Seething expanding geometry-new LQG view of reality

[marcus]

Seething expanding geometry--new LQG view of reality

A view of reality emerging from current LQG research (like 1004.1780) is somewhat analogous to the "seething vacuum" of quantum field theory. But it is a "seething geometry" in which bits of area and volume constantly come into existence and go out of existence.

I suppose that eventually the LQG geometry will seethe and boil not only with bits area and volume but also with matter. In other words, some elaboration of the math will be found so that particle fields living in the geometry can properly be added to the picture. (Several ways to do that have already been proposed, but it's still early days.)

The formalism uses a more complicated version of Feynman diagrams. A spin network is an instantaneous state of geometry: the nodes represent bits of volume and the links represent bits of area. Nodes and links are items which an observable---an operator measuring the volume of some physical region or the area of some physical surface---might or might not detect.

A spin foam is the history or "track" of a spin network as it evolves in time. To keep the terminology straight, different words are used to describe spin foams: vertex, edge, face. As a spin network evolves in time, its nodes describe edges, and its links describe faces. Geometric events (creation annihilation of some geometric element) occur at the vertices of the spin foam.

It is not hard to keep these few terms straight and it helps to use them consistently. Node and link (volume and area) at the instantaneous geometry level. Vertex, edge, face at the level of evolving geometry---the histories of a "sum over histories".

Because the spinfoam vertex is so critical---it is where important stuff like creation annihilation happens---the vertex amplitude practically defines the theory. The idea is that, amplitude being a complex number akin to probability, if you want to know the amplitude of evolving from geometry A to geometry B, you consider all the possible spinfoam diagrams that can get you from A to B and add up their amplitudes. The amplitude of each individual spin foam is the total amplitude of its parts (primarily the vertices.) So the vertex formula plays a key role in determining the dynamics of evolving geometry.

The "new Lqg" paper I mentioned ( http://arxiv.org/abs/1004.1780 ) gives rules to calculate these amplitudes for "geometry's Feynman diagrams." They aren't the only proposed rules--the situation is still being sorted out. Alternative ways of calculating amplitudes, which increasingly seem to give the same answers, are discussed in the paper.

There is a lot happening around this new version of LQG so it makes sense to have a thread to watch the proceedings. This week Rovelli will be teaching a minicourse on it at the Morelia QG school. The week after, starting 4 July there will be the triennial GR conference. Rovelli and others will be talking there. The April paper already has more citations than any other QG paper from the second quarter of 2010. So it seems reasonable to expect a research thrust along these lines.

The three main QG research areas I am personally most interested in watching are LQG, NCG, and EFT. NCG here means work by Connes and by Marcolli and her associates. EFT has become interesting because it appears that a UV fixed point exists (asymptotic safety) so that effective field theory of various types may afford a valid approach. Of these three (LQG, NCG and EFT) it seems that LQG is of particular interest because it proposes to identify microscopic geometric degrees of freedom.

The nature of these microgeometric degrees of freedom, if correctly identified, could help to explain why geometry enjoys the asymptotic safety of a UV fixed point.

 

[marcus]

If someone is looking for an introduction to the approach presented in the April "new Lqg" paper, this might be useful:
http://www.fuw.edu.pl/~jpa/qgqg3/CarloRovelli.pdf

These are lecture slides. In March 2010 there was a weeklong QG Workshop at the ski resort Zakopane. The talk covered the same material as the April paper, and focused on making the essentials accessible.

Here is the full list of talks:
http://www.fuw.edu.pl/~jpa/qgqg3/schedule.html

=============================================
EDIT TO REPLY TO NATY post #4 below.
Naty,
You mention the ideas of Jacobson and Verlinde. I would also include Padmanabhan, who does a good job presenting the same ideas.

The entropic force idea is compatible with the LQG picture and indeed challenges us to come up with something like LQG to explain the microscopic degrees of freedom which carry geometric temperature and entropy.

I probably don't need to say more than this to you, Naty, because you know the background. But in case others are reading I will recall the analogy that Padmanabhan made with Boltzmann's work back in the late 19th C, before people were sure there were even such things as atoms and molecules. As Padma says: If you can heat something it must have invisible microscopic degrees of freedom. Boltzmann fought for the acceptance of molecular theory of gas because you need molecules to absorb heat, have temp, and carry entropy.

Verlinde to explain entropic force uses the analogy of a long kinky polymer in a heat bath, that is a bath of Boltzmannic jiggle-balls constantly bumping the polymer. This puts more kinks in the polymer and makes it contract---presto! entropic force.

