Group-Theoretical Classification of Orientable Objects and Particle Phenomenology

The quantum description of relativistic orientable objects by a scalar field on the Poincaré group is considered. The position of the relativistic orientable object in Minkowski space is completely determined by the position of a body-fixed reference frame with respect to the position of the space-fixed reference frame, so that all the positions can be specified by elements q of the Poincaré group. Relativistic orientable objects are described by scalar wave functions f(q), where the arguments q=(x,z) consist of space–time points x and of orientation variables z from SL(2,C) matrices. We introduce and study the double-sided representation T(g)f(q)=f(gl−1qgr), g=(gl,gr)∈M, of the group M. Here, the left multiplication by gl−1 corresponds to a change in a space-fixed reference frame, whereas the right multiplication by gr corresponds to a change in a body-fixed reference frame. On this basis, we develop a classification of orientable objects and draw attention to the possibility of connecting these results with particle phenomenology. In particular, we demonstrate how one may identify fields described by polynomials in z with known elementary particles of spins 0, 12, and 1. The developed classification does not contradict the phenomenology of elementary particles and, in some cases, even provides a group-theoretic explanation for it.