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From: Tim May <tcmay.domain.name.hidden>

Date: Tue, 31 Dec 2002 10:24:11 -0800

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

Date: Tue, 31 Dec 2002 10:24:11 -0800

On Tuesday, December 31, 2002, at 07:02 AM, Joao Leao wrote:

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

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