  We propose a universal group theoretic description of the fermion production
through any type of interaction to scalar or pseudo-scalar. Our group theoretic
approach relies on the group $SU(2) \times U(1)$, corresponding to the freedom
in choosing representations of the gamma matrices in Clifford algebra, under
which a part of the Dirac spinor function transforms like a fundamental
representation. In terms of a new $SO(3)$ ($\sim SU(2)$) vector constructed out
of spinor functions, we show that fermion production mechanism can be analogous
to the classical dynamics of a vector precessing with the angular velocity. In
our group theoretic approach, the equation of motion takes a universal form for
any system, and choosing a different type of interaction or a different basis
amounts to selecting the corresponding angular velocity. The expression of the
particle number density is greatly simplified, compared to the traditional
approach, and it provides us with a simple geometric interpretation of the
fermion production dynamics. For the purpose of the demonstration, we focus on
the fermion production through the derivative coupling to the pseudo-scalar.
