Semiclassical transport equations of Dirac particles in rotating frames

We establish covariant semiclassical transport equations of massive spin-1/2 particles which are generated by the quantum kinetic equation modified by enthalpy current dependent terms. The purpose of modification is to take into account the noninertial properties due to the angular velocity of rotating frame which is equivalent to the fluid vorticity in hydrodynamical approach. We present the equations satisfied by the Wigner function components and by studying their solution in the semiclassical approximation we accomplish the transport equations. To acquire a three-dimensional kinetic theory, the relativistic kinetic equations in the comoving frame are integrated over the zeroth component of four-momentum. The resulting vector and axial-vector currents are calculated at zero temperature. There exists another three-dimensional formulation of Dirac particles which correctly addresses the noninertial features of rotating coordinates. We review it briefly and obtain the mass corrections to the chiral vector and axial-vector currents produced by this formulation.


I. INTRODUCTION
Quark-gluon plasma created in heavy-ion collisions form a novel phase of nuclear matter. The constituent quarks and gluons are deconfined, where the quarks can be considered as massless, hence can be right-or left-handed. In the systems composed of chiral fermions collective behaviour generates the vector and axial-vector current terms proportional to magnetic field known as the chiral magnetic effect [1][2][3] and the chiral separation effect [4,5]. In currents, there also exist terms which are proportional to the vorticity. They lead to the chiral vortical effect [6] and the local (spin) polarization effect [7][8][9]. In these effects magnetic field and the vorticity of fluids play similar roles. In heavy-ion collisions vortical effects generate the polarization of Λ as measured in [10]. The dynamics of chiral fermions can be investigated by means of the covariant quantum kinetic equation in Minkowski space [11,12]. However, this transport equation lacks the similarity between the magnetic field and the vorticity. It depends explicitly on electromagnetic field strength but the vorticity is incorporated in it through the fluid velocity [9,13,14] and the equilibrium distribution function [15,16]. Moreover, in the Wigner function approach noninertial effects like the Coriolis force, do not appear. To take into account vorticity from the start, in [17] we proposed and studied a modified quantum kinetic equation for chiral fermions.
To solve a transport equation as the initial-value problem to guarantee that the distribution function is acceptable, three-dimensional (3D) kinetic equation is preferred [18][19][20]. A methodical approach of introducing a 3D kinetic theory is to integrate the four-dimensional (4D) one over the zeroth component of four-momentum, p 0 [19,21]. To perform this calculation one should choose a frame, but not each possible choice yields an acceptable 3D kinetic theory. By integrating the 4D transport theory proposed in [17] we accomplished the 3D kinetic theory which is consistent with the chiral phenomena. It is the unique consistent 3D semiclassical chiral kinetic equation which does not depend explicitly on the 3D position vector x, and possesses the Coriolis force. There exists a similar 3D formulation [22] which has been established from the relativistic theory given in curved spacetime [23] by integrating over p 0 , but it depends on x explicitly. A different approach of defining relativistic kinetic equations and the related 3D transport theories were discussed in [24,25]. A consistent nonrelativistic kinetic formulation of chiral fermions both in the presence of electromagnetic fields and global rotations was given in [26] by working directly in 3D. Its starting point is the equivalence between fluid vorticity in hydrodynamical approach and angular velocity of fluid in the comoving frame [27]. The kinetic theory of [26] also depends position explicitly, but it possesses the characteristic feature of being acquired as the vanishing mass limit of the kinetic theory of Dirac particles. Furthermore, in this formulation similarity between the vorticity and magnetic field is explicit. Thus, it generates some new phenomena like the rotational analogue of Hall effect in nonlinear transport of chiral plasma [28].
Although in quark-gluon plasma the quarks are treated as massless, in reality they are massive. Thus it is crucial to understand how nonvanishing fermion mass affects the chiral transport phenomena. The quantum kinetic equation of Wigner function was originally introduced for massive Dirac particles. The Wigner function can be decomposed in terms of the Clifford algebra generators whose coefficients are scalar, pseudoscalar, vector, axial-vector and tensor fields. These fields satisfy a group of differential equations. In the massless case equations satisfied by the vector and axial-vector field components decouple from the rest. However, for massive fermions one can choose different sets of field components to construct kinetic equations. One of the possible choices was given in [29]. Another choice was presented in [30], where the vector and axial-vector fields were chosen as independent functions to construct kinetic equations whose massless limit can directly be acquired. An alternative way of constructing transport equations from the quantum kinetic equation was given in [31]. This approach has been extended to construct kinetic theory of massive fermions in curved spacetimes [32]. In the context of the semiclassical Wigner function method the study of massive fermions was started in [33,34]. By means of the solutions of Dirac equation they constructed Wigner function components at the zeroth-order in Planck constant. Then, inspired by the chiral one, they wrote the axial-vector field at the first-order in Planck constant and the properties of polarization as well as pseudoscalar condensation for massive fermions have been investigated. Mass corrections to chiral phenomena were also inspected by means of equal-time kinetic equations [35].
The aim of this work is to study massive spin-1/2 fermions in terms of two different semiclassical kinetic theory approaches whose common characteristics lie in the facts that the similarity between magnetic field and vorticity is manifest, and the noninertial features like the Coriolis force are addressed correctly. The first one is the 3D formulation presented in [26]. We study the vector and axial-vector currents by paying attention to the terms which are linear in the magnetic field and the angular velocity. We calculate mass corrections to chiral effects. The second one is a relativistic approach. We propose to modify the quantum kinetic equation of Dirac particles by means of some enthalpy current dependent terms guided by the chiral formalism [17]. But the modification of the quantum kinetic equation of Dirac particles differ from the massless case with the terms which vanish in the massless limit. To derive kinetic equations generated by the modified quantum kinetic equation we mainly follow the method of [31]. After accomplishing the kinetic equations of distribution functions we integrate them over p 0 . Thereby we establish the 3D transport equations in the helicity basis. This formulation leads to the dispersion relation which is in accord with the one resulting from the Dirac Hamiltonian in rotating coordinates. Moreover, the Coriolis force is generated. We study mass corrections to chiral phenomena.
In Section II, the 3D formulation of [26] will be reviewed briefly. The resulting vector and axial-vector currents in the helicity basis will be presented. Then the mass corrections to magnetic and vortical effects will be calculated at zero temperature. In Section III, we discuss the modified quantum kinetic equation and obtain the transport equations. In comoving frame we integrate them over p 0 after choosing the quantization direction appropriately. Thereby the 3D kinetic equations are established. We discuss the particle number currents deduced from them at zero temperature. In the last section we present discussions of our results and open problems.

