From: mcirvin@husc8.harvard.edu (Mcirvin)
Newsgroups: sci.physics
Subject: Re: Is Space a continuum?
Date: 14 Aug 92 17:59:09 GMT
Article-I.D.: husc8.mcirvin.713815149

dhg@sinix.UUCP (David Griffith) writes:

>Is space a continuum? In other words, does General Relativity suitably
>adjusted by Quantum Mechanics, treat space-time as infinitely
>sub-divisible? If it's not, how does this affect the mathematics, eg
>calculus which assumes deltas that tend to zero?

Nobody really knows, but it's hoped that quantum mechanics will
have some effect on these sorts of questions.  Presumably spacetime
doesn't behave like a smooth manifold on the Planck scale, but what
it does instead is not obvious.

The reason why such modification is to be hoped for has to do
with quantum field theory.  In classical field theory the components
of the field are treated as functions of spacetime.  In quantum
field theory the functions are replaced by function-like things
whose values are operators something like the ones encountered
in nonrelativistic QM, except that the commutation (or anticommutation)
relations don't involve finite [anti]commutators, but rather have
things like Dirac delta functions (not the kind of delta
you're talking about) in them.  

A Dirac delta "function" is the sort of thing that physicists
use far too cavalierly for the tastes of mathematicians.  It's
zero everywhere except at one point, where it's infinite; and
this infinitely tall, infinitely thin spike has a finite 
integral.  This isn't really a function, it's a distribution.  It
is really only defined in terms of what you get when you 
integrate it over various intervals, and it's unclear what
happens when you multiply, say, two Dirac deltas on the real
line that have their spikes in the same place.

Likewise, since these operator functions' commutation relations
have Dirac deltas in them, they're not really operator-valued
functions, but operator-valued distributions.  You only get
a legitimate quantum operator with finite values by integrating
these things over some volume of space.  The smaller the scale
over which you integrate them, the more outrageously the numbers
mount.  Yet interacting field theories involve multiplying
various numbers of these fields together at the same spacetime
point.  It's no wonder that hideous divergences show up in 
the process of calculating things like transition probabilities. 

Now, the means of getting finite answers out of quantum
field theory (though this is not all it's good for) is
called renormalization, and it typically involves two
steps.  The first step is to somehow modify the physics
at short distances by applying an arbitrary short-wavelength 
cutoff, or making the space discrete, or some such thing;
this is called regularization.  Then, all results are expressed
in terms of quantities having only to do with physics on the
distance scale actually being dealt with.  For some quantum
field theories (a family that fortunately includes the theories
of strong, weak, and electromagnetic interactions), 
this procedure will produce results that are actually
independent of the arbitrary means of regularization, which
can be removed from the final result.  But the regularization
was necessary to make things make sense in the first place.
It's treated as a kind of fiction that gets removed after
the renormalization is done, but you can't do calculations
without it.

One means of justifying renormalization as a legitimate
procedure is to say that some form of regularization actually
physically happens at very short distances, say, on the
Planck scale, where it no longer becomes correct to
speak of spacetime as a smooth manifold.  Then, as long as
the theory you're looking at is "renormalizable" -- that
is, the values of physical quantities are actually
independent of things that happen at arbitrarily short
distances-- you can use any regularization method you
find convenient and get correct results, knowing that
the true method of regularization is somehow buried in
quantum gravity.

More importantly, if you try in any straightforward way to
construct a theory of quantum gravity by turning general
relativity into a quantum theory, you find that it is
*not* renormalizable!  Physical quantities depend very
much on the method used to cut off divergences, if you
try to calculate them using the means normally used in
particle physics.  So the hope is that quantum gravity
itself will modify spacetime at short distances in such
a way as to provide a natural cutoff for these divergences.
Otherwise, it doesn't seem possible to make the theory
make sense.

-- 
Matt McIrvin, professional gradgrind, amateur Usenet drifter 
Newsgroups: sci.physics