ON A MODEL FOR QUANTUM FIELD THEORY

V ± = lim A ± (t)e~' tH °eitH(2) (V ± ate sometimes called wave operators).Then for the pair of operators Η and Η there exists a scattering operator S, which can be expressed in terms of V ± by the formula s = vy_.(3)In the case which will interest us the roles of the operators Η. and Η can be interchanged:The existence of the strong limits ( 2) and (2*) implies that the operators V ± are unitary.Thus, in die case which will interest us V* ± = F" 1 .Note that from (2) it is easy to obtain the relationThis relation, in turn, implies that the operators S and Η Q commute.We also note that, according to (2), the operators V ± ate only defined to within a factor of modulus unity.Obviously, the same applies to the scattering operator.B. Another definition of the scattering operator uses the asymptotic fields φ. and φ .This definition is a specific feature of field theory. 2)The field φ. = φ { (ζ) is an operator-generalized function.The parameter ζ runs through an arbitrary set with measure Μ (the set of quantum numbers of one particle).The field φ. {ξ) satisfies the following two conditions: 1.The operators φ.^) = ϊφ ία (ξ) f (ξ) άξ, φ\β) = Ιφ* η (ξ)[(ξ) άξ, where / is a squaresummable function, generate an irreducible family.2. The formula