But Verlinde does not conjecture what, in geometric thermodynamics, could be the microgeometric degrees of freedom doing the bumping and kinking. A theory like LQG could be just the thing needed to explain at a detail level just how Verlinde/Jacobson stuff works. Like the molecules that Boltzmann's contemporaries were unsure about but which eventually vindicated Boltzmann's overall thermo picture of how gasses etc. work.

The possibility I just mentioned was pointed out in January 2010 soon after Verlinde's paper appeared:
http://arxiv.org/abs/1001.3668
The paper already has around 40 citations--which is respectable for a paper that has only been for out some 6 months.
http://arxiv.org/cits/1001.3668

 

[MTd2]

There is one thing I`d like to understand. There is the case of UV point for a truncation with matter, but what is the argument that it holds for truncation for any divergence for any divergence at any level with or without matter?

 

[Naty1]

... it is a "seething geometry" in which bits of area and volume constantly come into existence and go out of existence.

I suppose that eventually the LQG geometry will seethe and boil not only with bits area and volume but also with matter

.


Perhaps I don't get the real drift here in this latest LQG view, but this just doesn't seem as fundamental nor as potentially insightful as Verlinde and Jacobsen's ideas. What is the source of the geometric bits?? Are they viewed as fundamental here?

When Jacobsen says:

. How did classical General Relativity know that horizon area would turn out to be a form of entropy, and that surface gravity is a temperature? In this letter I will answer that question by turning the logic around and deriving the Einstein equation from the proportionality of entropy and horizon area together with the fundamental relation
_Q = TdS connecting heat Q, entropy S, and temperature T

and Verlinde:

. Gravity is explained as an entropic force caused by changes in the information associated with the positions of material bodies. A relativistic generalization of the presented arguments directly leads to the Einstein equations. When space is emergent even Newton's law of inertia needs to be explained. The equivalence principle leads us to conclude that it is actually this law of inertia whose origin is entropic.

Given a choice between bits of geometry on one hand or an information entropic basis for everything around us as an alternative, the latter seems more appealing...but maybe we are not ready to make such a big leap yet?? Or maybe the latter is just seems more "mysterious".

Interesting, Thanks Marcus.

 

[marcus]

Naty, I replied to your post in an edit to my post #2, above.
To continue talking about Rovelli's "new LQG" paper, if you have looked at it you will see that he has consistently adopted the combinatorial approach. This makes for a presentation of LQG which is quite different from what many of us are used to.

For one thing, the Hilbertspace of states is separable. That is also true for several other developments of LQG which have appeared over the past 3 or 4 years, but you still find people who don't know very much about LQG claiming that the state space is non-separable.
Separable simply means having a countable basis---technically easier to work with. But that is only a small part of what the combinatorial formulation accomplishes.

It is a lean, spare development that requires less hardware. The spin networks are abstract graphs, Γ , not embedded graphs. So you don't need a manifold to embed graphs in.

You might want to look at equation (1) on page 1, and at equations (2) and (3) on the next page. They are not hard to understand and form the basis for the discussion. Each distinct graph Γ is given its own "graph Hilbertspace" H_Γ.

The overall space of quantum states for the theory is approached by first taking a direct sum of all the individual graph Hilbertspaces, and then factoring down some by an equivalence relation to squeeze out a bit of redundancy.

So a major step towards understanding the state space (and subsequently the operators defined on it) is to first understand how the graph Hilbertspace H_Γ is defined, for one simple graph. Like, for example, the graph with two nodes joined by 3 links.
The recipe is given by equations (2) and (3).

I have to be off briefly but will return to this later. Explanations of the combi approach basics from anyone who cares to contribute---or simply comments---are of course welcome!

 

[MTd2]

So, you don't know how to answer my question, marcus?

 

[marcus]

...
So a major step towards understanding the state space (and subsequently the operators defined on it) is to first understand how the graph Hilbertspace H_Γ is defined, for one simple graph. Like, for example, the graph with two nodes joined by 3 links.
The recipe is given by equations (2) and (3).

I have to be off briefly but will return to this later. Explanations of the combi approach basics from anyone who cares to contribute---or simply comments---are of course welcome!

Comments especially welcome if they directly relate to the topic, which is the presentation of "new" LQG in the April paper http://arxiv.org/abs/1004.1780. The key thing that I want to focus on is the combinatorial approach---so I'm looking at eqns. (2) and (3).

