Why Poincare symmetry is a good approximate symmetry in particle theory

Abstract

As shown in the famous Dyson's paper "Missed Opportunities", even from purely mathematical considerations (without any physics) it follows that Poincare quantum symmetry is a special degenerate case of de Sitter quantum symmetries. Then the question arises why in particle physics Poincare symmetry works with a very high accuracy. The usual answer to this question is that a theory in de Sitter space becomes a theory in Minkowski space in the formal limit when the radius of de Sitter space tends to infinity. However, de Sitter and Minkowski spaces are purely classical concepts, and in quantum theory the answer to this question must be given only in terms of quantum concepts. At the quantum level, Poincare symmetry is a good approximate symmetry if the eigenvalues of the representation operators $M_{4\mu}$ of the anti-de Sitter algebra are much greater than the eigenvalues of the operators $M_{\mu\nu}$ ($\mu,\nu=0,1,2,3$). We show that explicit solutions with such properties exist within the framework of the approach proposed by Flato and Fronsdal where particles that are considered elementary in the standard theory are bound states of two Dirac singletons.