below) and can therefore translate step by step the framework [4, 5, 6, 7, 8] developed for gauge fields to arrive at the Higgs field Hilbert space.”

If π := i√det(q) η is defined as momentum conjugate to η, integration by parts makes the symplectic structure equal to ∫ d^3x [π ̇η + 1/4 πη E^i_a ̇E^a_i], implying the real-valued gravitational connection A^i_a acquires a purely imaginary correction term -1/4 πψ E^i_a = -i/4 ̄ηη e^i_a, which leads to difficulties because techniques break down for complex valued connections.

“Assume now that we would define π := i √ det( q ) η to be the momentum conjugate to η . Upon an integration by parts we see that the symplectic structure becomes up to a total time derivative equal to ∫ Σ d 3 x [ π ˙ η + 1 4 πηE i a ˙ E a i ] where E i a denotes the inverse of E a i . But since E a i is the momentum conjugate to A i a this implies that the real-valued gravitational connection A i”

The techniques of [5,6,7,8,3] can be applied to write down a kinematical inner product for the gravitational and Yang-Mills sector that faithfully incorporates all the reality conditions, yielding a Hilbert space L^2(A_SU(2) × A_G, dμ_AL,SU(2) ⊗ dμ_AL,G).

“We can immediately apply the techniques of [5, 6, 7, 8, 3] to write down a kinematical inner product for the gravitational and YangMills sector that faithfully incorporates all the reality conditions. We get a Hilbert space L 2 ( A SU (2) ×A G , dµ AL,SU (2) ⊗ dµ AL,G ) where the index 'AL' stands for AshtekarLewandowski measure and the group index indicates to which gauge group the AshtekarLewand”

A/G is the projective limit of spaces (A/G)_γ, one for each graph γ, isomorphic with G^n where n is the number of edges of γ.

“This is the precise analogue of the result [8] that A / G is the projective limit (rather than direct product) of spaces ( A / G ) γ , one for each graph γ , which is isomorphic with G n , n the number of edges of γ .”

U is contained in a measurable set of μ_U measure zero.

“while U is topologically dense in U it is contained in a measurable set of µ U measure zero.”