Clifford Algebras and Lorentz Group

Finite-dimensional representations of the proper orthochronous Lorentz group are studied in terms of spinor representations of the Clifford algebras. The Clifford algebras are understood as an `algebraic covering' of a full system of the finite-dimensional representations of the Lorentz group. Space-time discrete symmetries P, T and PT, represented by fundamental automorphisms of the Clifford algebras, are defined on all the representation spaces. Real, complex, quaternionic and octonionic representations of the Lorentz group are considered. Physical fields of the different types are formulated within such representations. The Atiyah-Bott-Shapiro periodicity is defined on the Lorentz group. It is shown that modulo 2 and modulo 8 periodicities of the Clifford algebras allow to take a new look at the de Broglie-Jordan neutrino theory of light and the Gell-Mann-Ne'emann eightfold way in particle physics. On the representation spaces the charge conjugation C is represented by a pseudoautomorphism of the complex Clifford algebra. Quotient representations of the Lorentz group are introduced. It is shown that quotient representations are the most suitable for description of the massless physical fields. By way of example, neutrino field is described via the simplest quotient representation. Weyl-Hestenes equations for neutrino field are given.