Quantum Gravity

THE GEOMETRODYNAMICAL WAVE FUNCTION 145analogous to (3.86)—for parametrized field theories—and (3.49)—for the bosonicstring.A simple analogy to (5.19) is Gauss’ law in QED (or its generalizations tothe non-Abelian case; see Section 4.1). The quantized version of the constraint∇E ≈ 0readsi ∇δΨ[A] =0, (5.23)δAfrom which invariance of Ψ with respect to gauge transformations A → A + ∇λfollows.We have seen that the wave functional Ψ[h ab ] is invariant under infinitesimalcoordinate transformations (‘small diffeomorphisms’). There may, however, exist‘large diffeomorphisms’, that is, diffeomorphisms which are not connected withthe identity, under which Ψ might not be invariant.This situation is familiar from Yang–Mills theories (see e.g. Huang 1992).The quantized form of the Gauss law (4.30) demands that Ψ[A i a ]beinvariantunder infinitesimal (‘small’) gauge transformations; cf. the QED-example (5.23).We take the Yang–Mills gauge group G as the mapS 3 −→ SU(N) ≡ G, (5.24)where R 3 has been compactified to the three-sphere S 3 ; this is possible since itis assumed that gauge transformations approach a constant at spatial infinity.The key role in the study of ‘large gauge transformations’ is played byπ 0 (G) ≡G/G 0 , (5.25)where G 0 denotes the component of G connected with the identity. Thus, π 0counts the number of components of the gauge group. One can also writeπ 0 (G) =[S 3 ,G] ≡ π 3 (G) =Z , (5.26)where [S 3 ,G] denotes the set of homotopy classes of continuous maps from S 3 toG. 7 The ‘winding numbers’ n ∈ Z denote the number of times that the spatial S 3is covered by the SU(2)-manifold S 3 . 8 This, then, leads to a vacuum state for eachconnected component of G, called ‘K-vacuum’ |k〉, k ∈ Z. Astate|k〉 is invariantunder small gauge transformations, but transforms as |k〉 →|k + n〉 under largegauge transformations. If one defines the central concept of a ‘θ-vacuum’ by|θ〉 =∞∑k=−∞e −ikθ |k〉 , (5.27)with a real parameter θ, the transformation of this state under a large gaugetransformation reads7 Two maps are called homotopic if they can be continuously deformed into each other. Allhomotopic maps yield a homotopy class.8 The SU(N)-case can be reduced to the SU(2)-case.