Dimensional regularization

From Wikipedia, the free encyclopedia

In theoretical physics, dimensional regularization is a particular way to get rid of infinities that occur when one evaluates Feynman diagrams in quantum field theory. One assumes that the spacetime dimension is not four but rather d which need not be an integer. It often turns out that the integrals extrapolated to a general dimension converge. The divergences are then parameterized as quantities proportional to 1 / ε whose coefficients must be cancelled by renormalization to obtain physical quantities.

The hypersurface area of a d-1 sphere of radius r is where Γ is the gamma function when d is a positive integer. We can assume by fiat that this equation also holds when d isn't an integer.

If we wish to evaluate a loop integral which is logarithmically divergent in 4 dimensions, like

we first generalize this equation to an arbitrary number of dimensions, including nonintegral dimensions like d=4-ε. When ε is positive, this integral converges and we take the limit as ε approaches zero.

This gives

[edit] See also

• Regularization (physics)
• Pauli-Villars regularization