ABSTRACT
This chapter reveals the relation between supernumber spaces and physics. It describes the construction of a symmetry mixing bosonic and fermionic fields. Superalgebras present a natural generalization of the concept of Lie algebras. Another algebraic operation, often used in applications, is anticommutation of operators. For example, to formulate particle dynamics in quantum field theory, one has to choose commutation relations for integer spin particles and anticommutation relations for half-integer spin particles. There are three basic differences between ordinary Lie algebras and super Lie algebras. First, Lie algebras are endowed with the operation of multiplication by ordinary numbers only, while for super Lie algebras it is defined for arbitrary c-numbers. Secondly, generators of Lie algebras satisfy commutation relations, while generators of super Lie algebras are subject to commutation relations. Finally, generators of a Lie algebra are elements of the algebra. The chapter focuses on massless unitary representations of the Poincare superalgebra.