II. 3D SEMICLASSICAL KINETIC THEORY
In [26] dynamical features of Dirac particles are studied by means of semiclassical wave packets formed by the particle or antiparticle solutions of Dirac equation u e χ (p, E); e = ±1, where E = p 2 + m 2 , is the energy of free particle of mass m, and ξ e χ are some constant coefficients. χ = ±1 indicates spin of the solutions: where n χ e is the spin quantization direction in the rest frame of particle and Σ = σ 0 0 σ , is the spin matrix. x and p denote the position and momentum of wave packet center. The symplectic two-form which describes this Hamiltonian formalism possesses the similarity between the magnetic field B and the angular velocity of the rotating frame ω. Moreover, it depends on the Berry curvature derived from the free particle solutions u e χ (p, E), which can be expressed by Once the underlying symplectic form is given one can construct the volume form and accomplish the phase space measure ζ χ e as well as the velocities of phase space variables as follows × ν χ e . e denotes the effective electric field: We set the electric charge to unity, so that q e = e. The dispersion relation is obtained from the Hamiltonian of charged Dirac particle under the influence of external electromagnetic fields, in a rotating frame as Its derivative with respect to momentum yields the velocity Although in this formalism distribution function can be a matrix with elements f χ ′ χ , we let it be diagonal, adequate to deal with the polarization which may be called the helicity basis. In this basis the Berry curvature (1) is simplified: Thus, the dispersion relation (2) turns out to be The collisionless transport equation in the helicity basis is Now one can define particle number and current densities as By setting The vector and axial vector currents are defined as They possess linear terms in B and ω which can be expressed by After performing the angular parts of the momentum space integrals we acquire where The upper and lower signs correspond to axial-vector and vector currents, respectively. These produce the correct results for m = 0, as they have been calculated explicitly in [28]. For simplicity let us deal with µ R = µ L = µ. Then, the vector current coefficients vanish: σ B,ω V = 0. By performing the integrals in (7) and (8) at zero temperature one gets For small mass we may approximately set ν ≈p and deal with f F D A,V , from the start. At T = 0, this approximation yields These are in accord with the results acquired by field theoretic calculations using Kubo formula for Dirac particles [37].

