Classical particles with internal structure. II. Second-order internal spaces

Abstract

A systematic study of classical relativistic particles with internal structure, initiated in a previous paper, is continued and a study of second-order internal spaces (SOS) is presented within the framework of the Lagrangian form of constrained dynamics. Such internal spaces Q are those for which a phase-space treatment must necessarily use the cotangent bundle T^∗Q. The large variety of possible SOS's-ten discrete cases and two one-parameter families-is separated into those capable of a manifestly covariant description, and those for which special methods based on the transitive action of SL(2,C) on a coset space are needed. The concept of the isotopy representation plays an important role in this context. Seven of the possible discrete SOS's are shown to be describable in a manifestly covariant way; two discrete and one one-parameter family of SOS's are shown to be unphysical in the sense that no Lagrangians can be written in which the internal and the space-time position variables are nontrivially coupled; and the remaining single discrete and one one-parameter family are shown to be physical though not describable in a manifestly covariant way. General phase-space methods uniformly applicable to all SOS's are developed; and as an illustrative example a Lagrangian model for an SOS in which the internal space is spanned by two orthonormal spacelike unit vectors is presented.