2. THE CLASSICAL MODEL The characteristic feature of the Bianchi IX model is the existence of the simply transitive isometry group. The infinitesimal generators of this group are the three linearly independent spacelike killing vectors, ξ$_{i}$, which obey where i , j and k runs from 1 to 3. The generators of these transformations are One may write the metric of Bianchi IX in terms of forms , where N is the lapse function, h$_{ij}$ is the 3-metric and the σ$_{i}$ are the left invariant 1-forms of SU (2): with 0 ≤ ψ ≤ 4 π , 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2 π . Without lose of generality, we can assume that the metric h$_{ij}$ is diagonal. Using the Misner variables to parameterize the metric where α and β are functions of time only. The matrix with the property T rβ = 0, ensures that the 3-volume of the hypersurface depends only on the conformal factor α . The variables β$_{+}$ and β − describe the anisotropy of the spacetime, 3 β$_{ij}$ = diag [ β$_{+}$ $_{+}$√$_{3}$$_{β}$$_{−}$$_{, β}$$_{+}$ − √$_{3}$$_{β}$$_{−}$$_{,}$ − 2 β$_{+}$ ] , (6) h$_{ij}$ = e 2 $^{α}$( e 2 $^{β}$)$_{ij}$ , (5) σ 1 = cosψdθ + sinψsinθdφ, σ 2 = − sinψdθ + cosψsinθdφ, (4) σ 3 = dψ + cosθdφ, ds 2 = N $^{2}$dt2 − h$_{ij}$σ $^{i}$σ$^{j}$, (3) ξ$_{1}$ = x $^{2}$∂$_{3}$ − x $^{3}$∂$_{2}$ , ξ$_{2}$ = x $^{3}$∂$_{1}$ − x $^{1}$∂$_{3}$ , (2) ξ$_{3}$ = x $^{1}$∂$_{2}$ − x $^{2}$∂$_{1}$ . [ ξ$_{i}$, ξ$_{j}$ ] = C$_{ij}$ $^{k}$ξ$_{k}$ , (1)