The scattering matrix for locally noncommutative interactions

Abstract

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.

Diffractive pion production at COMPASS—First results on 3π final states—neutral mode

“…E852 [6], VES [7], and Crystal Barrel [8]. The π 1 (1600) was observed by both E852 and VES in the decay channels: ρπ [9,10], η π [7,11], f 1 π [12,13], and ωππ [13,14]. Especially the observations of π 1 (1600) into ρπ based on analyses of π − π + π − final state events are controversially discussed [15,16].…”

High statistics light meson spectroscopy with the COMPASS spectrometer

“…The π 1 (1400) was observed by E852 [6] and by VES [7] in the reaction π − N → ηπ − N, and by Crystal Barrel [8,9] in pn → π − π 0 η and pp → 2π 0 η Dalitz plot analysis. Another 1 −+ state, the π 1 (1600), decaying into ρπ [10,11], η π [12,13], f 1 (1285)π [14,15], and ωππ [16,15] was observed in peripheral π − p interactions in E852 and VES. The resonant nature of both states, however, is still heavily disputed in the community [17,15].…”

Observation of aJPC= 1−+exotic signal in theπ−π0π0system diffractively produced at COMPASS, and comparison to the charged decay mode

“…The observations in the ρπ decay channel analysed in 3π final states are still controversially discussed in the community. Especially the resonant nature of the observed signals in the exotic J PC = 1 −+ partial-wave reported by the E852 at BNL and the VES experiments [2,3] in diffractively produced π − π + π − final states are questioned, in later publications previous conclusions were withdrawn [4] and re-analyses of the (3π)…”