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].

The adaptive immune system functions through the combined action of antigen-presenting cells (APCs) and T cells. Specifically, class I major histocompatibility complex antigen presentation to CD8(+) T cells is limited to proteosome-generated peptides from intracellular pathogens while the class II (MHCII) endocytic pathway presents only proteolytic peptides from extracellular pathogens to CD4(+) T cells. Carbohydrates have been thought to stimulate immune responses independently of T cells; however, zwitterionic polysaccharides (ZPSs) from the capsules of some bacteria can activate CD4(+) T cells. Here we show that ZPSs are processed to low molecular weight carbohydrates by a nitric oxide-mediated mechanism and presented to T cells through the MHCII endocytic pathway. Furthermore, these carbohydrates bind to MHCII inside APCs for presentation to T cells. Our observations begin to elucidate the mechanisms by which some carbohydrates induce important immunologic responses through T cell activation, suggesting a fundamental shift in the MHCII presentation paradigm.

We calculate the Wigner function for massive spin-1/2 particles in an inhomogeneous electromagnetic field to leading order in the Planck constant . Going beyond leading order in we then derive a generalized Boltzmann equation in which the force exerted by an inhomogeneous electromagnetic field on the particle dipole moment arises naturally. Furthermore, a kinetic equation for this dipole moment is derived. Carefully taking the massless limit we find agreement with previous results. The case of global equilibrium with rotation is also studied. Finally, we outline the derivation of fluid-dynamical equations from the components of the Wigner function. The conservation of total angular momentum is promoted as an additional fluid-dynamical equation of motion. Our framework can be used to study polarization effects induced by vorticity and magnetic field in relativistic heavy-ion collisions.

Fermions become polarized in a vortical fluid due to spin-vorticity coupling. Such a polarization can be calculated from the Wigner function in a quantum kinetic approach. Extending previous results for chiral fermions, we derive the Wigner function for massive fermions up to the next-toleading order in spatial gradient expansion. The polarization density of fermions can be calculated from the axial vector component of the Wigner function and is found to be proportional to the local vorticity ω. The polarizations per particle for fermions and anti-fermions decrease with the chemical potential and increase with energy (mass). Both quantities approach the asymptotic value ω/4 in the large energy (mass) limit. The polarization per particle for fermions is always smaller than that for anti-fermions, whose ratio of fermions to anti-fermions also decreases with the chemical potential. The polarization per particle on the Cooper-Frye freeze-out hyper-surface can also be formulated and is consistent with the previous result of Becattini et al..