What is the significance of the Sahlmann algebra in loop quantum gravity?

[marcus]

I've been looking at papers by the current postdocs, often they will be by two or more postdocs, or several plus a senior person.
People age so fast in mathematical physics research already
Ashtekar, Rovelli, even Baez?, Lewandowski, Thiemann are old
and they have to bring up a new generation.

Daniele Oriti (a guy, Italians spell Daniel that way)
Hanno Sahlmann
Martin Bojowald
Etera Richard Livine (dont know if he or she)
Robert Oeckl
Florian Conrady (new name, Heidelberg and Berlin)
Andrzej Okolow (only the one paper that i know of)
Sergei Alexandrov

Funny thing, when you see an online photo of someone like
Rovelli he looks really young. Baez too.

I do not think the list is complete or representative but I notice some things: like there seem to be a lot of people from
Cambridge, Potsdam (MPI and U. Berlin), Marseilles, Lyon, Warsaw. People from Madrid and Mexico City also come up, though as it happens I haven't listed them.

May be a thought here, maybe not. The fact that Rovelli seems to have moved back from Pittsburgh etc to Marseilles. Maybe the weather is nicer.

 

[marcus]

what mental image comes to mind?

In a previous post you asked what mental image comes to mind regarding, I think, the "electric field" or "inverse densitized triad" E.

It is a good kind of question to consider. As you say "how to get one's mind around" the Ashtekar new variables (A, E)

It is interesting that A gets integrated along curves and E gets integrated over 2D surfaces so that what one ends up dealing with are not (A,E) but (holonomies, fluxes).

Notations like h_e[A] for integrate the connection A along the edge e.

And like E_{S, ƒ}[E] for flux of E thru surface S sampled by function ƒ.

So in the end it is cylinder functions (holonomies) and derivations (made using fluxes)

This generates an urge to turn around and go back to the roots.
What is the physical meaning of A,E?
Connections are pretty conceptual and graphic---not much trouble visualizing a tangent vector veering as it is moved about.
But how to imagine E?
What kind of feel do you get about the manifold from knowing E, what kind of grip on it do you get.
the fact that E is essentially an inverse triad seems important.
something that will take a vector and write it out for you in a special basis or frame
and that it remembers the metric
well the events of the day are tugging, must go

 

[marcus]

more intuition about A, E

from Ashtekar's not-too-technical 2002 paper "Q. Geom. in Action"
http://arxiv.org/math-ph/0202008
page 5

"Let me now turn to specifics. It is perhaps simplest to begin with a Hamiltonian or symplectic description of general relativity. The phase space is the cotangent bundle. The configuration variable is a connection, A on a fixed 3-manifold Σ representing 'space' and (as in gauge theories) the momenta are the 'electric field' 2-forms E, both of which take values in the Lie-algebra of SU(2)."

Notice he has them be 2-forms valued in G' from the start (others might begin with vector densities valued in G'* and then take the dual and get 2-forms to integrate over surfaces, but he is just a little more direct) Then he refers to the "orthonormal triad" interpretation, to give a little geometric intuition about them.

"In the present gravitational context, the momenta acquire a geometrical significance: their Hodge-duals _*E can be naturally interpreted as orthonormal triads (with density weight 1) and determine the dynamical, Riemannian geometry of Σ. Thus, (in contrast to Wheeler's geometrodynamics)
the Riemannian structures on Σ are now built from momentum variables.

The basic kinematic objects are holonomies of A, which dictate how spinors are parallel transported along curves, and the 2-forms E, which determine the Riemannian metric of Σ.

(Matter couplings to gravity have also been studied extensively [2, 1].)..."

 

[marcus]

E describes "the flux lines of area"

Continuing the quote from Ashtekar on page 5 of the same paper, there is another bit of intution about E.

"...In the quantum theory, the fundamental excitations of geometry are most conveniently expressed in terms of holonomies [3, 4]. They are thus one-dimensional, polymer-like and, in analogy with gauge theories, can be thought of as 'flux lines of the electric field'. More precisely, they turn out to be flux lines of areas: an elementary flux line deposits a quantum of area on any 2-surface S it intersects. Thus, if quantum geometry were to be excited along just a few flux lines, most surfaces would have zero area and the quantum state would not at all resemble a classical geometry.

