Approaches to Quantum Gravity

302 L. FreidelThe inverse group Fourier transform is then explicitly written∫d˜φ(g) 3 X=R 3 8πκ φ(X)⋆e 13 2κ tr(Xg−1 )∫d 3 X=8πκ φ(X)√ 1 − κ 2 P 2 (g)e 13 2κ tr(Xg−1) . (16.34)R 3This Fourier transform intertwines the ⋆-product with the group convolutionproduct •˜φ 1 ⋆φ 2 (g) = ˜φ 1 • ˜φ 2 (g). (16.35)Finally this Fourier transform is an isometry between L 2 (SO(3)) and C κ (R 3 )equipped with the norm∫||φ|| 2 κ = dXφ⋆φ(X). (16.36)8πκ3 The non-commutative space-time structure and the fact that the space of fieldsC κ (R 3 ) have bounded momenta expresses the fact that there exists a minimallength scale accessible in the theory. This is clear if one looks at thenon-commutative delta function defined byδ 0 ⋆φ(X) = φ(0)δ 0 (X). (16.37)It is given byδ 0 (X) = 2κ J ( |X|)1 κ, (16.38)|X|with J 1 the first Bessel function, it is clear that δ 0 (X) is concentrated around X = 0but has a non-zero width.16.6 Effective non-commutative field theoryNow that we are equipped with this star product we can write the Fourier kernel of(16.21) as a producte 2κ 1 tr(X vG v ) = ⋆ e ɛv(e)2κ tr(X v g e )(16.39)∂e∈vand the amplitude (16.21) reads∫ ∏ dX v∏I Ɣ =dg8πκ 3 e ˜K me (g e ) ∏ (v∈Ɣ e∈Ɣ v∈ƔThe effective Feynman propagator is given by∫K m (X) = i dg⋆ e ɛv(e)2κ tr(X v g e )v∈Ɣe 12κ tr(Xg)P 2 (g) − ( sin κmκ). (16.40)) 2. (16.41)