On the use of Hilbert Spaces to represent states

[the_pulp]

Sorry if you've received the impression that your thread was being hijacked. The posts were indeed mostly relevant to your topic, but clearly that was not obvious. [Although... perhaps the stuff on Naimark's theorem should be moved to a different thread.]

Did you get (or study) the point I was trying to make in my earlier reference (post #36) to Ballentine's derivation of quantum angular momentum spectra?

No, I did not study it (I thought it was related to another thing, but it seems that it wasnt!). I am going to do it today at night. In the meantime if you can give me a short summary of what is the relation between that and my doubt, very welcome.

 

[strangerep]

No, I did not study [Ballentine] (I thought it was related to another thing, but it seems that it wasnt!). I am going to do it today at night. In the meantime if you can give me a short summary of what is the relation between that and my doubt, very welcome.

Ballentine ch1 explains how we have physically meaningful observable quantities, and that we wish to construct probability measures over them.

Doing this in a linear space rather than a nonlinear space is simply easier. There's no point using a technically less-convenient space if we don't need to. Similarly, a metric-topological space has lots more nice properties than more general topological spaces, so we don't use the latter unless the physics can't be modeled satisfactorily in the former. Rigged Hilbert space is a step towards a more general structure, but we don't need to use that for simple cases.

My point about the quantum angular momentum spectrum was about how a basic Hilbert space framework and Hermitian operators already gives an important physical result: the half-integral angular momentum spectrum.

Not sure what else I can usefully say here. Maybe after reading the earlier chapters of Ballentine, you should re-articulate whatever issues remain unresolved...

 

[TrickyDicky]

Helloooo! I opened this thread because I was asking about linearity!

Another possible way I think one could motivate linearity is to note that elementary particles in quantum theory are considered point particles so they have no size (this also happens in the linear classical mechanics and electrodynamics theories and their idealized point particles) . Of course if that assumption was not physically exact I suppose it would introduce some non-linearity in the picture and QM as we know it would be just a linear approximation for a point.

We do know that we have to recur to QFT perturbative methods to have good approximations to certain experimental results (I'm thinking about the Lamb shift or the anomalous magnetic moment of the electron that are not so well approximated by either relativistic or NRQM).

 

[yoda jedi]

Yes, that was sort of my route. I was not very sure about:
1) The linearity of the states (I think that when someone says to me that the state space should be linear because of the superposition principle it sounds like "the state space should be linear because the state space should be linear").
2) The use of complex numbers

And, related to the born rule, I was not using Gleasons theorems because I was thinking about Saunder's paper, but it is more or less the same (I will read your paper). Now, Ballentine and the link you sent me about complex numbers I think I've got the ideas a little bit more ordered.

Thanks!

there is nonlinear quantum theory.

 

[TrickyDicky]

there is nonlinear quantum theory.

Not in any of the various QM textbooks I've consulted, do you have some reference?

 

[yoda jedi]

Not in any of the various QM textbooks I've consulted, do you have some reference?

Diffeomorphism Groups and Nonlinear Quantum Mechanics
http://iopscience.iop.org/1742-6596/343/1/012006/pdf/1742-6596_343_1_012006.pdf

Nonlinear quantum mechanics, the superposition principle, and the quantum measurement problem
http://www.ias.ac.in/pramana/v76/p67/fulltext.pdf



.