Group-theoretical classification of orientable objects and particle phenomenology

In our previous works, we have proposed a quantum description of relativistic orientable objects by a scalar field on the Poincar\'{e} group. This description is, in a sense, a generalization of ideas used by Wigner, Casimir and Eckart back in the 1930's in constructing a non-relativistic theory of a rigid rotator. The present work is a continuation and development of the above mentioned our works. 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 space-fixed reference frame, and can be specified by elements q of the motion group of the Minkowski space - the Poincar\'e group $M(3,1)$. Quantum states of relativistic orientable objects are described by scalar wave functions $f(q)$ where the arguments $q=(x,z)$ consist of Minkowski space-time points x, and of orientation variables z given by elements of the matrix Z∈ SL(2,C). Technically, we introduce and study the so-called double-sided representation T(g)f(q)=f(gₗ⁻¹qgᵣ), g=(gₗ,gᵣ)∈ M, of the group M, in the space of the scalar functions $f(q)$. Here the left multiplication by gₗ⁻¹ corresponds to a change of space-fixed reference frame, whereas the right multiplication by gᵣ corresponds to a change of body-fixed reference frame. On this basis, we develop a classification of the orientable objects and draw the attention to a possibility of connecting these results with the particle phenomenology. In particular, we demonstrate how one may identify fields described by linear and quadratic functions of z with known elementary particles of spins $0$,1/2, and $1$. The developed classification does not contradict the phenomenology of elementary particles and, moreover, in some cases give its group-theoretic explanation.