ΛEPRL quantiz'n of cosmological horizon area in Planck units

[marcus]

The ΛEPRL spin foam model presented 25 November at ILQGS by Haggard and Riello achieves an interesting quantization of the cosmological constant. Basically this is done on slide #10 around minute 15 of the audio.
http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.pdf
http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.wav

I want to talk about the basic philosophy of their generalization of Regge using simplices with homegeneous curvature, that they call "Λ Regge" and how the quantization comes about.

But before giving motivation and overview, I will just very briefly indicate what is shown on slide #10 and said in the audio for a few seconds starting around minute 15:30. You can drag the time button and listen selectively to that section of the talk, if you wish.

In their notation R_Λ is what in Jorrie's Lightcone calculator is called R_∞, the longterm asymptotic value of the Hubble radius. The Hubble radius, c/H, is currently about 14.4 Gly and is expected to grow and approach the limit of 17.3 Gly. The practical effect of the positive cosmological curvature constant Λ (a kind of residual vacuum curvature) is that the percentage distance expansion rate H (now about 1/144 % per million years) is not expected to decline to zero, but to decline ever more slowly and level off at 1/173% per million years. This places a bound on the growth of its reciprocal, c/H, the Hubble radius.

So in their equation on slide 10 they refer to the eventual area of the Hubble sphere 4π R_Λ². And they give a LQG theoretical argument, using their ΛEPRL spin foam model, that this area must be an integer multiple of the Planck area, scaled by the Immirzi parameter, namely γl_P².

So the picture is there is a big sphere with radius 17.3 billion lightyears, which will eventually be our cosmological event horizon, with distances larger than 17.3 Gly growing faster than light and objects near or on the horizon being redshifted without limit. so that time for them appears to stop.

And the theoretical result is that this large spherical area comprises an integer number of little Planck sized areas. In LQG it is not uncommon for the Planck area l_P² to be multiplied by the Immirzi gamma number γ and that happens here.

So you can see the equation on slide 10 saying 4π R_Λ² ∈ γl_P²ℕ
In other words the eventual cosmological horizon area is some natural number times gamma by the Planck area.

 

[marcus]

The important thing to realize here is that the cosmological constant Λ is a CURVATURE, and in Regge calculus, the circa 1960 ancestor of SIMPLICIAL approaches to General Relativity, one uses FLAT simplices. Flat 3d simplices called tetrahedra and flat 4d simplices called pentachorons.

But if there is a residual constant "vacuum curvature" throughout space and time, shouldn't we be using simplices with constant homogeneous curvature?
The other curvature over and above that constant baseline can be represented by "DEFICIT ANGLES" going around points in 2d, or going around lines in 3d, or going around triangles in 4d. You know how it works: a triangulated 2d surface will "pucker" at a point if the triangles that meet at that point don't provide the full 360 degrees of angle. That's a deficit. That's the sort of thing which in GR is caused by MATTER.

What Haggard and friends are doing is generalizing Regge to LAMBDA-Regge, where this cosmological constant vacuum curvature is built into the simplices!
The Pents and Tets actually have curved sides and curved edges! I don't know if they got the idea from somewhere or who had it first, it is just a beautiful idea. The cosmological curvature is not obviously caused by any thing, and it is uniform, homogeneous. So why make the simplices FLAT? Give them all this homogeneous curvature, and let the additional curvatures that are CAUSED by matter and stuff show up as deficit angles in accordance with the simplicial version of the GR equation.

 

[marcus]

They present this step by step (starting from Regge's building block approach to GR). I think they do this in a nice understandable way.
==quote slide 6==
Regge’s philosophy:
construct a manifold out of flat building blocks + (d - 2)-dimensional defects

At the quantum level,
flatness is implemented via BF dynamics in the bulk of building blocks, while defects are created by using only geometric boundary states[thus breaking BF symmetries]

GR = BF + geometricity constraints
==quote slide 7==
ΛRegge’s philosophy:
construct a manifold out of homogeneously curved building blocks + (d-2)-dimensional defects [Bahr&Dittrich]

At the quantum level,
the homogenous curvature is implemented via BF - (Λ/6) BB dynamics, and defects are created as in the flat case

ΛGR = BF - (Λ/6) BB + geometricity constraints
==endquote==
I'll interject here that EPRL is very much a boundary formalism where you want to calculate amplitudes for a given boundary surrounding a spacetime region or process. One can think of the boundary as specifying initial and final geometry and the amplitude as a "transition amplitude" (or a path integral or a sum over histories) However you think of it, if the authors want to define a version of EPRL with Λ, a "ΛEPRL", they have to be able to calculate amplitudes of boundary geometry. They just finished telling us about ΛBF theory, i.e. BF theory into which Λ has been introduced. So on the next slide they introduce the CHERN SIMONS functional on the boundary, and they say:
==quote slide 8==
For boundary connection functionals, ΛBF in the bulk is equivalent to CS on the boundary
==endquote==
Then in slide 9 there is some hocus-pocus involving the Chern Simons LEVEL, which they denote by lowercase h.
They find the level is complex and that its real part must be an integer.
And then, as I mentioned at the start, on slide 10 they conclude that the cosmological constant Λ is quantized.

