Applications Of Wigner Function Method In Quark Matter

The scientific goal of high energy heavy-ion collisions at RHIC and LHC is to search for and study the quark-gluon plasma,a new state of matter in strong interaction which may have existed in the early universe.In non-central collisions,a huge orbital angular momentum as well as a strong magnetic field are produced along the same direction.How such a huge orbital angular momentum is distributed into the hot and dense matter is the core question we are going to answer.What is the effect of the strong magnetic field on the behavior of particles is another associated question.It was proposed that the huge angular momentum is converted to the spin polarization of quarks through the spin-orbital coupling.The local shear flow of the fluid which gives rise to the vorticity will polarize the quarks in the fluid.In the later stage of the collisions,quarks and gluons will experience hadronization,through which the global polarization of quarks will be transferred to the polarization of hadrons along the direction of the orbital angular momentum.So we can explore the vortical structure of the hot and dense matter through the measurement of the hadron polarization.The strong magnetic field can lead to chiral magnetic effect which is closely related to the chiral anomaly.The experimentalists are looking for the observables of the chiral magnetic effect at RHIC and LHC.The Wigner function method is a powerful tool to study the polarization effect as well as the chiral magnetic effect.The Wigner function is the quantum mechanical ana-logue of the single-particle distribution in the classical theory.Due to the uncertainty principle in quantum mechanics,the phase space variables of the Wigner function do not represent the position or momentum of one particle.We introduce the definition of the Wigner function in non-relativistic quantum mechanics as well as in quantum field theory.The ensemble average of Wigner function operator is related to the macroscop-ic quantities in thermodynamical equilibrium which can be expressed by the integra-tion over the phase space.We investigate the Wigner function for the Dirac field in a background electromagnetic field in details.We derive the equation of motions for the Wigner function and those for its sixteen components.The main purpose of the thesis is to study the polarization effect from vorticity and the the pseudoscalar condensate induced by anomaly and vorticity in the Wigner function approach.We have extended our previous works on the Wigner function for chiral or mass-less fermions to that for massive fermions.The Wigner function at the leading order is derived from its definition by setting the gauge link to 1 and by expanding the free form of the fermionic quantum fields in momentum space.Then all components of the Wigner function can be extracted by projecting the corresponding Dirac matrices and taking traces.The axial vector component at the next-to-leading order for massive fermions can be obtained by extending that for massless fermions and satisfies the re-quired equations.We have shown that the axial vector component behaves like a spin pseudo-vector in phase space up to a factor 1/2 non-relativistic case.The polarization density can be computed by integration of the axial vector component over momentum.We have found that the polarization per particle for fermions is always less than that for anti-fermions due to the finite baryonic chemical potential,which is qualitatively con-sistent with results of the Lambda and anti-Lambda polarization measured bt the STAR collaboration.We have also formulated the polarization per particle for fermions with the specific momentum on the Cooper-Frye freezeout hypersurface in a hydrodynamic description,which is consistent to the previous result of Becattini et al..We derive the pseudoscalar condensate induced by the anomaly and vorticity from the Wigner function for massive fermions in homogeneous electromagnetic fields.The pseudoscalar component of the Wigner function can be obtained from the axial vector component.By directly calculating the space-time divergence of the chiral current,we can determine the pseudoscalar condensate which has an anomalous E · B term and an E · ? term.The E · ? term can also be regarded as a force-vorticity coupling since there is a sign difference in its prefactor between the fermion and antifermion sector.The force-vorticity part of the pseudoscalar condensate is the new term.As a mass effect,the pseudoscalar condensate is linearly proportinal to the fermion mass when the mass is small.Such a pseudoscalar condensate is a general feature for a fluid of massive and charged fermions in a thermal and dense plasma with anomaly and vorticity.The neutral pion and eta meson condensates can also be derived from generalization of the single flavor to multi-flavor case,which depend on quark masses,quark chemical potentials and temperature.We can reproduce the previous result of the neutral pion condensate in vacuum induced by the anomaly,and our result also has a force-vorticity part.There are possible observables of pseudoscalar condensates related to the electromagnetic field and vorticity in heavy ion collisions such as collective flows of neutral pions and eta mesons.