Chapter 16 deals with the important problem of quantization with symmetries, that is, how to implement symmetries of the classical action in the corresponding quantum theory. The proposed solutions are based on methods like regularization by addition of higher order derivatives or regulator fields, or lattice regularization. Difficulties encountered in the case of chiral theories are emphasized. This may lead to obstacles for symmetric quantization called anomalies. Examples can be found in the case of chiral gauge theories. Their origin can be traced to the problem of quantum operator ordering in products. A non–perturbative regularization, also useful for numerical simulations, is based on introducing a space lattice. Difficulties appear for lattice Dirac fermions, leading the fermion doubling problem. Wilson’s fermions provide a non–chiral invariant solution. Chiral invariant solutions have been found, called overlap fermions or domain wall fermions.
Domain wall fermions on the brane
