Constructive Quantum Field Theory. The 1973 ''Ettore by G. Velo, A. S. Wightman

By G. Velo, A. S. Wightman

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Sample text

For large ~ x; and only occasion- Indeed, a typical T haves in some respects as if it were bounded; for example, if is any square-integrable T, function, the set of distributions bef(x) T such that < _d 2 )-~12 dX-~l + 1 T(x) f(x) is square-integrable Since is a set of ~o-measure one. S'(~ n) is not a locally compact space, measure theory on it is not quite standard. We will not give a systematic investiga- tion of the subject, but there are a few simple remarks which should make it seem less strange.

On $, $ ~ L p. e "t0A~ = 0 and by analyticity is strongly continuous on on L2° is strongly continuous on satisfy Since Ker (e "tA) = 10} and Lp q and let Then t > 0. Lp 1 ~ p ~ fi then - flip < Ile-tAf - ffl 2 is dense in are uniformly bounded on If We will con- L q, ~ = 0. (e "tA} Now~ for e't~ = 0 Thus The reader can easily check that the Lp" So we conclude that Then He-tA~ - Wlp _< lle-(t÷t°)~- e-t0Allp > 0 Ran (e "tA) is 31 as t -> 0 Ran e by the holomorphicity in the interior of (e'tOA) -tA is dense and the {e "tA] is strongly continuous on Warning.

Since are uniformly bounded~ we conclude that Lp. 2) on T, ( ,Zo, %) To see this one need only calculate that IIXA(Xo(~))II Z ~ z = ~O(A) S ( , 0,~0) but [I~tXA(xo(~))IILZ(~,Zo,%) = NP(t,Xo(~),A)NL1 = ~t(A) and for appropriate A, ~t(A) > ~o(A). e. ~t(A) = ~o(A) for all t and A) rather than just having stationary transition probabilities, then we would have LP-con tractivity. This is why the processes which Ed Nelson will construct will be generalizations of the so called Ornstein-Uhlenbeck process which is stationary.

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