By Martin Schottenloher

The first a part of this publication offers a self-contained and mathematically rigorous exposition of classical conformal symmetry in n dimensions and its quantization in dimensions. the second one half surveys a few extra complicated subject matters of conformal box theory.

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N+1 ) = (ξ 0 − ξ , b + ξ − b : ξ − 2ξ − b : ξ n+1 − ξ , b + ξ − b ), where ξ − = 12 (ξ n+1 − ξ 0 ). 3 The Conformal Group of R2,0 ϕ (ı(x)) = 31 N− x N+ x : x− x b : 2 2 = ı(ϕ (x)) for all x ∈ R p,q , N(x) = 0. The mapping ϕ is conformal, since ϕ = ψΛ with ⎛ ⎞ 1 − 12 b −(η b)T 12 b −b ⎠ ∈ SO(p + 1, q + 1). En Λ=⎝ b 1 T − 2 b −(η b) 1 + 12 b In particular, ϕ is a conformal continuation of ϕ . 7 ϕ : N p,q → N p,q of the type ϕ (ξ 0 : . . : ξ n+1 ) = γ (Λξ ) with Λ ∈ SO(p + 1, q + 1) having a conformal inverse ϕ −1 = ψΛ−1 .

In the same way as above the strong topology on Aut(P) is defined using δ replacing the norm. Observe that the strong topology on U(H) and U(P) as well as on Mu and Aut(P) is the topology of pointwise convergence. So, in contrast to its name, the strong topology is rather a weak topology. Since all these sets of mappings are uniformly bounded they are equicontinuous by the theorem of Banach–Steinhaus and hence the strong topology also agrees with the compact open topology, that is the topology of uniform convergence on the compact subsets of H (resp.

A global conformal transformation in the case of R p,q , p + q > 2, is a conformal transformation ϕ : M → R p,q , which is defined on the entire set R p,q with the possible exception of a hyperplane. By the results of Sect. ) Now, N 2,0 is isometrically isomorphic to the 2-sphere S2 (in general, one has p,0 N ∼ = S p , since S p × S0 = S p × {1, −1}) and hence N 2,0 is conformally isomorphic to the Riemann sphere P := P1 (C). 12. A M¨obius transformation is a holomorphic function ϕ , for which there is a matrix ab ∈ SL(2, C) such that cd ϕ (z) = az + b , cz + d = 0.