ABSTRACT
The notion of kinematical confinement introduced by Flato, Fronsdal, Sternheimer [1] and their coworkers must be considered as one of the most fascinating and powerful ideas of modern theoretical physics. Here we hope to sharpen the meaning of this remarkable mechanism for explaining quark confinement. We recall a well-known deformation of the Poincaré Lie algebra [2] which relates the this algebra to the Lie algebra of https://www.w3.org/1998/Math/MathML"> S O 0 ( 2,3 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq6696.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the anti-de Sitter group. We solve the defining equations of the deformation map for the Poincaré translation generators to obtain an isomorphic copy of the Poincaré Lie algebra in which the translation generators depend upon the deformation parameter [3]. Representations of this isomorphic copy of the Poincaré Lie algebra go over into Segal-Inönü-Wigner contractions [4] of the corresponding representations of the anti-de Sitter algebra as the deformation parameter, which is essentially the reciprocal of the radius of the anti-de Sitter space, goes to zero. We apply our results to the singleton representations of the anti-de Sitter algebra, and implications for kinematical confinement are then given.
