By Girotti H.O.

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**Extra info for Noncommutative quantum field theories**

**Example text**

2/ . 2/ V . 2/ . 2/ . ı Vj . k; n/ ı C kC1 . x kVj kC kC1 . x 1 0 1 kVj kC kC1 . R/ x/ k 1Cj max `D0;:::;kC1 n=2 kC kC1 . x x \Sj / x \Sj / kk`C kC1 . x 0 C kC1 . 2 C 2j; /kC kC1 . 2 C 2j; /kC kC1 . x x/ "jkC1 ` C kC1 . R/ kVj kC kC1 . k; n; j / x \Sj / Vj . 2 C 2j; / x \Sj / `D0;:::;kC1 x \Sj / "jkC1 C k. k; n/ x/ Á x/ x \Sj / kk`C kC1 . 12 we have k 1Cj n=2 kC kC1 . k/ kt 7! 2/ . ı Vj . R/ kVj kC kC1 . x x/ x \Sj / x \Sj / : C k. x max `D0;:::;kC1 x/ kk`C kC1 . R/ if j kk`C kC1 . x kk`C kC1 .

X / the C k -operator norm of K˙ is less than 1. P Thus the Neumann series j1D0 . K˙ /j converges in all C k -operator norms and id C K˙ is an isomorphism with bounded inverse on all C k . x ; E /. j / kC k . x kK˙ x/ Ä kK˙ k2C k . x Äı j 2 where ı ´ vol. x / kK˙ kC 0 . x x/ x/ vol. x /j 1 vol. x / kK˙ k2C k . x < 1. Hence the series 1 X . j / K˙ kK˙ kjC 02. x x/ x/ 56 2. The local theory converges in all C k . x x ; E E/. x/ . 9. 8. Then for each u 2 C 0 . u//: Proof. id C K˙ / 1 K˙ . x/Œ' defines a smooth section in E over x Fix ' 2 D.

X/. ˛/ > nC2. ˛ C 2; x/ classically. ˛/ > n C 2. Analyticity then implies (5) for all ˛. ˛ 2/ 2; x/i 32 1. ˛; x/: 2˛ (6) Let ' be a testfunction on . 0/Œ. x '/ ı expx D ı0 Œ. x '/ ı expx D .. x; x 0 / W Tx ! Tx 0 be a time-orientation preserving linear isometry. ˛/Œ. ˛/ is, as before, the Riesz distribution on Tx . x; x 0 / to depend smoothly on x 0 , then . 6. ˛; x// Ä n C 1 as well. ˛/ it is clear that the constant C may be chosen locally uniformly in x. x 0 ; /. 6. The remaining assertions follow directly from the corresponding properties of the Riesz distributions on Minkowski space.