Symmetries for Light-Front Quantization of Yukawa Model with Renormalization

In this work we discuss the Yukawa model with the extra term of self-interacting scalar field in D=1+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D=1+3$$\end{document} dimensions. We present the method of derivation the light-front commutators and anti-commutators from the Heisenberg equations induced by the kinematical generating operator of the translation P+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{+}$$\end{document}. Mentioned Heisenberg equations are the starting point for obtaining this algebra of the (anti-) commutators. Some discrepancies between existing and proposed method of quantization are revealed. The Lorentz and the CPT symmetry, together with some features of the quantum theory were applied to obtain the two-point Wightman function for the free fermions. Moreover, these Wightman functions were computed especially without referring to the Fock expansion. The Gaussian effective potential for the Yukawa model was found in the terms of the Wightman functions. It was regularized by the space-like point-splitting method. The coupling constants within the model were redefined. The optimum mass parameters remained regularization independent. Finally, the Gaussian effective potential was renormalized.