Download [Chapter] Introduction to Some Aspects of the Relativistic by Arthur S. Wightman PDF

By Arthur S. Wightman

Cargèse Lectures in Theoretical Physics

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The argument goes just as in reference so we will not repeat it here. : exp gcf> : can be defined by its power series as in the case m > 0. Although one cannot appeal to the reconstruction theorem because { , } is not definite one can prove directly the convergence of the scalar products between vectors of the form 14 and define the set of such vectors as Vg;. It is easy to check that : exp (gcf>) : satisfies {) : (j) lff} for all ) : (/)

As already remarked before, Johnson's formulae show that the fermions of the theory are infra-particles. 74) for the vacuum expectation values as a starting point and to seek to define a Hilbert space and operators which yield them. There is a very simple way to do this and at the same time make clear why the vacuum expectation values (4. 74) are so simple in structure. Let efi 1 be a mass zero free scalar field defined on V� equipped with a sesquilinear form { , h defined as in Example 2. ) cf> 2 is defined on VW equipped with a sesquilinear form { , h whose meaning will be made clear shortly.

Bilinear quantities such as = J d 2 xf(x) tf;+(x) iy/J. tf;(x) make sense without Wick ordering in this non-physical world, as one easily sees from their definition in ( P v . · Pn) = = = (J dQ0(p) dQ0(q) c(p )* c(q)f( - (p - q)) u+(p) iy/J. it(q))(n) x 1I' (p1, . . Pn ) � ( - l )i+1 J dQ0(q)f( - (p� - q)) u+ ( pj) iy/J. u(q) x i I x p (q , p1 , . · Pi• . . ,. ) ( 4. (x ) kl'- ( x ) 1T-1 12 a/J. p ( x ) , 1T-11 2 a,,. a (x ) = = R ELATIVISTIC D YNAMICS OF QUAN'l'IZED FIELDS 223 and one has similar infra-red troubles, which can be overcome by similar techniques.

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