On Tuesday, December 31, 2002, at 07:02 AM, Joao Leao wrote:
I don't agree with Tim's suggestion that infinite-dimensional Hilbert spacesI 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.
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.
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
