  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.
