  This work explores the quantum dynamics of the interaction between scalar
(matter) and vectorial (intermediate) particles and studies their thermodynamic
equilibrium in the grand-canonical ensemble. The aim of the article is to
clarify the connection between the physical degrees of freedom of a theory in
both the quantization process and the description of the thermodynamic
equilibrium, in which we see an intimate connection between physical degrees of
freedom, Gibbs free energy and the equipartition theorem. We have split the
work into two sections. First, we analyze the quantum interaction in the
context of the generalized scalar Duffin-Kemmer-Petiau quantum electrodynamics
(GSDKP) by using the functional formalism. We build the Hamiltonian structure
following the Dirac methodology, apply the Faddeev-Senjanovic procedure to
obtain the transition amplitude in the generalized Coulomb gauge and, finally,
use the Faddeev-Popov-DeWitt method to write the amplitude in covariant form in
the no-mixing gauge. Subsequently, we exclusively use the Matsubara-Fradkin
(MF) formalism in order to describe fields in thermodynamical equilibrium. The
corresponding equations in thermodynamic equilibrium for the scalar, vectorial
and ghost sectors are explicitly constructed from which the extraction of the
partition function is straightforward. It is in the construction of the
vectorial sector that the emergence and importance of the ghost fields are
revealed: they eliminate the extra non-physical degrees of freedom of the
vectorial sector thus maintaining the physical degrees of freedom.
