We derive a relativistic chiral kinetic equation with manifest Lorentz covariance from Wigner functions of spin-1/2 massless fermions in a constant background electromagnetic field. It contains vorticity terms and a 4-dimensional Euclidean Berry monopole which gives axial anomaly. By integrating out the zero-th component of the 4-momentum p, we reproduce the previous 3-dimensional results derived from the Hamiltonian approach, together with the newly derived vorticity terms. The phase space continuity equation has an anomalous source term proportional to the product of electric and magnetic fields (FσρF. This provides a unified interpretation of the chiral magnetic and vortical effects, chiral anomaly, Berry curvature, and the Berry monopole in the framework of Wigner functions. Introduction. -The Berry phase is a topological phase factor acquired by an eigen-energy state when it undergoes adiabatic evolution along a loop in parameter space [1]. It is in close analogy to the Aharonov-Bohm phase when a charged particle moves in a loop enclosing a magnetic flux, while the Berry curvature is like the magnetic field. The integral of the Berry curvature over a closed surface can be quantized as integers known as Chern-Simons numbers, which is similar to the Dirac magnetic monopole and has deep connection with the quantum Hall effect. The Berry phase is a beautiful, simple and universal structure in quantum physics and has many interesting applications, for a recent review of the Berry phase in condensed matter physics, see e.g. Ref.[2].

A power expansion scheme is set up to determine the Wigner function that satisfies the quantum kinetic equation for spin-1/2 charged fermions in a background electromagnetic field. Vector and axial-vector current induced by magnetic field and vorticity are obtained simultaneously from the Wigner function. The chiral magnetic and vortical effect and chiral anomaly are shown as natural consequences of the quantum kinetic equation. The axial-vector current induced by vorticity is argued to lead to a local polarization effect along the vorticity direction in heavy-ion collisions.PACS numbers: 25.75. Nq, 12.38.Mh, 13.88.+e Introduction. -Chiral anomaly is an important quantum effect which is absent at the classical level. Recently it has been shown that such a microscopic quantum effect can have a macroscopic impact on the dynamics of relativistic fluids, termed as the chiral magnetic and vortical effect (CME and CVE) [1][2][3] as manifested in currents induced by magnetic field and vorticity. Such effects and related topics have been investigated within a variety of approaches, such as AdS/CFT duality [4][5][6][7][8], relativistic hydrodynamics [9][10][11], and quantum field theory [2,[12][13][14][15][16][17]. However, it is still not clear how CME and CVE can emerge from a microscopic quantum kinetic theory.In this Letter we make a first attempt to derive both the CME and CVE from a quantum kinetic theory. A power expansion in space-time derivatives and weak external fields is used to determine the analytic form of vector and axial-vector components of the Wigner function that satisfies the quantum kinetic equation for spin-1/2 massless fermions. The CME and CVE appear naturally in the induced currents. Chiral anomaly and other conservation laws are also automatically satisfied. The axialvector current induced by vorticity depends quadratically on the temperature, baryonic and chiral chemical potential. So it should be present in both hot and dense matter, and can lead to a local polarization effect in heavy-ion collisions as proposed in earlier studies [18][19][20]. This provides another possible future experimental measurement of the CVE in high-energy heavy-ion collisions.The quantum kinetic approach can provide a bridge between the microscopic and macroscopic description of the CME and CVE and should be more suitable for future simulations of both effects in heavy-ion collisions. The power expansion method can also be applied to the calculation of other transport coefficients.

The chiral kinetic theory of Weyl fermions with collisions in the presence of weak electric and magnetic fields is derived from quantum field theories. It is found that the side-jump terms in the perturbative solution of Wigner functions play a significant role for the derivation. Moreover, such terms manifest the breaking of Lorentz symmetry for distribution functions. The Lorentz covariance of Wigner functions thus leads to modified Lorentz transformation associated with sidejump phenomena further influenced by background fields and collisions.Introduction.-Novel quantum transport processes in Weyl fermionic systems have been widely investigated, in particular for the so-called chiral magnetic and vortical effects such that charged currents are induced by magnetic and vortical fields [1-3]. These effects associated with quantum anomaly have been studied from different theoretical approaches including relativistic hydrodynamics [4][5][6][7], lattice simulations [8][9][10][11][12], and gauge/gravity duality [13][14][15]. These effects might be (in-)directly observed in heavy ion collisions [16] and in condensed matter systems such as Weyl semimetals [17].From both theoretical and experimental perspectives, it is imperative to understand these anomalous effects in non-equilibrium conditions. One promising approach is kinetic theory, which can delineate non-equilibrium transport of a particle when the interaction and background fields are sufficiently weak. Nevertheless, it is hard to incorporate anomalous effects through the standard Boltzmann equations [6]. The chiral kinetic theory (CKT), which describes anomalous transport of Weyl fermions, has been thus developed from the path-integral [18], Hamiltonian [19], and local-equilibrium quantum kinetic approaches [20,21]. In such formalism, the effective velocity and forces for a single particle are modified by the Berry curvature Ω p = p/(2|p| 3 ), where p represents the spatial momentum of the particle, which originates from the Berry phase in an adiabatic process [22]. Further generalization to massive Dirac fermions can be found in Ref. [23]. In order to bridge the semi-classical approaches [18,19] and quantum field theories, the CKT is also derived from Wigner functions in the high-density effective theory [24] (see also Ref. [25] for relevant study of the on-shell effective field theory.)However, there still exist potential issues in the chiral kinetic equation. First, the field-theory derivation in Refs.[24] and [20,21] are subject to a predominant chemical potential and local equilibrium, respectively. The derivation for more general systems beyond local equilibrium is thus needed. Second, the non-manifestation of Lorentz invariance in the chiral kinetic equation has been recently discussed in Refs. [26,27] from the semi-