Quantum Gravity

146 QUANTUM GEOMETRODYNAMICS∞∑k=−∞e −ikθ |k + n〉 =e inθ |θ〉 .The θ-states are thus labelled by Hom (Z,U(1)), the homomorphisms from Zto U(1). Different values of θ characterize different ‘worlds’ (compare the ambiguityrelated with the Barbero–Immirzi parameter in Section 4.3); θ is inprinciple a measurable quantity and one has, for example, from the limit onthe neutron dipole element, the constraint |θ| < 10 −9 on the θ-parameter ofQCD. Instead of the gauge-dependent wave functions (5.27), one can work withgauge-independent wave functions, but with an additional term in the action,the ‘θ-action’ (Ashtekar 1991; Huang 1992). A state of the form (5.27) is alsowell known from solid state physics (‘Bloch wave function’).One can envisage the states |k〉 as being ‘peaked’ around a particular minimumin a periodic potential. Therefore, tunnelling is possible between differentminima. In fact, tunnelling is described by ‘instantons’, that is, solutions of theclassical Euclidean field equations for which the initial and final value of thegauge potential differ by a large gauge transformation (Huang 1992).One does not have to restrict oneself to S 3 , but can generalize this notion toan arbitrary compact orientable three-space Σ (Isham 1981),|θ〉 = ∑ (k,g)θ(k, g)|k, g〉 , (5.28)where θ(k, g) ∈ Hom([Σ,G],U(1)) appears instead of the e −ikθ of (5.27). As itturns out, g ∈ Hom(π 1 (Σ),π 1 (G)).Instead of taking the gauge group as the starting point, one can alternativelyfocus on the physical configuration space of the theory. This is more suitablefor the comparison with gravity. For Yang–Mills fields, one has the configurationspace Q = A/G, whereA denotes the set of connections. In gravity, Q = S(Σ) =Riem Σ/Diff Σ, see Section 4.2. If the group acts freely on A (or Riem Σ), thatis, if it has no fixed points, thenπ 1 (Q) =π 0 (G) ,and the θ-structure as obtained from π 0 (G) can be connected directly with thetopological structure of the configuration space, that is, with π 1 (Q). As we haveseen in Section 4.2.5, Diff Σ does not act freely on Riem Σ, so S(Σ) had to betransformed into the ‘resolution space’ S R (Σ). Everything is fine if we restrictDiff Σ to D F (¯Σ) (this is relevant in the open case) and take into account thatS R (¯Σ) ∼ = S(Σ). Then,π 1 (S(Σ)) = π 0 (D F (¯Σ)) ,and one can classify θ-states by elements of Hom (π 0 (D F (¯Σ)),U(1)). Isham(1981) has investigated the question as to which three-manifolds Σ can yielda non-trivial θ-structure. He has found that