Penrose's original spin networks and 3-space

[S.Daedalus]

Recently, I read two early papers of Penrose on spin networks (made available online by John Baez http://math.ucr.edu/home/baez/penrose/" ), familiar today of course because of their use in loop quantum gravity.

In their original form, however, they seem to me to have been quite different beasts: as far as I see, the idea was to let a quantum system define its own geometry, via interaction through the exchange of 'n-units', i.e. subsystems carrying n units of spin 1/2. The spin networks then were just the representation of these exchanges.

Remarkably, it turned out that these spin networks reproduce the angles of Euclidean 3-space by the following procedure: you 'detach' a spin-1/2 unit from one large unit, and 'attach' it to another; the probability that the total spin of the second unit decreases then is given by the well-known $p = \cos^2\frac{\theta}{2}$; this is taken to define the angle between both units.

This much, I was aware of abstractly -- as in 'spin networks reproduce the angles of 3-space' --, but two conclusions Penrose drew seemed very noteworthy to me. One, that the background geometry some quantum system is placed in doesn't matter -- from the system's point of view, it will always 'live' in 3-space -- the system defines its own geometry. Two, and I think this is more hypothesized about, this may pose an explanation for some quantum mechanical weirdness -- an electron 'going both ways' through a double slit experiment doesn't split in its own geometry; it simply doesn't live in quite the same geometry the two slits do, but brings its own geometry with it. These things, I hadn't heard about previously. So, my questions are:

1) Is what I said so far about right?
2) What's the modern view on these things -- quantum systems creating their own geometry, etc.?
3) Has the work in this way, i.e. apart from loop gravity, been continued? I'd especially like to know about elaborations on the idea that the particle and the slits in a double slit experiment don't share the same geometry; Penrose cites it as an idea of Aharonov, but the only source is 'private communication'.

Thanks in advance!

 

[S.Daedalus]

Is nobody interested in this? Or do my questions not even make enough sense to address?

 

[yoron]

They are new to me, and quite interesting. But when you talk about a 'background' to it? I thought this was a way to do away with that concept? That it was the 'background' in itself? Or maybe that came later?

 

[Naty1]

Yes they were "different beasts".

You might find reading pages 946-955 of Penrose THE ROAD TO0 REALITY of interest.

Penrose explains his original motivation was to describe physics in "discrete combinatorail quantities" because of his belief that spacetime should "be based, at root, on discreteness, rather than continuity..." He goes on to say there is "...no actual place for gravity in the spin networks, as originally put forward." and he expresses initial surprise at finding spin networks playing such an important later role in quantum gravity theory.



There is quite a bit in those pages I don't understand, but he clearly states some major differences between what he put forward and those apin networks in use in quantum gravity:

..."the loop-variable networks acquire additional topological structure from their embedding in the manifold S.."
..."...our loop descriptions have given...just a static description, with no dynamics involved...the formalism seems not yet to have solved the more difficult problem of the dynaical evolution away from S (sometimes referred to as the 'Hamiltonian constraint"...

and goes on to say there is not yet an "...accepted solution to these difficult dynamical issues..."

 

[Naty1]

What's the modern view on these things -- quantum systems creating their own geometry, etc.?

Also new to me, and ditto...very interesting.

This recent paper from Rovelli (9/2011) does not seem to utilize such an approach within loop quantum gravity...he starts with Hilbert space...but I am far from "expurt" here.

"I describe a possible perspective on the current state of loop quantum gravity, at the light of the developments of the last years..."

http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.1780v2.pdf

I believe Marcus here follows some of this and I believe he originally posted this paper here...hopefully he will comment.

 

[S.Daedalus]

They are new to me, and quite interesting. But when you talk about a 'background' to it? I thought this was a way to do away with that concept? That it was the 'background' in itself? Or maybe that came later?

Yes, I believe that came with the realization that 'spin networks' can be used to parametrize solutions in loop quantum gravity; their earlier version is derived from the concrete picture of quantum systems exchanging spin-1/2 units, which defines an 'internal' geometry, i.e. a geometry any observer made of those units would experience.

and goes on to say there is not yet an "...accepted solution to these difficult dynamical issues..."

Oh yes, I had forgotten that Penrose talks about them in The Road To Reality. However, I'm not really interested in their application to gravity; though in the earlier papers I referenced, he seemed quite optimistic about a connection, because of the fact that two large units that only interact weakly -- i.e. only exchange a small number of spin units -- don't live in quite the same geometry, which seems to point naturally to a dynamic nature of geometry -- differences in interaction between those units causes differences in the geometry they share.

In that context, I had an idea I can't resist bringing up, even though it's probably stupid: if some set of quantum systems each live in their own, 'local' geometry, and only weakly interact, what about the global U(1) phase symmetry of QM? Wouldn't that have to be replaced by a local version, to reflect the local nature of the geometries?

By local geometry, I think I mean something like: say there's some background space in which all the quantum systems are embedded; picture some sturdy Newtonian thing for concreteness. Now, through the spin network picture, the quantum systems all define their own geometries, with a slightly different geometry for each quantum system -- i.e. the geometry changes with reference to the background space. A global U(1) transformation, meaning a transformation in the background space, will leave everything unchanged; but from the internal point of view, it wouldn't look like a global transformation -- it would look different here than it looks over there.

More concretely, if there's some coordinate transformation x'(x) connecting the background space's coordinates and the local geometry for one quantum system ('large unit') here, and another transformation x''(x) for a quantum system over there, then the global transformation, i.e. a multiplication with some phase factor $e^{i\chi}$ with constant $\chi$, will have a different form in the coordinates x' than it has in the coordinates x'' -- it will become an apparently 'local' transformation. Since the behaviour of the quantum systems nevertheless must be invariant, we must then introduce covariant derivatives to ensure this -- which, of course, means the introduction of an electromagnetic potential in this case. So, what gives? Does local geometry localize global symmetries?