A Quantum and Inhomogeneous Description of Chiral and $U_A(1)$ Symmetry Breaking in Electromagnetic Fields

We study quantum effect on chiral and $U_A(1)$ symmetry breaking in external electromagnetic fields in the frame of equal-time kinetic theory. We derive the transport and constraint equations for the spin and isospin components of the quark Wigner function in a Nambu--Jona-Lasinio model. By taking semi-classical expansion of the equations, we obtain the chiral condensate at classical level and neutral pion condensate at the first order in $\hbar$. As an example of off-shell effect, there exist strong quantum oscillations induced by multi couplings among the spin and isospin components.