The analytic structure of the quark propagator in Minkowski space is more complex than in Euclidean space due to the possible existence of poles and branch cuts at timelike momenta. These singularities impose enormous complications on the numerical treatment of the nonperturbative Dyson-Schwinger equation for the quark propagator. Here we discuss a computational method that avoids most of these complications. The method makes use of the spectral representation of the propagator and of its inverse. The use of spectral functions allows one to handle in exact manner poles and branch cuts in momentum integrals. We obtain model-independent integral equations for the spectral functions and perform their renormalization by employing a momentum-subtraction scheme. We discuss an algorithm for solving numerically the integral equations and present explicit calculations in a schematic model for the quark-gluon scattering kernel.

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Using elements of symmetry, as gauge invariance, aspects of field theories represented in symplectic space are introduced and analyzed under physical bases. The states of a system are described by symplectic wave functions, which are associated with the Wigner function. Such wave functions are vectors in a Hilbert space introduced from the cotangent-bundle of the Minkowski space. The symplectic Klein-Gordon and the Dirac equations are derived, and a minimum coupling is considered in order to analyze the Landau problem in phase space.

In this work we analyse the Casimir effect for bosons and fermions on a hypertorus for both cases at zero and finite temperature. We find that at zero temperature there is a transition from negative to positive values of pressure which depends heavily on the magnitude of the compactification parameters. When low temperatures are considered, a change in the relation between the pressure behaviour and the compactification parameters is observed. In contrast, at high energies, due to the dominance of the black-body radiation term in the pressure, this change becomes insignificant. We then use these results to consider a non-interacting massless QCD model and estimate the possible contribution of the Casimir effect for the critical temperature of quarks deconfinement.

In this paper, we explore a bi-dimensional nonrelativistic strong interaction system that represent the bound state of heavy quark-antiquark, where we consider a Cornell potential which it has Coulombian and linear terms. For this purpose, we solve the Schrödinger equation in the phase space with the linear potential. The eigenfunction found is associated with the Wigner function via Weyl product using the representation theory of Galilei group in the phase space. We find that the Wigner function offers an easier way to visualize the properties of cc, bb, bc mesons systems than the wavefunction does.

In this paper, we study within the structure of Symplectic Quantum Mechanics a bidimensional nonrelativistic strong interaction system which represent the bound state of heavy quark-antiquark, where we consider a Cornell potential which consists of Coulomb-type plus linear potentials. First, we solve the Schrödinger equation in the phase space with the linear potential. The solution (ground state) is obtained and analyzed by means of the Wigner function related to Airy function for the c c ¯ meson. In the second case, to treat the Schrödinger-like equation in the phase space, a procedure based on the Bohlin transformation is presented and applied to the Cornell potential. In this case, the system is separated into two parts, one analogous to the oscillator and the other we treat using perturbation method. Then, we quantized the Hamiltonian with the aid of stars operators in the phase space representation so that we can determine through the algebraic method the eigenfunctions of the undisturbed Hamiltonian (oscillator solution), and the other part of the Hamiltonian was the perturbation method. The eigenfunctions found (undisturbed plus disturbed) are associated with the Wigner function via Weyl product using the representation theory of Galilei group in the phase space. The Wigner function is analyzed, and the nonclassicality of ground state and first excited state is studied by the nonclassicality indicator or negativity parameter of the Wigner function for this system. In some aspects, we observe that the Wigner function offers an easier way to visualize the nonclassic nature of meson system than the wavefunction does phase space.