III. MODIFIED QUANTUM KINETIC EQUATION
Quantum kinetic equation of relativistic fluids has been derived for charged Dirac particles in the presence of external electromagnetic fields [11,12]. It explicitly depends on the electromagnetic field strength F µν . But the vorticity tensor of fluid which plays a role similar to electromagnetic field strength, shows up through the solution of kinetic equation. For massless particles we proposed to modify the quantum kinetic equation by adding some terms which depend on the enthalpy current, W µ = hu µ , where u µ = dx µ /dτ is the fluid four-velocity and h is the internal energy (enthalpy). By virtue of the modified relativistic formulation we established a 3D chiral kinetic theory which is consistent with anomalous chiral effects and possessing a Coriolis like force. To incorporate the noninertial features of fluid vorticity into the Wigner function formalism of Dirac particles, we would like to modify their quantum kinetic equation in a similar manner. However, the modification of quantum kinetic equation for massive particles differs from the chiral case with some terms which vanish in the massless limit. We will show that the proposed modification yields the Coriolis force and the dispersion relation of the Dirac particle coupled to electromagnetic field in a rotating frame.
We present the modified quantum kinetic equation in a frame moving with the four-velocity v µ ; v µ v µ = 1, whose linear acceleration vanishes: v ν ∂ ν v µ = 0. We would like to modify the quantum kinetic equation with frame dependent terms adequate to consider the non-inertial features of relativistic vorticity. Thus we propose as the quantum kinetic equation in a rotating frame of reference with where ∂ µ ≡ ∂/∂x µ , ∂ µ p ≡ ∂/∂p µ , and In this section the electric charge e is suppressed. j 0 (x) and j 1 (x) are spherical Bessel functions in ∆ ≡ 2 ∂ p · ∂ x . The space-time derivative ∂ µ contained in ∆ acts on [F µν + w µν ] , but not on the Wigner function. In contrary ∂ pν acts on the Wigner function, but not on [F µν + w µν ] .
The decomposition of the Wigner function is written through the 16 generators of the Clifford algebra as where the coefficients C ≡ {F , P, V µ , A µ , S µν } , respectively, are the scalar, pseudoscalar, vector, axial-vector, and antisymmetric tensor components of the Wigner function. These fields can be expanded in powers of Planck constant: We deal with the semiclassical approximation where only the zeroth-and first-order fields are considered. Thus, to derive the equations which they satisfy, instead of (11), (12), we only need to deal with By plugging the decomposed Wigner function (14) into the modified quantum kinetic equation (10), one derives the equations satisfied by the fields C, whose real parts are and the imaginary parts are Some of these equations can be employed to express a portion of the fields in terms of the others. Hence, depending on the choice of independent set of fields one can follow different routes of studying the transport equations. Whatever the choices are, in the semiclassical approach one should first discuss the general solutions at the zeroth-order in Planck constant and then at the higher orders.

A. General solutions at O( )
Employing the solutions of Dirac equation, u e χ (p, E), the Wigner function components at the zeroth-order in Planck constant have been obtained in [31,33,34] as follows is the dipole-moment tensor. Obviously, (25) solve (15)-(24) by ignoring -dependent terms. We choose the distribution functions in accord with (3), so that the scalar functions f 0 V , f 0 A are given as s µ = e θ(ep 0 )s µ e (p, n e ), is the spin quantization direction four-vector satisfying s 2 = −1, p · s = 0. It can be written in terms of the spin quantization direction in the rest frame of particle (antiparticle) n e , as [36] s µ e (p, n e ) = p · n e m , en e + ep · n e m(E + m) p .
Observe that the dipole-moment tensor satisfies Σ µν Σ µν = 2, and the axial-vector component can be expressed as Multiplying (15), (16), respectively, with m, p µ , and keeping the O ( ) terms yield Thus the scalar component of the Wigner function at the first-order in , can be written as where f 1 V is a scalar function. Multiplying (17), (18) , respectively, with ǫ µσγκ p σ , m and joining them by employing (21), (22), (25), lead to

It is solved by
where Σ 1 γκ can be thought of as the -order contribution to dipole-moment tensor. It satisfies the constraint equation which follows from (22) by keeping terms at O ( ): By substituting the axial-vector with (29) in (19) one finds To acquire the vector field V µ at the -order, we insert (25) and (30) into (16): By plugging (31) into (17) one gets the axial vector field at O( ), as whereF µν = 1 2 ǫ µναβ F αβ andw µν = 1 2 ǫ µναβ w αβ , are the dual tensors. We solved a part of the equations (15)- (24) to write the Wigner function components up to -order in terms of the undetermined distribution functions f V , f A and spin quantization direction n. Now, by using the rest of (15)-(24), we will establish the kinetic equations which they satisfy.

B. Semiclassical Kinetic Equations
By expanding (20) and (24) up to the first-order in , one derives the kinetic equations [31]: By inserting (32) into (20) and employing the commutator we obtain the following kinetic equation, where We can express the last term of (37) in terms of the delta function, instead of its derivative: Let us multiply the zeroth-order equation (36) Σ ν]β = 0, and then take its derivative with respect to momentum. Contracting the resultant equation with momentum results in Thus the kinetic equation (37) can equivalently be written as There are four unknown functions in the general solutions of Wigner function components. Hence, we need to obtain some other kinetic equations: At the 2 -order (24) yields By summing and subtracting (38) and (41) we get (42) C 1 , C 2 designate the terms which do not contain derivatives of f A or f V : The zeroth-order particle distribution functions given in (26), (27), can be extended as follows, An alternative form of kinetic equations is given in Appendix A.

