Poincaré Group. Relativistic Theories

This chapter is devoted to relativistic theories in algebraic approach.In these theories, the Poincar´e group acts as a group of automorphisms on associative algebra with involution A. Assuming that A is asymptotically commutative we prove that the scattering matrix of elementary excitations of Poincar´e invariant state is Lorentz-invariant. We discuss the classification of unitary representations of Poincar´e group and show that in the algebraic approach,one can use this classification to construct theories corresponding to free field theories without using the notion of field. For scalar bosons, electrons,1 and photons we relate this construction to the standard construction in terms of the Klein-Gordon equation, Dirac equation, and Maxwell equations. We review shortly the theory of interacting fields, in particular, quantum electrodynamics.For local theories the algebraic approach can be based Araki-Haag-Kastler axioms. In these axioms one assigns an algebra to every bounded domain in Minkowski space. The union of these algebras is strongly asymptotically commutative if masses in the theory are bounded below by a positive number ( there exists a mass gap).