The physical state condition in the BRST quantization of Chern-Simons field theory is used to derive Gauss law constraints in the presence of Wilson loops, which play an important role in explicitly establishing the connection of Chern-Simons field theory with 2-dimensional conformal field theory.When we discuss knot invariants in terms of Chern-Simons theory [1][2][3][4][5][6] and the relationship between Chern-Simons field theory and conformal field theory [7,8] , an important relation is Gauss law constraint in presence of Wilson line, which was first given in ref.[1].This relation plays an important role in proving that states of Chern-Simons theory satisfy the Knizhnik-Zamolodchikov equation [9][10][11][12][13] . In this letter we intend to derive the Gauss law constraints from BRST quantization of Chern-Simons field theory. We think this investigation is significant since in some sense BRST quantization formulation is defined better than a formal manipulation without gauge fixing [14] . We will show that when Wilson lines exist the physical state condition in BRST quantization will lead to Gauss law constraints † ICSC-World Laboratory, Switzerland

Differential regularization is used to investigate the one-loop quantum corrections to Chern-Simons-Maxwell spinor and scalar electrodynamics. We illustrate the techniques to write the loop amplitudes in coordinate space. The short-distance expansion method is developed to perform the Fourier transformation of the amplitudes into momentum space and the possible renormalization ambiguity in Chern-Simons type gauge theories in terms of differential regularization is discussed. We also stress that the surface terms appearing in the differential regularization should be kept along for finite theories and they will result in the finite renormalization ambiguity.Differential regularization is a relatively new regularization scheme [1]. The basic idea of this regularization is quite simple. It works in coordinate space and is based on the observation that the UV divergence reflects in the fact that the higher order amplitude can not have a Fourier transform into momentum space due to the short-distance singularity. Thus one can regulate such an amplitude by first writing its singular parts as the derivatives of the † ICSC-World Laboratory, Switzerland

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