For definiteness let's think of our graph as having two nodes joined by 3 links. In the notation of the paper N = 2 and L = 3. So we have a cartesian product SU(2)^L = SU(2)³. This tells the possible things that can happen (as per a connection) when you travel along these three links. Darn, have to go again. OK back now:

I guess we are all familiar with the Hilbertspace of square-integrable complex valued functions on some space X with a measure. The notation is L²(X). And a common notation for a complex-valued function on X is ψ(x). Quantum state spaces are often defined this way: "wave-functions" defined on a set X of possible positions or configurations. It's simple, we just have to avoid confusing the notation L² with the set L of links of the graph Γ.

Letters U and V are used for elements of the transformation group SU(2). The represent the kind of thing that can happen to you as you travel along a link "l" between two nodes----veering, rolling, slewing, skidding, tumbling---whatever. They also can represent what happens to you when you come in on one link, and pass thru the node, and go out on another link---the reorientation that might happen. (but that is results from arbitrary choice of orientations in how things have been set up. It is, as they say, "gauge" or unphysical, and will get factored out.)

OK so U and V are elements of the Lie Group, and a typical element of the cartesian product SU(2)³ is the "vector" or 3-tuple of group elements:
(U₁, U₂, U₃)

Now I need to pay a bit closer attention. The graph Hilbertspace H_Γ for the graph Γ is not simply functions on SU(2)³. It is that modulo a certain factoring out of arbitrary unphysical detail. Rovelli writes it this way in equation (2)
Hilbertspace H_Γ = L²(SU(2)^L/SU(2)^N-1)
= L²(SU(2)³/SU(2)¹)

Two square-integrable functions ψ and ψ' defined on SU(2)³ are going to be considered the same if they agree modulo the action described in equation (3). If you can get one by using the other but messing slightly with the argument first before you apply the function. In the illustration, to keep it simple I will assume all three links have the orientation: the same source and the same target. So I will just write the two nodes as "s" and "t". I pick some element (V_s, V_t) of SU(2)² and what has to happen to identify the two functions ψ and ψ' is that:

ψ(U₁, U₂, U₃) = ψ'(V_sU₁V_t^-1, V_sU₂V_t^-1, V_sU₃V_t^-1)

The thing is something you have seen hundreds of times in basic algebra. It is the transformation U goes to VUV^-1, we used to call it "U conjugated by V". You go in with V, and then you do U, and then you go back out the way you came. The only novelty here is there are more subscripts, and more of it happening at once, because the map involves 3 links and 2 nodes in a certain relation to them.
It looks more complicated than simple "U conjugated by V" because you are letting the graph set the pattern for a simultaneous conjugation job. Idea is simple but more items involved.

I still didn't address the comment Rovelli makes about one node always turning out to be redundant so that if N is the number of nodes the cartesian product that you are factoring by is one less---it is SU(2)^N-1 instead of SU(2)^N, which is what I naively expected. Leave that for later, if we get around to it.

So that is the graph Hilbertspace for a given graph. It is a plain ordinary L² function space so it is separable (countable orthonormal basis). Now the set of all combinatorial graphs is countable. Basically a graph is just two finite sets, the nodes and links, so the possibilities are quite limited. Now we can get the big Hilbertspace for the whole theory just by taking a DIRECT SUM of all the individual graph Hilbertspaces (and modding out a bit of redundancy that creeps in).

Have to go but will try to do more of this later. Actually you might want to read page 2 of the paper! The page is clearly written. It defines the Hilbertspace of the theory and then shows you how some operators are constructed on it:
1. the gravitational field operator ("flux", see the explanation there)
2. the area operator (an observable that measures the area of a surface).
3. the volume operator (observing the volume of a region).

 

[Naty1]

Marcus:

I probably don't need to say more than this to you, Naty...

good heavens, don't make that assumption(lol) ...always feel free to opine, I can use all the help I can get...!

When I first came across spin foam in popular physics books and Penrose's ROAD TO REALITY, I thought "really,really cool"..and they remain so...but...

Not fully understanding the math, and especially all the implicit assumptions underlying the math, it's difficult for a layman like me to determine what's input and what's output...for example, First step in the Rovelli paper in this thread starts with the selection of Hilbert space, and I think "who asked for that??"...picking a math format that fits our Eucledean space ...


from that point on it seems like geometrical interpretations are dependent on that pick...that bothers me: would it not be stupendous if something like space popped out from an independent formulation...that's why I like the concept of information or entropy as a starting point...


