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Quantum Kinetic Theory with Vector and Axial Gauge Fields

In this paper we introduce the axial gauge field to the framework of the quantum kinetic theory with vector gauge field in the massless limit. Treating axial-gauge field on an equal footing with the vector-gauge field, we construct a consistent solution to the kinetic equations up to the first order in gradient expansion or equivalently the semi-classical expansion. The intuitive extension of quantum kinetic theory presented in this work provides a natural generalization, and turns out to give rise to the covariant anomaly and the covariant currents. The corresponding consistent currents can be obtained from the covariant ones by adding the Chern-Simons current. We use the consistent currents to calculate various correlation functions among currents and energy-momentum tensor in equilibrium state.