| dc.description.abstract | This thesis presents a systematic study of internal structures possible for a classical relativistic indecomposable particle without spatial extension. Group theoretic and differential geometric methods are used and the Lagrangian formalism of dynamics adopted. Based on the assumed indecomposability, the internal spaces Q are kinematically classified in terms of the coset spaces of the group SL(2, C) with respect to its distinct closed continuous subgroups. The allowed spaces Q are separated into First Order Spaces (FOS's) (of which there are two), and Second Order Spaces (SOS's) (of which there are ten distinct ones and two one-parameter families). The former Q’s admit an SL(2, C)-invariant symplectic structure while the latter do not. A search is then made for those cases in which Lagrangians can be constructed involving nontrivial coupling between the space-time coordinates and the internal variables. The resulting Lagrangians describe Regge trajectories, i.e., a family of particles characterized by a mass-spin relationship, in the classical limit.
In the introductory Chapter I, the motivations for studying such internal structures and a review of the earlier work in this subject are presented.
The physical principles underlying the study are enumerated in Chapter II. These include indecomposability, Poincaré symmetry, and reparametrization invariance of the action. The last mentioned leads, in the canonical formalism, to a primary constraint on the phase space variables which under suitable conditions is the mass-spin relationship. The separation of the Q’s into FOS's and SOS's is carried out. The forms of the most general Lagrangians, in both cases, are then described in terms of a maximal set of independent Lorentz scalars constructed out of the space-time and internal invariants.
FOS's are studied in Chapter III. They are related to orbits in the Lie algebra of SL(2, C). Their topological structure and properties are analyzed and a Lagrangian description for particles with such internal spaces is shown to exist in both the cases. The similarities and differences in the description of the (constrained) Hamiltonian dynamics and the space-time trajectories for the two cases are highlighted. An interesting feature distinguishing them is brought out by means of a calculation of the magnetic moment for particles lying on the Regge trajectory.
The SOS's are taken up in Chapter IV. They are separated into those which admit a manifestly covariant description and those which do not. The former are easily seen to allow a Lagrangian description of dynamics with nontrivial coupling between the space-time and internal variables. The latter are analyzed using the concept and methods of the isotopy representation which are set up in detail. It is shown that in three cases it is impossible to couple the internal and the space-time variables in a physically interesting way. Our study reveals that two other cases can be partly, but not completely, described in a manifestly covariant way. An example of the manifestly covariant spaces is worked out displaying the structure of the Lagrangian.
Chapter V presents a coordinate-independent differential geometric description of the work of the previous chapters. This is done to provide a different way of looking at the various concepts and to develop a global point of view.
In the concluding Chapter VI, the various models existing in the literature are discussed and related to the general classification scheme presented in the thesis. Some interesting problems worth pursuing in the field are also pointed out.
In Appendix A, the closed continuous subgroups of SL(2, C) (up to conjugation) are enumerated. In Appendix B, we give a simple-minded proof using local coordinates of an important theorem of Kostant-Kirillov-Souriau which has been used in Chapter III. Some results in the geometry of Lie groups and Lie algebras, necessary for Chapter V, are given in Appendix C. | |
