  The formalism based on the equal-time Wigner function of the two-point
correlation function for a quantized Klein--Gordon field is presented. The
notion of the gauge-invariant Wigner transform is introduced and equations for
the corresponding phase-space calculus are formulated. The equations of motion
governing the Wigner function of the Klein--Gordon field are derived. It is
shown that they lead to a relativistic transport equation with electric and
magnetic forces and quantum corrections. The governing equations are much
simpler than in the fermionic case which has been treated earlier. In addition
the newly developed formalism is applied towards the description of spontaneous
symmetry breakdown.