That is only one of the fun exciting things happening in this paper. There are 26+5 slides in the ILQGS set, and the audio has observations not explicitly noted in the slides.
http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.pdf
http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.wav

 

[marcus]

It's pretty clear this is a breakthrough paper, that they are reporting on in seminar.
===quote from slide #26 http://relativity.phys.lsu.edu/ilqgs/haggardriello112514.pdf ===
Conclusions

SL(2,C) Chern-Simons theory is a tool for substituting BF theory with BF- (Λ/6)BB
and leads to a quantized cosmological constant

Can reconstruct curved geometries in 3 and 4D from the resulting eqs of motion

• Phase space and quantized curved geometries

Connect Chern-Simons theory and the cosmological constant in 4D ---clarifies the role and origin of quantum groups
==endquote==
It was important to incorporate the cosmological curvature constant into the very simplices of the EPRL spin foam model
(based as it is on BF theory).
The way Λ was brought into EPRL earlier was through a quantum group: a quantum variant of the group. It wasn't clear why it should enter in that purely algebraic way. Now they have shown a deeper geometric reason. As they point out, their work "clarifies the role and origin of quantum groups" in this QG application.

 

[marcus]

My take on this is that it's one of those cases where a significant development in mathematics enables a series of advances in physics.
For an important math development think about the 1974 discovery of the Chern–Simons 3-form given by

This is where A is a Lie algebra valued 1 form---could be a "connection" --- and F is the corresponding curvature 2-form dA + A∧A.
So if you multiply it out you get dA∧A and an extra A∧A∧A and the trace becomes of dA∧A + (2/3) A∧A∧A, which may be more familiar. It is often written that way with the slightly puzzling 2/3. But the way I originally stated it is easier to swallow: two 3-forms: one made of the CURVATURE of the connection (wedged with itself to get a 3-form) and also, on the other hand, the "VOLUME" of the connection, multiplied by 1/3, or more exactly a volume associated with it.
Now we are interested in the EPRL PATH INTEGRAL or "sum over histories" approach so something 4d made of 4-simplices. (Pents, not Tets, "Pent"is short for pentachoron, the word for 4 simplex analogous to tetrahedron). So why are we interested of integrating this 3-form, the CS form?

Because the boundary of a 4d triangulation is a 3d triangulation, made of Tets. EPRL is very focused on the boundary of a history, a "transition amplitude" calculated from the information living on the boundary.

So Haggard and Riello take the important first step. They say let's just look at one Pent, topologically it is a 4-ball and it boundary is S³, the three-sphere or hypersphere. The five tetrahedral faces of the Pent. And let's now assume a homogeneous innate CURVATURE infuses all these things--a curvature intrinsic to 4d world geometry.
Instead of FLAT simplices (as in Regge calculus and usual triangulations) we have curved ones. We're going to do a new Regge and a new EPRL with these chunks of homogeneous constant curvature space-time. [And BTW, not sure about this but I think the Pent only has the constant Λ curvature BUT its boundary Tets could have additional curvature over and above Λ. Have to check this by taking another look at the ILQGS slides.]

And when we want to study the boundary, whether of a single curved 4-simplex or of a whole simplicial complex history dual to a foam, we can integrate the Chern-Simons form over the boundary and relate that to information residing in the bulk.

 

[marcus]

Incidental information: It's interesting that within a week of this ILQGS presentation by two of the four co-authors, another of the four gave a seminar at Erlangen on the same research, namely Muxin Han (Florida Atlantic U)
==quote http://www.gravity.physik.fau.de/seminars/groupseminar.shtml ==
IQG, FAU Erlangen
"Chern-Simons Theory, Flat Connections and 4d Quantum Geometry"
This talk explains the relation between a class of flat connections on a 3-manifold and the constant curvature simplicial geometries in 4-dimensions. The quantization of 4d simplicial geometry can be carried out via the quantization of flat connection on 3-manifold. Such a procedure gives a relation between quantum SL(2,C) Chern-Simons theory on 3-manifold and Loop Quantum Gravity on 4-manifold.

