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
, where the arguments
consist of space–time points
x and of orientation variables
z from
matrices. We introduce and study the double-sided representation
,
of the group
. Here, the left multiplication by
corresponds to a change in a space-fixed reference frame, whereas the right multiplication by
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,
, 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.
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