This paper is devoted to formulating the interaction-picture dynamics of the self-dual field minimally coupled to fermions. As a preliminary, we quantize the free self-dual model by means of the Dirac-bracket quantization procedure. The free self-dual model turns out to be a relativistically invariant quantum field theory whose excitations are identical to the physical (gauge-invariant) excitations of the free Maxwell–Chern–Simons theory. The interacting model is also quantized through the Dirac-bracket quantization procedure. One of the self-dual field components is found not to commute, at equal times, with the fermionic fields. Hence, the formulation of the interaction-picture dynamics demands the elimination of that component. This procedure brings, in turn, two new interactions terms, which are local in space and time while nonrenormalizable by power counting. Relativistic invariance is tested in connection with the elastic fermion–fermion scattering amplitude. We prove that all the noncovariant pieces in the interaction Hamiltonian are equivalent to the covariant minimal interaction of the self-dual field with the fermions. The high-energy behavior of the self-dual field propagator confirms that the coupled theory is nonrenormalizable. The self-dual field minimally coupled to fermions bears no resemblance to the renormalizable model defined by the Maxwell–Chern–Simons field minimally coupled to fermions.
Canonical quantization of classical mechanics in curvilinear coordinates. Invariant quantization procedure