Wed, 19 Nov
16:00-17:00 SR 02.729 Prof. Muxin Han
==endquote==
I don't know when their paper is going to appear, I hope within a month or so. We need to keep track of all four of the co-authors: Haggard, Han, Kaminski, Riello (first names Hal, Muxin, Wojciech,Aldo/locations Bard, Florida Atlantic, Warsaw, Perimeter)

 

[inflector]

Can't quite follow all of that yet, but I'm quite pleased with the ways that pentachorons are showing up in many places of late. They first appeared, at least in my purview, in CDT quite a few years back and seem to be used much more in the new exciting progress of the last couple years.

I always found simplices to be a more natural foundation than rectilinear hypercubes / tesseracts. Highly unlikely the universe is built on blocks that aren't as simple as possible and simplices are that, if nothing else.

 

[marcus]

Can't quite follow all of that yet, but I'm quite pleased with the ways that pentachorons are showing up in many places of late. They first appeared, at least in my purview, in CDT quite a few years back and seem to be used much more in the new exciting progress of the last couple years.

I always found simplices to be a more natural foundation than rectilinear hypercubes / tesseracts. Highly unlikely the universe is built on blocks that aren't as simple as possible and simplices are that, if nothing else.

I agree! I've watched CDT over the years, ten years now, I guess, and really admire that approach. "Triangulation" is the key word, where you approximate some natural shape with simplices. Renate Loll (a fine expositor among her other accomplishments) has a graphic of a RABBIT, approximated by a triangulation.

Atyy just today pointed out a new CDT paper (about the theory's spontaneous dimensional reduction). It is by a frequent Loll collaborator, Jurkiewicz and a new person named Dan Coumbe.
It verifies that if you take 4d CDT and measure the dimensionality by a diffusion process (the higher the dimension the easier it is to get lost and never wander back home) then of course it comes out 4d at usual scale but then if you go DOWN AND DOWN in the scale you are running the diffusion process at the dimensionality actually shrinks continuously down to BELOW 2d! You might enjoy taking a look, the paper is nicely written. You could probably search "Coumbe" at arXiv and get it.
Yes, that gives:
http://arxiv.org/find/grp_physics/1/au: coumbe/0/1/0/all/0/1 And today's paper is at the top of the list.
I'll try googling "coumbe dimensional reduction" and see if that gets it. Yes!
Dan Coumbe got his PhD at Glasgow in the UK just last year, I see.

I agree that simplexes are beautifully simple. In any given dimension D you just take D+1 points which the least number of points needed to describe a nontrivial volume and form the convex body defined by those points. A triangle if it is D=2, and so on.

Any idea if the "Lambda Regge" idea that Haggard Han Kaminski Riello have applied to spinfoam would make sense in CDT?
I mean, instead of making all the identical pents that CDT uses FLAT (the way they usually do) make them all with a very slight curvature Lambda?
I wonder if that idea would interest a CDT researcher, haven't thought about it at all.

 

[atyy]

What's up with the rabbits? Leporello in the other thread and a bunny triangulation here. Seems we might have a major competitor to turtles all the way down! :D

 

[marcus]

Loll used this triangulated rabbit in one of her popular articles

 

[marcus]

Do you know the opera D.G.? Leporello is one of the great comic servant characters of all time, I think.
In the Spanish original the libertine's valet was "Catalinon" and in the French version by Moliere he was "Sganarelle". Not sure, but I suspect Mozart and Da Ponte created the character we know as Leporello. Hard to know what they had in mind.

 

[atyy]

It's not one I've ever seen in full, but I know the famous soprano arias, and have seen almost all the versions of the famous final scene on Youtube in which the stone statue comes to live! I didn't know what Leporello meant until I saw your remark, but I think Little Rabbit is just about right for Mozart and Da Ponte's comic character.

 

[inflector]

Loll used this triangulated rabbit in one of her popular articles

A year or so back, I had a couple-hour discussion on physics ideas with David McConville, chairman of the Buckminster Fuller Institute, and one of the principals of elluminati who produces spherical projection systems for museums among other things. One of the more interesting aspects of the discussion was learning how much 3D graphics programming ideas represent simplest cases for more complex representations.

Decomposing a surface into triangles is a staple of graphics programming and modeling. It's the simplest way to get the job done. Very flexible and very fast. The graphics chips are optimized for operations (like shading) on lists of triangles, for instance.

Any idea if the "Lambda Regge" idea that Haggard Han Kaminski Riello have applied to spinfoam would make sense in CDT?
I mean, instead of making all the identical pents that CDT uses FLAT (the way they usually do) make them all with a very slight curvature Lambda?

I can't see how this would be too hard to do from a computer modeling perspective, once you had a regular CDT model running. It might take a lot longer to run the simulations but it shouldn't be too great an addition. Seems like a great idea for a post-doc who wants to make her mark.