What is the connection between quantum mechanics and geometry?

[selfAdjoint]

In http:/arxiv.org/hep-th/0104028, a paper titled What is Quantum Gravity begins by addressing the question What is Quantum Mechanics. Building on several years of work (cited in the paper as reference [1]) the authors derive non-relativistic quantum mechanics from two principles:

I. Quantum observations are statistical. That is each observation defines a probability distribution, and the space of such distributions, with a natural (Fisher) metric is the world of quantum observation.

II. The kinematics of quantum mechanics is a Hamiltonian flow on the space of distributions.

So far, the ideas could have been stated in the 19th century. But the geometric ideas necessary to flesh out the principle only became available in the later 20th century. The upshot is this:

The above geometric structure describing canonical QM, beautifully tested in numerous experiments, is also very robust from the purely geometric point of view [1]. A consistent generalization of QM would doubtlessly be interesting from both the experimental and
theoretical viewpoints. Unlike various generalizations proposed in the past (which in many instances have lead to difficult conceptual problems) the one put forward in [1] extends the kinematical structure so that it is compatible with the generalized dynamical structure! The quantum symplectic and metric structure, and therefore the almost complex structure become fully dynamical.


By the way, in this approach the complex projective n-space arises naturally, leading us perhaps a little way toward twistor space.

 

[marcus]

Originally posted by selfAdjoint
In http:/arxiv.org/hep-th/0104028, a paper titled What is Quantum Gravity begins by addressing the question What is Quantum Mechanics...

I think there's a typo in the link and it should be
http://arxiv.org./hep-th/0401028

by Minic and Tze

 

[marcus]

I see Minic and Tze thank Tom Banks in the acknowledgment section.
And one of the papers they cite is a stringy one by Leonard Susskind and Tom Banks.

And the paper was written for a Santa Barbara conference, I think while they visitors at Santa Barbara.

But this is not an explicitly stringy paper. It looks to me more like
axiomatic development of quantum theory. They hold with background independence and quantizing General Relativity itself, or so it appears.

The main thing they cite seems to be earlier papers by themselves (or by Minic and others) going back a while which they have been publishing in, like, Physical Review Series D. I will fetch a couple of recent links.

http://arxiv.org/abs/hep-th/0305193
"Background Independent Quantum Mechanics and Gravity"

http://arxiv.org/abs/hep-th/0309239
"A general theory of quantum relativity"

Right now I can't tell, but does anybody want to try explicating?
It could be interesting.

 

[selfAdjoint]

Originally posted by marcus
I think there's a typo in the link and it should be
http://arxiv.org./hep-th/0401028

by Minic and Tze

Yes thank you. I refrained from going back through the links (some of which did look stringy), because the overall logic of their development was what fascinated me. "Statistical" observations plus Hamiltonian dynamics necessitates a complex sphere in distribution space, hence via Hopf fibration a complex projective space and thence quantum mechanics. Who knew?

 

[marcus]

Originally posted by selfAdjoint
... "Statistical" observations plus Hamiltonian dynamics necessitates a complex sphere in distribution space, hence via Hopf fibration a complex projective space and thence quantum mechanics...

BTW when CP(n), the complex projective sphere of n dimensions, came up I could not immediately see where the dimension number n came from. Must have overlooked something obvious.

What shall we do with this miniseries of Minic papers? Do you think it would be worthwhile for you to explicate them a bit and make them more accessible? I don't know how much help I would be, or even what I think of the papers so far---they are very new to me and "neither fish nor fowl"

 

[selfAdjoint]

Marcus, I can't guarantee anything but I'll try to dig into the earlier papers and see what I can come up with. I've got two other physics things taking my time; I'm in a quantum mechanics course on another site (Sakurai), and I'm studying a wonderful reference on BRST approaches that I found on hep-th. Even though I'm a retired widower with lots of time and few distractions, I'm not the workaholic I was way back in grad school.

 

[marcus]

Originally posted by selfAdjoint
... widower with lots of time and few distractions, I'm not the workaholic I was...

a lot of things matter more than physics, well, some do
I can see where widower could be tough
and I hope your daughter(s) is/are good company and don't live too far away, like outside Wisconsin

of COURSE no guarantees and promises! that is the nice thing
about a board like PF

I'm not even sure Minic merits our attention! (doesnt that sound
funny) and I am urging you to be the guineapig because
you brought Minic up and I want to hear how you explain things
like "Hopf fibration" and "complex projective sphere" to
Average PF Joe. Better you than me.

 

[jeff]

Originally posted by marcus
But this

http://arxiv.org/abs/hep-th/0305193
"Background Independent Quantum Mechanics and Gravity"

is not an explicitly stringy paper.

But they are using matrix theory hamiltonians to generate dynamics.

Originally posted by marcus
It looks to me more like
axiomatic development of quantum theory.

But it does propose genuinely novel physical ideas, and along with similar papers reminds us that requiring background independence needn't lead to LQG.

 

[selfAdjoint]

Originally posted by marcus
I want to hear how you explain things
like "Hopf fibration" and "complex projective sphere" to
Average PF Joe. Better you than me.

Well, I don't know about "average PF Joe", but here's a discussion of the Hopf fibration from http://csunix1.lvc.edu/~lyons/notes/hopf_reprint.pdf that is aimed at people with precalculus and a little linear algebra.

[edit] Aimed at high school teachers (I think), it has several "investigations" where the student is encouraged to find out some fact by working on it. The complex projective sphere is in there, if not by name; the work is mostly done with quaternions (a nice short intro to which is included).

 

[zack_vt]

Minic

Hello everyone,

I just found PF this evening and am excited to someday join in your conversations(fascinating stuff going on here). Marcus posed the question, "What shall we do with this miniseries of Minic papers?", and I find myself asking a similar question as I am currently intending to study with/under him. This thread hasn't been touched in a few weeks, so I don't know if you have come to a conclusion or not, but am very interested in your take on Minic's papers.

-newbie grad student with delusions of grandeur

P.S. Be warned, I'm going to start PM'ing folks in a few days.

 

[selfAdjoint]

zack, good to hear from you! Maybe you could suggest to Prof Minic that he might post here too? If you look on the Strings, Branes and LQG forum you will see that we do have professional physicists, even some well-known ones, who post on occasion.

I think what we need here, as Marcus suggested, is to have the more difficult concepts broken down for the interested non technical reader. I personally am very enthusiastic about this way of deriving quantum mechanics (and also its application to string physics). I hope we can get something going here.

[afterthought:] I think we can explain basic Hamiltonian theory to this audience. Would anyone care to comment on that? Also, even if we can't break it down for everyone, it would be good to discuss the pros and cons among ourselves.