A proof of the unitarity ofS-matrix in a nonlocal quantum field theory

Abstract

Using the formfactors which are entire analytic functions in a momentum space, nonlocality is introduced for a wide class of interaction Lagrangians in the quantum theory of one-component scalar field φ(x). We point out a regularization procedure which possesses the following features:

• 1. The regularizedS ^δ matrix is defined and there exists the limit

  $$\mathop {\lim }\limits_{\delta \to 0} S^\delta = S.$$

• 2. The Green positive-frequency functions which determine the operation of multiplication in$S \cdot S^ + \mathop = \limits_{Df} S \circledast S^ +$ can be also regularized ⊛^δ and there exists the limit

  $$\mathop {\lim }\limits_{\delta \to 0} \circledast ^\delta = \circledast \equiv .$$

• 3. The operator$J(\delta _1 ,\delta _2 ,\delta _3 ) = S^{\delta _1 } \circledast ^{\delta _2 } S^{\delta _3 + }$ is continuous at the point δ₁=δ₂=δ₃=0.

• 4.
  $$S^\delta \circledast ^\delta S^{\delta + } \equiv 1at\delta > 0.$$

  Consequently, theS-matrix is unitary, i.e.

  $$S \circledast S^ + = S \cdot S^ + = 1.$$