In the standard formulation of a local quantum field theory, the scattering matrix is written as a T-exponential:(1)In doing this, it is supposed that the interaction Lagrangian s is a relativisticly covariant and self-adjoint operator, suchwhich satisfies the local commutativity conditionHere s is a transform from the proper inhomogeneous Lorentz group, the symbol "+" denotes the operation of Hermitian conjugation, and [.,.] is a commutator. The success in describing electromagnetic and electrically weak interactions of elementary particles in the quantum field theory is associated in the main with the scattering matrix (1). However, the matrix elements relevant to the interactions (3) involve ultraviolet divergences. ha the past few years, first of all, due to the need to solve the problem of ultraviolet divergences, interest in the interactions for which the condition (3) is not fulfilled (locally noncommutative interactions) has quickened. This is an inherent property, in particular, of models with a nonlocal interaction Lagrangian, with a Lagrangian depending separately on positive-frequency and negative-frequency field components, and of various generalizations of the standard formulation of the local quantum field theory (for details see, e.g., [1][2][3]). Formal application of the scattering matrix (1) to locally noncommutative interactions violates the condition of relativistic covariance. One way around this problem is to change the Tordering for relativisticly covariant chronological ordering. This problem was considered by a number of researchers [4][5][6]. However, the chronological orderings they propose give rise to a scattering matrix which, even in the third order of the perturbation theory, either does not satisfy the unitarity condition [5, 6] or does not go over into (1) for the locally commutative interactions (3) [4]. One possible expression for a relativisticly covariant, unitary, and causative S-matrix, which goes over into (1) provided that (3) is fulfilled, was found in [7, 8]. To do this, postulates had been formulated [7] which must be satisfied by the S-matrix in the quantum field theory with a locally noncommutative interaction.Siberian Medical University.
