Approaches to Quantum Gravity

276 A. Perezwhere M = ×R (for an arbitrary Riemann surface), ω is an SU(2)-connectionand the triad e is an su(2)-valued 1-form. The gauge symmetries of the action arethe local SU(2) gauge transformationsδe = [e,α] , δω = d ω α, (15.3)where α is a su(2)-valued 0-form, and the “topological” gauge transformationδe = d ω η, δω = 0, (15.4)where d ω denotes the covariant exterior derivative and η is a su(2)-valued 0-form.The first invariance is manifest from the form of the action, while the second is aconsequence of the Bianchi identity, d ω F(ω) = 0. The gauge symmetries are solarge that all the solutions to the equations of motion are locally pure gauge. Thetheory has only global or topological degrees of freedom.Upon the standard 2 + 1 decomposition, the phase space in these variables isparametrized by the pull back to of ω and e. In local coordinates one can expressthem in terms of the two-dimensional connection A i a and the triad field E b j=ɛ bc e k c δ jk where a = 1, 2 are space coordinate indices and i, j = 1, 2, 3aresu(2)indices. The Poisson bracket is given by{A i a (x), E b j (y)} =δ b a δi j δ (2) (x, y). (15.5)Local symmetries of the theory are generated by the first class constraintsD b E b j = 0, Fab i (A) = 0, (15.6)which are referred to as the Gauss law and the curvature constraint respectively.This simple theory has been quantized in various ways in the literature [5], here wewill use it to introduce the spin foam representation.15.3.2 Spin foams from the Hamiltonian formulationThe physical Hilbert space, H phys , is defined by those “states in H kin ”thatareannihilated by the constraints. As discussed in the chapter by Thiemann (see also[2; 4]), spin network states solve the Gauss constraint – ̂D a Ei a |s〉 =0–astheyare manifestly SU(2) gauge invariant. To complete the quantization one needs tocharacterize the space of solutions of the quantum curvature constraints ̂Fab i ,andto provide it with the physical inner product. As discussed in Section 15.2 wecan achieve this if we can make sense of the following formal expression for thegeneralized projection operator P:∫∫P = D[N] exp(i Tr[N ̂F(A)]) = ∏ δ[ ̂F(A)], (15.7)x⊂