4 - Dimensional regularization

Summary

In this chapter we discuss path integrals defined by dimensional regularization (DR). In contrast to the previous time slicing (TS) and mode regularization (MR) schemes, this type of regularization seems to have no meaning outside perturbation theory. However, it leads to the simplest set up for perturbative calculations. In fact, the associated counterterm V_DR turns out to be covariant, and the additional vertices obtained by expanding V_DR, needed at higher loops, can be obtained with relative ease (using, for example, Riemann normal coordinates).

Dimensional regularization is based on the analytic continuation in the number of dimensions of the momentum integrals corresponding to Feynman graphs (1 → D + 1 with arbitrary complex D, in our case). At complex D we assume that the regularization of ultraviolet (UV) divergences is achieved by the analytic continuation as usual. The limit D → 0 is taken at the end. Again one does not expect divergences to arise in quantum mechanics when the regulator is removed (D → 0), and thus no infinite counterterms are necessary to renormalize the theory: potential divergences are canceled by the ghosts.

To derive the dimensional regularization scheme, one can employ a set up quite similar to the one described in the previous chapter for mode regularization. The only difference will be the prescriptions of how to regulate ambiguous diagrams.