Quantum kinetic theory for spin-1/2 fermions in Wigner function formalism

Relativistic heavy-ion collisions provide a unique chance to create a new state of strong interaction matter called the quark-gluon plasma (QGP) which is a deconfined phase of nuclear matter under extreme conditions of high temperatures and densities. The complexity in relativistic heavy-ion collisions lies in the fact that the dynamical processes happen at very small space-time scale of the order 10 fm. Global properties of relativistic heavy-ion collisions can be well described by thermodynamic and hydrodynamic models which assume fast local equilibrium after collisions. However non-equilibrium and quantum effects may play an important role in such violent and microscopic processes at small time-scale. For example, coherent gluon fields called the color glass condensate may provide an initial condition for the formation of QGP far from equilibrium[1, 2, 3, 4, 5] . To investigate how these effects influence experimental observables is an important task which can be achieved with the help of the quantum transport theory.

Since 1980s, there have been tremendous efforts in constructing a quantum transport theory for the QGP in order to describe the non-equilibrium and quantum phenomena [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. These works can be grouped into those about Abelian gauge fields [18, 21, 25, 26, 27, 30, 31, 32] and non-Abelian gauge fields [6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 22, 24, 29]. Most of these works are based on covariant Wigner functions while a part of them are formulated in equal-time Wigner functions [25, 26, 27, 29, 30, 31, 32, 33]. For applications in astrophysics a generalization of Wigner functions in the curved space-time was made [13, 21, 24, 28, 34]. The reviews on these earlier devolvements can be found in Refs. 35, 36. These works was done in the era before RHIC was run at BNL. However, the Wigner function formalism has not been put into simulations in rigorous way to describe data at RHIC or LHC. One reason is that the formalism is rather complicated in which multiple components are entangled with each other and the transport equation are highly constrained. This made it very difficult to be solved in numerical simulation. Another reason is that most results on collective flows at RHIC or LHC energies can be well described by relativistic hydrodynamics with proper initial conditions [37, 38, 39, 40, 41, 42, 43, 44].

In recent years, the theoretical prediction [45, 46, 47] and the experimental observation by STAR Collaboration at RHIC [48, 49] of global polarization effect (GPE) have drawn great attention in the field, both experimentally and theoretically (see e.g. recent short reviews [50, 51, 52]). A series of other quantum effects in related aspects in heavy ion collisions have been extensively studied, such as the chiral magnetic effect (CME) [53, 54, 55], the chiral vortical effect (CVE) [56, 57, 58, 59], the chiral separation effect (CSE) [60, 61], and the local polarization effect (LPE) [62]. This opens a new window to probe the nature of the QGP in a different perspective. Since all these effects are associated with the particle’s spin degrees of freedom, the conventional relativistic hydrodynamics or Vlasov-Boltzmann equation have to be generalized to describe these quantum effects in a consistent way. The quantum transport theory is the appropriate tool for this goal.

The quantum kinetic theory based on Wigner functions was proposed to derive the CME, CVE, CSE and LPE in the Ref. 62, which is a success in describing these effects in a natural and consistent way. Since then, there have been a large number of works along this line on various chiral or spin effects in relativistic heavy ion collisions, condensed matter physics, and astrophysics. In this article, we will give an overview for recent developments of the Wigner function formalism [63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 85, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96]. Especially, we will focus on (a) Disentanglement of covariant Wigner functions into a single distribution function which satisfies a single kinetic equation for massless fermions or one distribution function and three spin distribution functions which satisfy four kinetic equations for massive fermions in background electromagnetic fields; (b)Derivation of various chiral and spin effects such as chiral anomaly, CME, CVE, etc.. There are a number of interesting topics that we do not cover in this review article, e.g., the smooth transition from the spin kinetic theory for massive fermions to chiral kinetic theory for massless fermions [78, 79, 80, 87, 89], the quantum kinetic theory in curved space [13, 20, 21, 24, 28, 34, 75, 83], the chiral kinetic equation in relaxation time approximation [71, 72], the chiral kinetic equation in strong magnetic field [82, 85], the chiral kinetic equation with vorticity or in rotating frame [63, 81, 88, 96, 97], and derivation of the spin hydrodynamics from Wigner equations [92, 93, 94, 91]. Also the numerical simulation of the chiral kinetic equation [98, 99, 100, 101] is not discussed. We refer readers who are interesting in these topics to the references listed above. In addition to the Wigner function approach, these chiral and spin effects can also be studied in other approaches, such as the AdS/CFT duality [102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112], relativistic anomalous and spin hydrodynamics [113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125], quantum field theories [54, 55, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139]. The chiral kinetic equation can also be derived from other methods such as semiclassical approaches [140, 141, 142, 143, 144, 145, 146, 147], effective field theories [148, 149, 150, 151], and the world-line formalism [152, 153, 154]. The reviews on these chiral and spin effects can be found in Refs. 155, 156, 157, 158, 51 .

The article is organized as follows. In Section 2, we review the Lorentz covariant and gauge invariant quantum transport theory for spin-1/2 fermions in background Abelian gauge fields. The spinor decomposition and semiclassical expansion are presented. This is the starting point for the following sections. In Section 3, we give a systematic method to disentangle multicomponent Wigner funcitons and their equations into independent distribution functions and their kinetic equations so that the number of degrees of freedom is greatly reduced. We will discuss the derivation of Wigner functions for massless and massive fermions separately. In Section 4, we present a consistent way of describing various chiral and spin effects in covariant Wigner functions. The summary and outlook are made in the final section.

