Abstract
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.
Highlights
Let us review briefly the identification of the physical degrees of freedom in quantum electrodynamics in four spacetime dimensions (QED4) at both zero and finite temperature where the interaction between matter and radiation is synthesized in the following Lagrangian [2]
Χ(1), χ(1), χ(2), χ(2), φ1, φ2, φ3, Σ1, Σ2, Σ3 (22 constraints). This relationship between constraints and physical degrees of freedom is crucial, and it is reflected both in its quantization of the generalized scalar Duffin-Kemmer-Petiau quantum electrodynamics (GSDKP), as we have seen in the previous section, and in describing its thermodynamic equilibrium, by means of the free energy and the equipartition theorem, as we shall see
Because we are concerned with the study of thermodynamic equilibrium, the above equations become stationary in the sense that they do not depend on time
Summary
One of the most important parts in the analysis of physical theories is the distinction between what is measurable and what is not due to the fact that we usually use non-physical objects to describe the world [1]. We know that when we are dealing with physical degrees of freedom in the configuration space we have 4 wave equations of second order for the propagation of the energy of particles and anti-particles, the equivalence between the dynamics in the physical degrees of freedom is maintained and no additional constraints must be added. This is, a consequence of the second-class nature of the spinorial sector where gauge fixing conditions are not neccessary.
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