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From: ran@ho95b.UUCP (RANeinast)
Newsgroups: net.sf-lovers,net.physics
Subject: Re: Renormalization (relativity, etc.)
Message-ID: <420@ho95b.UUCP>
Date: Thu, 23-May-85 13:44:18 EDT
Article-I.D.: ho95b.420
Posted: Thu May 23 13:44:18 1985
Date-Received: Fri, 24-May-85 22:38:54 EDT
Organization: AT&T-Bell Labs, Holmdel, NJ
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>From: Laurence R Brothers  <LAURENCE@SU-CSLI.ARPA>
>
>I'm curious: does anyone out there claim to understand the modern physics
>concept of "renormalization". As far as I can tell, this involves
>an equation (extremely simplified) of the form a + I = J where I and
>J are transfinite. a, however is a relatively normal number which relates
>to a mathematical feature of mass. Obviously in any normal terms,
>a is not a quantity which can be solved for, but it IS in renormalization.

A simple example of renormalization (even without "modern physics"):
Consider a point mass; the gravitational potential is V= -a/r.
["a" is some constant].  If you use this potential to find the potential
of an infinite line, you find yourself evaluating the integral

  I = int from 0 to inf { 1/sqrt(r^2+x^2)} dx,

which is infinite.  However, you realize that the potential has no
real physical significance, just differences between potentials
at different places, so you can subtract off a constant without affecting
anything.  So you replace I with

      int from 0 to L { 1/sqrt(r^2+x^2)} dx  - ln(L),

and let L go to infinity.  You've now subtracted two infinite numbers
to get a finite result, which is V= -b*ln(r), which is the answer
you'd have gotten if you'd just used Gauss' law on the line.


How does this apply to modern physics?  An example is the mass of the electron.
The "bare mass" is infinite, but the uncertainty principle allows
electron-positron pairs to pop in and out of vacuum (this is also
an infinite effect).  This effectively shields the original electron,
the two infinities cancel, and the measured mass of the electron
is perfectly finite.  Most physicists believe that this effect
is an artifact of the way the calculation is done (as above);
if we knew the "correct" way [the equivalent of Gauss' law, above]
(not perturbation theory--an approximation), we could do the whole
calculation without infinities.
-- 

". . . and shun the frumious Bandersnatch."
Robert Neinast (ihnp4!ho95b!ran)
AT&T-Bell Labs