We adopt following notation and conventions: $a\cdot b=a^{\mu}b_{\mu}$, $g_{\mu\nu}=\mathrm{diag}(+,-,-,-)$, $\epsilon^{0123}=-\epsilon_{0123}=1$, $\gamma_{5}=\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$, $\sigma^{\mu\nu}=(i/2)[\gamma^{\mu},\gamma^{\nu}]$, and summation over repeated indices is implied if not stated explicitly.

The classical transport theory describes the time evolution of the distribution $f(t,{\bf x},{\bf p})$, a measure of the particle number density in phase space, which satisfies the Vlasov-Boltzmann equation,

where ${\bf v}={\bf p}/E_{p}$ denotes the particle’s velocity with mass $m$ and energy $E_{p}=\sqrt{{\bf p}^{2}+m^{2}}$, ${\bf E}$ and ${\bf B}$ are the electric field and magnetic field respectively, and ${\cal C}[f]$ denotes collision terms. For notational simplicity, we have already absorbed the electric charge ‘$e$’ into ${\bf E}$ and ${\bf B}$.

In quantum theory, we use the Wigner function $W(x,p)$, a quantum anologue of $f(t,{\bf x},{\bf p})$, first introduced by E. Wigner in Ref. 159. In quantum electrodynamics, the Lorentz covariant Wigner function is defined as an ensemble average of the Wigner operator [18]

where $\psi_{\alpha}$ and $\bar{\psi}_{\beta}$ are Dirac fields with $\alpha,\beta$ running from 1 to 4 in spinor space and the phase factor $U$ is the so-called Wilson line or gauge link

which ensures the gauge invariance of the Wigner function. The operator ${\cal P}$ denotes a certain path ordering with respect to the parameter $s$. Under the requirement that the Wigner function measures the particle density with the physical kinetic momentum $p$, the path must be a straight line. Again we have absorbed the electric charge into the gauge potential $A_{\mu}$. For simplicity, we will restrain ourselves to the case of classical background fields that the gauge potential $A_{\mu}$ and field strength tensor $F_{\mu\nu}$ are classical functions of time and space. In this case the path ordering operator ${\cal P}$ is just a unit operator. In the background field approximation, there is no issue of the so-called BBGKY-hierarchy [160].

We note in particular that we define the ensemble average $\langle\cdots\rangle$ in Eq. (2) without taking the normal ordering of Dirac fields because the Dirac equation is satisfied by the Wigner function, but the normal ordering will destroy such a property. This is crucial [86] to reproduce the correct and universal coefficient of chiral anomaly and we will come to this point in Sec. 4.1.

In this section, we give a brief review on the main results presented in Ref. 18, which is the starting point for the quantum transport theory we are going to address in this review article. The covariant Wigner function is a $4\times 4$ matrix and can be expanded in terms of the 16 independent generators of the Clifford algebra

where all coefficients are real functions due to the property $W^{\dagger}=\gamma^{0}W\gamma^{0}$ and can be determined by taking traces with proper generators

In background fields, the equations of motion for Wigner functions can be derived from the Dirac equation,

where the operators $K^{\mu}$, $G^{\mu}$ and $\Pi^{\mu}$ are defined by

In the above, $j_{0}(z)$ and $j_{1}(z)$ are the zeroth and first order spherical Bessel functions, respectively, and the triangle operator $\Delta\equiv\partial_{p}\cdot\partial_{x}$ denotes the mixed derivative. Note that $\partial_{x}$ in the operator $\Delta$ acts only on $F^{\mu\nu}$ to its right but not on other functions. Substituting the decomposition (4) into Eq. (10), we obtain coupled equations for components of the Wigner function

These equations can be further decomposed into real and imaginary parts. The real parts of the above equations read

which are linearly proportional to the particle mass on the right-hand side of the equations, while the imaginary parts read

without explicit mass dependence.

We see that there are 16 components of the Wigner function which satisfy above 32 equations. In this sense quantum transport theory is much more complicated than classical one in which only the phase space distribution $f(t,{\bf x},{\bf p})$ and the corresponding Vlasov-Boltzmann equation (1) are involved. Therefore it is important to disentangle the coupled equations for different components of the Wigner function as much as possible to reduce the number of independent variables and equations. The key method is to use the semiclassical expansion in the Planck constant in which all operators and functions are expanded in powers of $\hbar$. The expansions of the Wigner function $W$ and the operators $\Pi^{\mu}$ and $G^{\mu}$ are

where $k$ denote nonnegative integers, $W$ can be replaced of any of its components: $\mathscr{F}$, $\mathscr{P}$, $\mathscr{V}_{\mu}$, $\mathscr{A}_{\mu}$ and $\mathscr{S}_{\mu\nu}$. The expansion of $\Pi^{\mu}$ and $G^{\mu}$ can be obtained by an expansion of Bessel functions and contains only even-order terms

To $O(\hbar^{2})$ these operator are

We note that $\hbar$ expansion of Bessel functions is equivalent to expansion in $\Delta$ for the operators $\Pi^{\mu}$ and $G^{\mu}$, which implies that the condition for the validity of the expansion is

where $\Delta R_{F}$ denotes the spatial scale for the variation of the electromagnetic field $F^{\mu\nu}(x)$, while $\Delta P_{W}$ denotes the momentum scale for the variation of the Wigner function.

The quantum transport theory in covariant Wigner functions are characterized by a set of coupled equations for their components. However one can show that the number of independent functions and the corresponding equations can be reduced in semiclassical expansion. In this section, we will show how to disentangle the components of Wigner functions and their equations in semiclassical expansion in powers of $\hbar$. We will first look at the case of massless fermions and then that of massive fermions. The results for massless fermions are mainly from Refs. 70 and 86 while those for massive fermions are from Refs. 18 and 76.