On Tuesday, December 31, 2002, at 07:02 AM, Joao Leao wrote:
> I don't agree with Tim's suggestion that infinite-dimensional Hilbert
> spaces
> are somewhat "ancilliary" in QM and that all systems are calculable in
> finite dimensional modes. In fact infinite sets of spaces are the rule
> in
> QM and
> the finite dimensional subspaces only serve as toy systems.
I said it is often done. Many of the details of the infinite case are
just not needed. And QM is often taught this way, with no loss of
rigor, provided any subtleties are pointed out to the student.
For example, here are some fairly typical lecture notes for a course on
QM:
"2.2. Hilbert Space
Hilbert spaces are mentioned in most textbooks on quantum mechanics and
functional analysis [3] . Therefore we will only mention some features,
which are not found almost everywhere. We will also not have to go into
the subtleties of topologies, continuous spectra, or unbounded
operators, because throughout this course, we can assume that all
Hilbert spaces are finite dimensional. Modifications in the infinite
dimensional case will be mentioned in the notes. Our standard notation
is <\phi ,\psi > for the scalar product of the vectors \phi ,\psi \in
H, ||\phi ||=<\phi ,\phi >1/2 for the norm, and B(H) for the algebra of
bounded linear operators on H. Of course, all linear operators on a
finite dimensional space are bounded anyway, and the B is used mostly
for conformity with the infinite dimensional case. "
http://www.imaph.tu-bs.de/qi/lecture/qinf21.html
In nearly every area of physics, the issue of "infinity" is phrased in
terms of sequences or structures approaching or growing towards the
infinite or infinitesimal. For example, a test mass is assumed to be
small enough not to perturb the curvature tensor. But actual infinite
spaces are not needed, not even in thermodynamics.
This dispenses with a lot of the mathematical cruft, alluded to above
(continuous spectra, compactness, etc.). That cruft contains a lot of
beautiful math, but physics just doesn't need it, at least not very
often.
I have no axe to grind on this. For those who want to study only the
completely general, infinite-dimensional cases, cool. But a good
understanding of finite-dimensional vector spaces (e.g., the Halmos
book) provides the math one needs for QM, especially at the level we
usually discuss it at here. (As many here perhaps already know, Halmos
was Von Neumann's assistant, writing up his lectures, when he wrote his
book.)
Provided the complex space is normed, and is complete, which all
finite-dimensional vector spaces are, the math works. No infinities are
needed, which is good.
--Tim May
Received on Tue Dec 31 2002 - 13:26:55 PST