Re: Renormalization

jsolomon_at_mail.com
Date: 03/16/05 

Date: Wed, 16 Mar 2005 16:19:00 +0000 (UTC)

Arnold Neumaier wrote:
> jsolomon@mail.com wrote:
>
> > Renormalization is required because, when standard perturbation
theory
> > is used, it is found that the photon has a divergent mass.
However,
> > unlike the electron, there is no mass term to absorb this
divergence.
>
> Are you speaking about the infrared divergence,
> or about the photon self-energy?

I guess you could say I am talking about the photon self energy. Or to
be more specific I am talking about the polarization tensor. This
quantity appears during the calculation of the photon self-energy. The
thing that bothers me is that when the polarization tensor (PT) is
calculated non-gauge invariant terms appear in the result. This is a
problem because the theory is suppose to be gauge invariant. An
example of this is given in Peskin and Schroeder "An Introduction to
Quantum Field Theory" (1995). Equation 7.79 on page 248 of their book
gives an expression for the PT. As they point out this expression
contains non-gauge invariant terms, or, in their words, it violates the
Ward identity. In fact you can look in many textbooks, or articles,
and find that, when standard perturbation theory is used, calculations
of the vacuum current and polarization tensor contain non-gauge
invariant terms. A while ago I posted a paper on arXiv.org that gives
a number of references to papers that calculate these quantities (See
quant-ph/0306071). In all cases non-gauge invariant terms appear.

Now the fact that these terms appear is a problem because the theory is
"suppose" to be gauge invariant. Therefore these terms must be removed
in order to make the result physically correct. This is the process of
regularization. Different authors use different methods. Some
physicists simply examine the result and remove the offending terms by
hand. I call this regularization by eraser. Others use more
sophisticated mathematical techniques such as Paul-Villars
regularization or dimensional regularization.

I have a couple of problems with all this. First, why does a theory
that is suppose to be gauge invariant produce a non-gauge invariant
result? I have never found a satisfactory answer in the literature to
this question. In fact, as far as I can tell, it is never discussed.
Second, the methods of regularization appear "ad hoc" to me in the
sense that they have no physical basis. That is these mathematical
regularization methods do not derive from the basic theory but are
invented after the fact, i.e., to eliminate a problem.

- Dan Solomon