Descriptions of Relativistic Dynamics with World Line Condition

In this paper, a generalized form of relativistic dynamics is presented. A realization of the Poincaré algebra is provided in terms of vector fields on the tangent bundle of a simultaneity surface in R 4 . The construction of this realization is explicitly shown to clarify the role of the commutation relations of the Poincaré algebra versus their description in terms of Poisson brackets in the no-interaction theorem. Moreover, a geometrical analysis of the “eleventh generator” formalism introduced by Sudarshan and Mukunda is outlined, this formalism being at the basis of many proposals which evaded the no-interaction theorem.

Algebra Versus Its Anti-symmetric Algebra

For a given algebra A= 〈A,+,·〉, we can define its anti-symmetric algebra A-= 〈A-,+,[ , ]〉 using the commutator [ , ] of A, where the sets A and A- are the same. We show that there are isomorphic algebras A1 and A2 such that their anti-symmetric algebras are not isomorphic. We define a special type Lie algebra and show that it is simple.