Group-theoretical classification of orientable objects and particle phenomenology

TL;DR

This paper develops a group-theoretic framework using the Poincaré group to classify relativistic orientable objects and connects this classification to elementary particle phenomenology, providing a new perspective on particle types and spins.

Contribution

It introduces a novel scalar field representation on the Poincaré group for orientable objects and links this to particle phenomenology, extending previous theoretical work.

Findings

Classification of orientable objects using group theory.

Identification of fields with spins 0, 1/2, and 1.

Group-theoretic explanation of elementary particle phenomenology.

Abstract

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 $Z\in SL(2,C)$. Technically, we introduce and study the so-called double-sided representation $\boldsymbol{T}(\boldsymbol{g})f(q)=f(g_l^{-1}qg_r)$, $\boldsymbol{g}=(g_l,g_r)\in \boldsymbol{M}$, of the group $\boldsymbol{M}$, in the space of the scalar functions $f(q)$. Here the left multiplication by $g_l^{-1}$ corresponds to a change of space-fixed reference frame, whereas the right multiplication by $g_r$ 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 $0$,$\frac{1}{2}$, and $1$. The developed classification does not contradict the phenomenology of elementary particles and, moreover, in some cases give its group-theoretic explanation.