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F o ' f 1 , f 2 , ... ,In £ C• (2) *-operation: (3) product: with a unit element 1 It is easily checked that (! + 9:,) x (! x 9:,) x h ! = x (9:, x h) and (! h x 9:,) {l,O,O, ... } = (! x * '" = 9:, h) x f + 9:, x '" . • h, Each Sn becomes a topological space with respect to a basis of continuous norms given by sup X £ (1+ R4n Ilx11 2 ) N/2\ oCl fn (x) I, N a non-negative integer. lalsN We use here the conventional multi-index notation of distribution theory [34] with x = (x 1 " " ' x ) II xl1 2 = n 2 +2 n I Ei =l (xi + xi)' a (al, ..

Austr. 42, 9 (1975). 13. H. R. A. Shepp, Ann. Phys. ), 88, 588 (1975). 14. J. Eachus, L. Streit, Rep. Math. Phys. 4, 161 (1973). 15. T. Hida, L. Streit (in preparation) . 16. G. Sudarshan 17. R. Klauder, L. Streit, W. Wyss, unpublished. 18. Discussions with V. Enss were very helpful in clarifying this point. 19. C. Hegerfeldt, Contribution to the ZiF-Symposium Dec. 75, and K. C. Hegerfeldt, Is the Wick Square Infinitely Divisible 7, ZiF preprint,1976. 20. C. Hegerfeldt, Corom. Math. Phys. 45, 133 (1975).

The field tends toward a pseudofree (unless y 3 = 2) massless limit [25] if g lies in the domain of attract- * ion of gy. ) 1 S (A) Va > o The interaction Hamiltonian is V(~)ren given by the structure function g [24], [25]: g"/g = X (x+l) 2A 2 + V (A) so that the above considerations determine the ultralocal Hamiltonians of a given universality class y. It should be clear that the construction of long range scaling limits is only one of many capabilities of the general fixed pOint strategy. Different physical questions even about the same dynamical system will give rise to different mappings T, and possibly to different fixed points C (a).