# Current Problems in Elementary Particle and Mathematical by P. Urban (auth.), Prof. Dr. Paul Urban (eds.)

By P. Urban (auth.), Prof. Dr. Paul Urban (eds.)

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Definition Consider ~ = {1,Sl'S2' ••• 'Sn' ••• } an extension of ~ to S. ) 1; £ ~ 0 for f £ 1+ S3: Each Sn is symmetric; S4: t:~ and f(i i, x ~(Aa,l» ~ £ 1· S (i) f(~) for all a ~ 0 £ 4 R This notation differs from that in [4] and conforms to usage in constructive field theory. States satisfying S(P+) ~ 0 are called strongly positive or Nelson-Symanzik positive in contrast to positive states for which S(1+)~0. Due to 8 invariance we have that S is real and the underlying relativistic theory 0 invariant.

Due to 8 invariance we have that S is real and the underlying relativistic theory 0 invariant. This restriction is a consequence of strong positivity [4]. As of yet interesting conditions on f which guar- antee an extension to a Schwinger state are not known. There is a partial result for positive extensions. Define a Hilbert norm on S of the form with 55 and notice that (N I , N2 ) is independent of n. This is the key to Proposition (Challifour-Slinker [6]). Let 1~(f)1 < (N N2) - q II (f) for all f e: 5 and some pair of non-negative --0 integers (N I ,N 2 ).

12]). It seems worth while to note that there is an alternate cutoff-free construction of characteristic functions 28 which has the potential to deal with each of the four types of singular behaviour listed above. For the pathintegral formulation of (imaginary time) nonrelativistic potential scattering H. R. Shepp (EKS, [13]) have recently proposed an interesting alternative. f. g. ref. [13]) functional integrals of the form T -gJV(x(t»dt o (gV) where ~B is the probability measure of the Wiener process B formally related to white noise X through B(t) X(t) Instead of this weighted average over Brownian motion paths B with weight function p g (x) = Ng exp {- TVdt} 0 EKS propose to employ a direct, unweighted average over a distorted set of paths Y B B The dynamics is incorporated not in a weight function but in a transformation of the paths which occur in the averaging.