Quantum Kinetic Equation With Arbitrary Mass For Fermions

Spin is one of the most fundamental degrees of freedom of elementary particles and plays a very important role in the modern physics.In recent years,there has been a considerable amount of work on studying the spin physics in relativistic heavy ion collisions.Lots of strange effects were emerged in that process,for example,chiral magnetic effect(CME),chiral vortex effect(CVE),global polar-ization effect(GPE)and so on.These spin effects can reveal the deeper and finer properties of the theory and provide us with a new experimental probe to study quantum chromodynamics.The complexity in relativistic heavy ion collisions is that the quark gluon plasma produced by collisions is not a static equilibrium system but a transient non-equilibrium system.The quantum kinetic theory is an ideal theoretical tool to describe both spin effect and non-equilibrium effect.The quantum kinetic theory for massless fermions—chiral kinetic theory has been well established through great efforts in the past decade.In recent years,great progress has been made in the quantum kinetic theory for massive fermions.However,different from the chiral kinetic equations at chiral limit,the quantum kinetic equations for massive fermions have been derived in many different forms.Hence it is still very worthwhile to obtain a quantum kinetic equation for fermions with arbitrary mass,which can be reduced into the chiral kinetic equations in the massless limit as much as possible.The quantum kinetic theory with arbitrary mass is supposed to be able to achieve smooth transition from massive fermion to massless fermion,which is very significant to understand and study the kinetic evolution in the process of chiral spontaneous breaking during the relativistic heavy ion collisions.The quantum transport theory based on the Wigner function is self-consistent formalism and is derived from the first principle—quantum field theory.For Dirac fermion system,the Wigner function has 16 independent components which satisfy the 32 Wigner equations and they can be classified into the scalar,pseudo-scalar,vector,axial-vector and antisymmetric tensor.In recent years,using this Wigner function approach,the chiral kinetic theory of massless fermions in the background field had been successfully derived.The quantum kinetic theory for the massive fermions has also been given in this approach but the equations can be put in very different forms.In our dissertation,we will also employ the Wigner function approach to derive quantum kinetic equation in another different form,which appears very similar to the chiral kinetic equation and can be regarded as a more natural generalization.Firstly,in order to keep consistent with the chiral kinetic equation as much as possible,we also decompose the original Wigner equation into the right-handed/left-handed part.We find that the right-handed and left-handed parts do not decouple with each other which is different from the chiral kinetic theory.These functions are entangled with other Wigner functions.Similar to the mass-less case,we also decompose both the right-handed/left-handed Wigner functions and antisymmetric Wigner functions into time-like part and space-like part and rewrite the Wigner equations in time-like and space-like forms correspondingly.Next,we choose the time-like component of right-hand/left-hand Wigner func-tion Jsn(s?±1)as two basic independent variable and find that these two left-handed and right-handed distribution functions can not make the Wigner equations closed by themselves and we need at least two extra Wigner functions to close the equations.In this dissertation,we can choose the magnetic-moment components of anti-symmetry tensor Wigner functioc M? as another basic in-dependent component,which can be demonstrated after we calculate the zeroth and first order Wigner equation by semi-classical expansion.We find that 12 equations of 32 Wigner equations just express the other 12 Wigner functions as the function of the basic variables.It is remarkable that another 12 Wigner equa-tions are satisfied automatically.We are only left with 4 basic Wigner function which satisfies the remained 8 Wigner functions.At last,the component of the M? can be reduced by considering the relation-ship of Jsn(s=±1)and?M? and the final independent variables turn out to be Jsn(s=±1)and M??.M?? is the transverse magnetic-moment components of anti-symmetry tensor.From the on-shell conditions,we can further express the original Wigner function by the terms proportional to Dirac ? function and its derivatives.We choose the 4 unknown functions fs and M?? before the Dirac? function as the final basic variables corresponding to the original basic vari-ables Jsn and M??,respectively.Then we get the zeroth order kinetic equation and first order kinetic equation for fs and respectively.The total kinetic equation can be obtained by summing them up.With these final basic variables,we have put the quantum kinetic equation for fermions with arbitrary mass in 8-dimensional phase space into very similar form to chiral kinetic equation.This quantum kinetic equation in 8 phase space can achieve a smooth transition from massive fermions to massless fermions.We also show the explicit expression of other Wigner function in the form of the final basic function fs and M??.The key mathematical trick in our work is semi-classical expansion about h.The final Wigner function in 8-dimension phase space is calculated up to the first order of h.However in order to obtain the Wigner equations up to the first order,we need to calculate some Wigner function up to the second order of h.