In quantum field theory when dealing with divergent integrals, particularly in calculating corrections to scattering amplitudes, what is often done to render the integrals convergent is to add a regulator, which is some parameter $\Lambda$ which becomes the upper limit of the integrals instead of $\infty$.
The physical explanation of this is that the quantum field theory is just an approximation of the "true" theory (if one exists), and it is only valid at low energies. We are ignorant of the processes that occur at high energies, and so it makes no sense to extend the theory to that regime. So we cut off the integral to include the low energy processes we are familiar with only.
My problem with this explanation is, even if we don't know what happens at high energies, is it okay to just leave those phenomena out of our calculations? In effect we are rewriting our integral as $\int_{-\infty}^{\infty} = \int_{-\infty}^{\Lambda} + \int_{\Lambda}^{\infty}$, and then ignoring the second term. Is this really okay? And even if we were able to ascertain the "true" theory and calculated the contribution to the correction from high energy processes, I feel like it would come out to be large, since looking at it from our effective field theory viewpoint it already appears to be infinity. Do we have any right to say it is negligible?
Answer
It took the insights of Wilson and Kadanoff to answer this question. Universality. It doesn't matter all that much what the precise details in the ultraviolet are. Under the renormalization group, only a small number of parameters are either relevant or marginal. All the rest are irrelevant. As long as you take care to match up the relevant and marginal parameters, the precise regulator you choose doesn't matter. Even if it differs from the actual underlying physics, in the infrared, it still gives the same answers.
No comments:
Post a Comment