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We propose an approach to the quantum-mechanical description of relativisticorientable objects. It generalizes Wigner's ideas concerning the treatment ofnonrelativistic orientable objects (in particular, a nonrelativistic rotator)with the help of two reference frames (space-fixed and body-fixed). A technicalrealization of this generalization (for instance, in 3+1 dimensions) amounts tointroducing wave functions that depend on elements of the Poincare group $G$. Acomplete set of transformations that test the symmetries of an orientableobject and of the embedding space belongs to the group $\Pi =G\times G$. Allsuch transformations can be studied by considering a generalized regularrepresentation of $G$ in the space of scalar functions on the group, $f(x,z)$,that depend on the Minkowski space points $x\in G/Spin(3,1)$ as well as on theorientation variables given by the elements $z$ of a matrix $Z\in Spin(3,1)$.In particular, the field $f(x,z)$ is a generating function of usual spin-tensormulticomponent fields. In the theory under consideration, there are fourdifferent types of spinors, and an orientable object is characterized by tenquantum numbers. We study the corresponding relativistic wave equations andtheir symmetry properties.

Field on Poincare group and quantum description of orientable objects