Kiefer C. Quantum gravity

THE GEOMETRODYNAMICAL WAVE FUNCTION 145
analogous to (3.86)—for parametrized field theories—and (3.49)—for the bosonic
string.
A simple analogy to (5.19) is Gauss’ law in QED (or its generalizations to
the non-Abelian case; see Section 4.1). The quantized version of the constraint
∇E ≈ 0reads
i ∇δΨ[A] =0, (5.23)
δA
from which invariance of Ψ with respect to gauge transformations A → A + ∇λ
follows.
We have seen that the wave functional Ψ[h ab ] is invariant under infinitesimal
coordinate transformations (‘small diffeomorphisms’). There may, however, exist
‘large diffeomorphisms’, that is, diffeomorphisms which are not connected with
the 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 ]beinvariant
under infinitesimal (‘small’) gauge transformations; cf. the QED-example (5.23).
We take the Yang–Mills gauge group G as the map
S 3 −→ SU(N) ≡ G, (5.24)
where R 3 has been compactified to the three-sphere S 3 ; this is possible since it
is assumed that gauge transformations approach a constant at spatial infinity.
The key role in the study of ‘large gauge transformations’ is played by
π 0 (G) ≡G/G 0 , (5.25)
where G 0 denotes the component of G connected with the identity. Thus, π 0
counts the number of components of the gauge group. One can also write
π 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 to
G. 7 The ‘winding numbers’ n ∈ Z denote the number of times that the spatial S 3
is covered by the SU(2)-manifold S 3 . 8 This, then, leads to a vacuum state for each
connected component of G, called ‘K-vacuum’ |k〉, k ∈ Z. Astate|k〉 is invariant
under small gauge transformations, but transforms as |k〉 →|k + n〉 under large
gauge 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 gauge
transformation reads
7 Two maps are called homotopic if they can be continuously deformed into each other. All
homotopic maps yield a homotopy class.
8 The SU(N)-case can be reduced to the SU(2)-case.