  The transition from a classical to quantum theory is investigated within the
context of orthogonal and symplectic Clifford algebras, first for particles,
and then for fields. It is shown that the generators of Clifford algebras have
the role of quantum mechanical operators that satisfy the Heisenberg equations
of motion. For quadratic Hamiltonians, the latter equations are obtained from
the classical equations of motion, rewritten in terms of the phase space
coordinates and the corresponding basis vectors. Then, assuming that such
equations hold for arbitrary path, i.e., that coordinates and momenta are
undetermined, we arrive at the equations that contains basis vectors and their
time derivatives only. According to this approach, quantization of a classical
theory, formulated in phase space, is replacement of the phase space variables
with the corresponding basis vectors (operators). The basis vectors,
transformed into the Witt basis, satisfy the bosonic or fermionic
(anti)commutation relations, and can create spinor states of all minimal left
ideals of the corresponding Clifford algebra. We consider some specific actions
for point particles and fields, formulated in terms of commuting and/or
anticommuting phase space variables, together with the corresponding symplectic
or orthogonal basis vectors. Finally we discuss why such approach could be
useful for grand unification and quantum gravity.