Semi-classical geometries can result only if a huge number of these elementary excitations are superposed in suitably dense configurations [13, 14]. The state of quantum geometry around you, for example, must have so many elementary excitations that 10⁶⁸ of them intersect the sheet of paper you are reading, to endow it an area of 100 cm².

Even in such states, the geometry is still distributional, concentrated on the underlying elementary flux lines; but if suitably coarse-grained, it can be approximated by a smooth metric. Thus, the continuum picture is only an approximation that arises from coarse graining of semi-classical states..."

 

[selfAdjoint]

Thanks Marcus, for all these explanations. I am going to carefully go over the Ashtekar 2002 paper you linked to. 2-forms with values in G' at first sound more reasonable than vector densities with values in the dual space of G', but in fact - what is a 2-form? It's something times dx/\dy (being cheesy about the coordinates). Is he saying that for every such dx/\dy we have an associated map into G' (linear? preserving product?)

Or consider that tensor product from O-L, τ^* {X} ∂ (suppressing indexes). Can we look on that as a (dual) 2-form?

BTW I think the relation of a vector density to a 2-form is via that completely antisymmetric Levi-Civitta density. A vector density of weight one changes sign on reflection (because the determinant of the Jacobian of a reflection is -1).

 

[marcus]

Originally posted by selfAdjoint

BTW I think the relation of a vector density to a 2-form is via that completely antisymmetric Levi-Civitta density. A vector density of weight one changes sign on reflection (because the determinant of the Jacobian of a reflection is -1).

I agree, Levi-Civita usually written with epsilon

I just happened to have the Sahlmann/Thiemann paper (gr-qc/0303074) open to page 3 where it says:

E_S,ƒ = ∫_S E^a_i ƒⁱε_abcdx^bdx^c

 

[marcus]

selfAdjoint, your question "what mental image" related to E or tilde-E has gotten me focussed on getting some intuition. After lunch I was thinking one thing space is doing all the time is expanding----the metric, distances between pairs of stationary objects like galaxies, is increasing all the time...so does this show up in E?

What happens to the electric field or the "densitized inverse triads" or the fluxes thru sample surfaces?

All this stuff at some level is fairly straightforward and I think there is a straightforward answer----E gets bigger.

The area and volume operators---selfadjoint operators on the hilbert space, observables---they are calculated from the E's

There is a derivation, we did not go thru it here yet but it is
in several papers.

So as the E's get bigger all the areas and volumes of things will get bigger. because E is at the root of those observables.

E is something, on the tongue-tip, it is an idea---cant quite think of the word. But it is understandable, with some intuitive content. Maybe something will occur to you

 

[marcus]

Something has happened to Thomas Thiemann's expository
style. His October 2002 "Lectures on Loop Quantum Gravity"
is loaded with intuitive perceptions about the basic
elements of the theory with even some almost cartoon-like sketches and is considerably more readable than his
October 2001 "Introduction to Modern Canonical Quantum General Relativity" (written for LivingReviews).

One might try to dispose of the difference by saying he's targeting
a different audience but I doubt that explains even half
of it

Here's the link to "Lectures" in case anyone wants to check it out
http://arxiv.org/gr-qc/0210094

 

[selfAdjoint]

It's not just the audience but the occasion. The lectures are, well, _lectures_. To what looks lkike a bunch of bright grad students. The living reviews piece is a high-falutin' non-working, get it right or rue it megillah.

BTW I found a quick and dirty intro to these matters in Week 7 of Baez's This week's finds.

 

[marcus]

Originally posted by selfAdjoint
It's not just the audience but the occasion. The lectures are, well, _lectures_. To what looks like a bunch of bright grad students. The living reviews piece is a high-falutin' non-working, get it right or rue it megillah.

BTW I found a quick and dirty intro to these matters in Week 7 of Baez's This week's finds.

Glad you mentioned Week 7 of Baez finds. I had looked at it some time ago but got more out of it when I went back to it yesterday

I will prefer any day Thiemann hobnobbing with the grad students to Thiemann making pronouncements ex cathedra like he is the pope (in an ecumenical spirit I have broadened the ethnic scope----stuffed shirts are universal---but must concede the metaphor of the megillah is hard to beat)