Re: Renormalization


Arnold Neumaier wrote:
> jsolomon@xxxxxxxx wrote:
>
>
> To get a mathematical rigorous version of the P/S argument, put it
the
> following way (to simplify typography, write p for \ell_E):
>
> If d is a positive integer and f(p^2) is integrable (i.e. decays fast
> enough), then standard Lebesgue integration gives the formula
> integral (dp/2 pi)^d f(p^2) = C_d integral_0^inf dr
r^{d-1}f(r^2)
> (*)
> where C_d has the value (7.81). We observe that the formula (7.81)
> makes sense for arbitrary complex d with positive real part, and
> that therefore for
> f(s)=r^2j/(r^2+Delta)^n, n>j+d/2.
> the well-defined right hand side of (*) is an expression I(d,j,n)
> which depends analytically on d,j,n. In particular, the cases j=0
> and j=1 lead to the expressions (7.85/86). A similar reasoning
> produces (7.87) and the rules on p.807, which, together with the
> Feynman trick (6.39) can be used to evaluate integrals of
> arbitrary rational Lorentz-invariant expressions provided that
> they decay fast enough.
>
>
> Note that this formula is valid only for n>j+d/2 (which ensures
> sufficiently fast decay at infinity to make the Lebesgue integral
> well-defined) and integral d.
>
> For other values the computations are meaningless, and any
> contradiction derived from it (as the formal manipulations
> in your mail) is therefore irrelevant. As irrelevant as the
> well-known fact that a convergent alternating infinite sum can
> be given any value whatsoever by formal rearrangements.
>
> Remarkably, however, I(d,j,n) (and the analogous formulas on
> p.807) can be analytically continued to the interesting case
> d=4-eps. This allows us to _define_ an ''extended Lebesgue integral''
> by the formula
> integral (dp/2 pi)^d r^2j/(r^2+Delta)^n:= I(d,j,n) for d:=4-eps.
(**)
> and similar expressions for arbitrary rational Lorentz-invariant
> expressions. Moreover, if these expressions happen to have
> good limits for eps to 0 (which cannot happen for (**) but for
> suitable linear combinations) they define the value also for d=4.
>
> The derivation ensures that it gives the correct results in all
> cases where the integral makes sense in the traditional (Lebesgue)
> way.
>
> Thus we have defined a consistent extension of the Lebesgue
> integral of Lorentz-invariant expressions to the singular case.
> This is similar in spirit to Lebesgue's extension of the Riemann
> integral to the Lebesgue integral.
>
> The theory of renormalization now shows that all integrals
> occuring in the expressions for S-matrix elements for renormalizable
> theories have a well-defined extended Lebesgue integral for d=4.
> This is all that is required for consistency.
>
>
> Arnold Neumaier

I understand what you are saying and I understand how regularization
works and that it gets us the correct S-matrix and all that. The
question I have is this - Is it covering up some flaw in the underlying
theory? That is shouldn't we be looking for a theory that doesn't
require us to use ''extended Lebesgue integrals''.

To understand what I mean consider the following example. Suppose I am
trying to come up with a mathematical model to describe some system. I
observe the system and formulate a mathematical model based on educated
guesses. Now I have to check my mathematical model with experiment.
Suppose the result of some experiment is Rexp=-(1/2). Suppose my
calculated result is the expression,

Rcalc = integral [1 to oo](LdL)

This is the integral of the quantity L integrated from 1 to infinity
with respect to L. Now my calculated result is this divergent
integral. My experimental result is minus one-half. One could argue
that my mathematical model is wrong and I should go back and change it.
In answer to that I say - why not use regularization. I note that if
I replace the integrad L by L^d then I can write Rcalc as a function of
d, i.e., Rcalc(d). The original Rcalc is recovered when d=1. I notice
that for d<-1 the Rcalc(d) is defined and equals,

Rcalc(d) = -1/(1+d)

Now even though this quantity was original valid for d<-1 I then extend
this result to the entire complex plane. I then recall that d=1 and my
final result is,

Rcalc(d=1) = -1/2

This, of course, is my experimental result. So my theory is saved
thanks to the magic of regularization. I guess the question is how do
we ever know if some theory is wrong. Given enough effort and thought
a divergent integral can propably be regularized into about anything.

Dan Solomon

.

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