Symplectic Density Matrix

Abstract

The quasi amplitude of probability, a concept associated with the Wigner function, is used to introduce the notion of symplectic density matrix, $$\rho$$ . The Liouville-Von Neumann equation is derived in phase space by using symmetry. The physical content of $$\rho$$ is analyzed in detail, in particular with the study of the classical limit. The first application addresses the problem of quark-antiquark interaction, described by the so-called Cornell potential. The matrix $$\rho$$ is calculated, providing a phase space description of the quark-antiquark state. For the model considered here, the quark confinement is derived, such that the distance of confinement is about 4 GeV $$^{-1}$$ . This result improves theoretical calculations in the literature, approaching to the order of experimental results. Another application is the calculation of $$\rho$$ considering mixed state systems. Since $$\rho$$ is a density of probability, the entropy is defined without ambiguity, and as such the Gibbs ensemble is introduced for a quantum system in the phase space representation. Then a harmonic oscillator system at finite temperature is studied. The second mixed state system is a maximally entangled bipartite state. In these applications, when it is possible, the nature of $$\rho$$ is compared with the Wigner function.