Re: Renormalization

Matthew Nobes wrote:
> Looking over this thread, it seems that a lot of people have rather
> old-fashioned ideas about renormalization. I mean, Heitler? 1954?
> Our understanding of quantum field theory has advanced a fair amount
> since 1954, there's even been a few Nobel prizes handed out which
bear
> directly on this ('t Hooft/Veltman for, among other things,
dimensional
> regularization, and Wilson for the renormalization group)
>

OK - I will use a more modern reference. Consider "Quantum
Electrodynamics of Strong Fields" (1985) by W. Greinger, B. Muller, and
J. Rafelski. In chapter 14 they calculate the vacuum polarization
tensor PI. In equation 14.43 they obtain the expression,

PI = PI1 + PI(sp)

where the first term PI1 is gauge invariant and the second term PI(sp)
is not gauge invariant. According to Greiner et al "this latter term
violates the gauge invariance of the theory". They also write "Thus in
a satisfactory formulation of the theory of charged particles, such a
term must not appear. WE NOW EXPLICITLY SHOW THAT IT IS INDEED
PRESENT, INDICATING THAT PERTURBATIVE QED IS NOT A COMPLETE THEORY."

Griener et al then write "As this circumstance cannot be emphasized
enough, we now proceed to calculate the spurious term in the vacuum
polarization."

They then proceed to calculate the term PI(sp) and show that it is not
zero and highly divergent (see equation 14.53). They then write "It is
important to appreciate here that this term does not arise as a
consequence of some illegitimate manipulations of Fourier transforms
... It is possible to rederive it in configuration space without using
Fourier transforms"

OK, to sum up, Greiner et all, in their book, present a treatment of
QED. I would say that this treatment is fairly standard and
straightforward. They introduce the usual "machinery" of QED, i.e.,
field operators, creation and destruction operators, define the vacuum
state, the Dirac Hamiltonian, define the current operator, all the
usual stuff. With this mathematical base they calculate certain things
using, what they believe to be, proper mathematical techniques. They
get an answer that suprises them. They find the vacuum polarization
tensor PI is not gauge invariant. Why is this a suprise? Because they
thought their theory was gauge invariant, therefore it is kind of
upsetting to get a non-gauge invariant result. What is wrong? They
don't know but they defend their mathematics. They insist that their
mathematics is correct indicating, in their words, that "QED is not a
complete theory".

Now what is their solution to this quandry? They drop the offending
term from the polarization tensor leaving a gauge invariant result. Of
course one might argue that they should use regularization. By why
bother. They achieved the same result by simply identify the "bad"
term and dropping it from the solution.

Now this leaves me with two questions. The first is should we consider
regularization an acceptable part of a QFT or is it a "cheat"? That
is, does it cover the fact that our theory is incomplete and until we
complete it we must use this method but expect it to go away at some
point.

The other question is this. Greiner et al formulate a theory that they
believe to be gauge invariant. When they make a calculation in this
theory they obtain a non-gauge invariant result. The question is where
in their chain of reasoning did they go wrong and does this point to an
inconsistency in the formulation of QED?

- Dan Solomon

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