Abstract
The covariant path integral quantization of the theory of the scalar and spinor fields interacting through the Abelian and non-Abelian Chern - Simons gauge fields in 2 + 1 dimensions is carried out using the De Witt - Fadeev - Popov method. The mathematical ill-definiteness of the path integral of theories with pure Chern - Simons' fields is remedied by the introduction of the Maxwell or Maxwell-type (in the non-Abelian case) terms, which make the resulting theories super-renormalizable and guarantees their gauge-invariant regularization and renormalization. The generating functionals are constructed and shown to be the same as those of quantum electrodynamics (quantum chromodynamics) in 2 + 1 dimensions with the substitution of the Chern - Simons propagator for the photon (gluon) propagator. By constructing the propagator in the general case, the existence of two limits; pure Chern - Simons and quantum electrodynamics (quantum chromodynamics) after renormalization is demonstrated.
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