FOM: Connections between mathematics, physics and FOM
Jeffrey John Ketland
Jeffrey.Ketland at nottingham.ac.uk
Tue Feb 1 12:28:34 EST 2000
Mark Steiner wrote:
> Discovery is very important; a recent paper by a mathematical physicist
> argues that for physics we don't need differentail equations over the
> reals at all, we could use difference equations over the rationals,
> given that we can represent all possible measurements by rational
> numbers. But, the physicist goes on, we could never make discoveries
> without the mathematical form of the differential equations themselves.
My view is that kind of approach is fundamentally flawed, both
technically and philosophically. The continuum model of spacetime
is really fundamental to theoretical physics. One occasionally
hears that people are trying to replace the manifold structure by
something countable, or discrete or what have you. I think it won't
work. There are many reasons, of which two are:
(i) One reason comes from quantum mechanics and the basic
commutation relations for the self-adjoint operators Q and P which
represent position and momentum. The Dirac commutation
relations [Q, P] = ih have no countable representations and the
spectra of these operators must be the whole real line (actually it
can be shown that neither Q nor P have eigenfunctions in the
Hilbert space L^2[R]). Roughly, you can't find two countable
matrices A and B whose commutator is proprortional to the identity
matrix. One can find a brief mention of this in Weyl's 1931 book
"Group Theory and Quantum Mechanics". One cannot simply
replace the real line by the ratios or something like that. That
doesn't seem to work. (I read a recent (1999) paper by a young
French author discussing some of these mathematical points on
the Los Alamos preprint site, called "Mathematical Surprises in
Quantum Mechanics". I'll find the details if anyone is interested).
(ii) The other reason is that serious progress in *real* theoretical
physics has been obtained by trying to replace the basic
differentiable manifold structure by a more complicated manifold
structure. Eg, one moves to higher dimensions with
compactification (as in Kaluza-Klein theories, supergravity and
superstrings) or one considers various fibre bundle structures over
the base manifold (Yang-Mills guage theories). Recent
speculations, about "spacetime foam", make the structure even
more complex, not less compex.
Replacing the reals by the ratios is primarily motivated by
*epistemological* considerations, about what we (finite beings) can
*experimentally measure* (ratios). But I'm very sceptical that this
program of eliminating the reals from physics works though. (In
fact, I think it is reactionary and anti-progressive).
Jeff Ketland
Dr Jeffrey Ketland
Department of Philosophy, C15 Trent Building
University of Nottingham, University Park,
Nottingham, NG7 2RD. UNITED KINGDOM.
Tel: 0115 951 5843
Fax: 0115 951 5840
E-mail: <Jeffrey.Ketland at nottingham.ac.uk>
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