IV. 3D KINETIC EQUATIONS IN THE COMOVING FRAME
In Section III, we modified the quantum kinetic equation in terms of w µν given in (13), where v µ is an arbitrary frame velocity. Let us now deal with the comoving frame by setting v µ = u µ . As we have already mentioned, we consider vanishing linear acceleration, u ν ∂ ν u µ = 0. Let us choose the distribution function f χ as Now one can observe that to satisfy the zeroth-order equations (34), (35), the fluid velocity should fulfill the condition ∂ ν u µ = −∂ µ u ν . Therefore we can express the vorticity tensor as where ω µ is the vorticity of fluid. The internal energy is h = u · p. Thereby (13) can be written as follows where we employed the Schouten identity: u ν ǫ µασρ = −(u µ ǫ ανσρ + u α ǫ νµσρ + u σ ǫ ρνµα + u ρ ǫ νµασ ).
In order to obtain a 3D transport equation by integrating (42) over p 0 , we need to specify s µ in accord with the equation (36). Because of expressing it as in (28), this is equivalent to solve (36) for n. Although the solution will be in the form n = n(p, F, w), to have an idea about the 3D transport equations let us ignore its F as well as w dependence and choose it adequate to the helicity basis: n e =p. Therefore, the spin quantization direction four-vector becomes Because of this choice when we integrate the kinetic equation (42) over p 0 , we need to ignore the terms which are quadratic in fields. To perform the integral one should first solve the mass-shell condition for p 0 : We work in the frame u µ = (1, 0), ω µ = (0, ω), hence by substituting s µ with (43), one can easily see that (44) dictates the dispersion relation This is in accord with (4). Thus, we establish the 3D transport equation To acquire the vector and axial-vector currents, we insert (45) into the definition (6). Now the terms which are parallel to B and ω can be shown to be (9). For µ R = µ L = µ, at T = 0, the vector current coefficients vanish and the axial-vector current coefficients can be calculated asσ These reflect the fact that in this formulation the zero mass limit cannot be achieved directly.

V. DISCUSSIONS
We studied the semiclassical kinetic theories of Dirac particles within two different approaches. First, the 3D formalism of Dirac particles [26], which manifestly exhibits the magnetic field, vorticity similarity is studied in helicity basis. We derived the axial-vector and vector currents for massive spin-1/2 particles. One of the distinguishing properties of this 3D formalism is the fact that massless limit can be reached effortlessly. Actually, one can easily observe that the currents (7), (8), generate the chiral magnetic and vortical effects correctly. We calculated the axialvector current at zero temperature. In the small mass limit it is consistent with the Kubo formalism based calculations [37].
Dirac fermions in external electromagnetic fields can be considered as relativistic fluids described by the Wigner function satisfying the quantum kinetic equation which involves the electromagnetic field strength. For being able to take into account the non-inertial characteristics of vorticity we modified this kinetic equation with terms depending on the four-velocity of rotating frame. We studied the equations satisfied by the Wigner function components and established the semiclassical relativistic kinetic equations of the scalar functions and spin degrees of freedom. Kinetic equations of the scalar functions are integrated over p 0 in a comoving frame by choosing the spin direction adequate to helicity basis. It is shown that the Coriolis force and the correct energy dispersion relation are generated. In addition, the coefficients of magnetic and vortical terms in axial-vector and vector currents are calculated at zero temperature. Unfortunately, the massless limit cannot be acquired directly within the approach which we have adopted to establish the semiclassical kinetic equations. Obviously, the resulting 3D model relies on the choice of spin quantization direction given in (43). For having a better understanding of 3D kinetic equations resulting from the relativistic ones, one should study the solution of (36) without ignoring the external electromagnetic fields and vorticity.
The original quantum kinetic equation was derived from the Dirac equation of charged particles coupled to the electromagnetic vector field A µ (x). Hence, to consider the noninertial features of vorticity one can think to add a term proportional to the four-velocity u µ (x) into the Dirac equation. It can be added only with a coefficient possessing the dimension of mass. Although internal energy, h, has mass dimension, in general it depends on momentum p µ . The unique possibility is to choose the coefficient equal to m. However, this choice does not yield an enthalpy current which we need to take into account the non-inertial properties correctly. Hence, the modification which we propose does not seem to be generated by some gauge field like terms coupled to Dirac particles.