But what really gets to the heart of my intuitive discomfort, which I tried to expresss in post #4 is in the Rovelli paper, pg 4:

Such geometrical pictures are helps for the intuition, but there is no microscopic geometry at the Planck scale and these pictures should not be taken too literally in my opinion…...These geometrical pictures can play a very useful role in various situations, but what the theory is about is expectation values of physical observables, not mental pictures of the geometry of individual states.

I don't know exactly what that means in terms of the math, but THATS WHAT I"M TALKING ABOUT...

interesting ideas, and as always, thanks for bringing these to our attention...

 

[marcus]

Naty, I would like to get back to equation (2) of the paper and look at a very simple case. the graph should have two nodes and two links. So the graph looks like a circle or oval formed by the two links. Then L=2 and N=2.

The intuitive picture of combi LQG that I have is that geometry is made of these bits of volume and area flickering into and out of existence. The nodes are chunks of volume and the links are specks of area (where chunks of volume "meet" so to speak). So geometry is dynamic it can change by having new chunks of volume flicker into existence and new specks of area---over in one place (where new nodes and links are appearing)---and over in another place bits of volume and area can be disappearing.

A spinfoam can describe the evolution of a spinnetwork graph by having new nodes and links appear and others disappear. That is what the foam picture is good for describing. A foam tells the history of the appearance and disappearance of bits of geometry.

So I want to look at equation (2) in this very simple case where the graph is just an oval made of two links (upper and lower) between two dots, two nodes (left and right).

WHAT FOLLOWS IS GOING TO BE PITCHED TOO SIMPLE FOR ANYONE WITH AN ENGINEERING or college physics background. Sorry about that. If you are already familiar with the Hilbertspace of squareintegrable functions please be patient with the basic level, or just skip over the rest of the post.

The group SU(2) is basically just unitary 2x2 matrices and for brevity I will call it G.
It has a nice uniform well-known measure called "Haar measure" on it we could call "dg".
It is all ready for doing integral calculus. People have known about that for a long time.
And any cartesian product GxGxG...xG also has Haar measure and you can define functions on it and integrate functions. In our case since L = 2 we just look at GxG, the set of pairs
(U,U') or (U,V). Pairs of these 2x2 matrices, pairs of group elements.

One of the most common things in math is the space of square-integrable functions defined on some set with a measure. If X is the base set, the usual notation is L²(X).

People can use the 2 as a subscript and write L₂(X), as the authors do, but I am more comfortable writing L². The first thing we look at is L²(GxG).

Square integrable functions are intuitively like "wave functions" on a range of positions or configurations. Say we have a particle that can be anywhere in the interval [0,1]. If we were doing probability theory we might describe its location by a probability distribution on [0,1]. But we could also look at L²[0,1]. the functions defined on the interval which if you square the function it is integrable. (or if it is complex valued you take the square of the absolute value). Requiring that the functions be not only integrable, but their squares integrable insures that all the stuff you want to define later (like operators on them) will be well defined. The time-honored L² is the first nontrivial example of a Hilbertspace that most people encounter.

So the ordered pairs (U, V) which are the elements of GxG say something about the classic geometry of this simple "oval" graph. They are two matrices associated with the upper and lower links, joining the left and right nodes.

Perhaps since U and V are matrices they tell something about what happens to a little man who is moved along the links, the rolling around of his frame of reference. Something. It doesn't matter what.

Now we introduce indeterminacy by NOT picking a definite element of GxG, consisting of two definite matrices (U,V), but INSTEAD picking a wave function ψ(U,V) out of L²(GxG). A wave function that goes all over GxG is more indefinite than just picking one isolated point (U,V) in the set GxG.

To put it briefly, GxG tells something classic about the classic geometry and ψ tells something quantish about the quantized geometry. It is (almost, starting to look like) a quantum state of geometry.

But not quite yet, because equation (2) says that really the Hilbertspace of quantum states of geometry is not the raw L²(GxG) but instead is that cooked down by a certain equivalence relation.
You take the raw thing and you identify certain orbits or equivalence classes. You amalgamate bunches of elements which are "essentially the same" geometry because you can get one from the other by a bogus shuffle that doesn't really change anything.

So you get rid of that redundancy and the actual space is written L²(GxG/G).

Look at equation (2) and you will see what I mean, because the number of nodes N = 2, so N-1 = 1. So what I'm writing is L²(G^L/G^N-1) = L²(G²/G¹).

The bogus shuffles that don't really change anything (the "gauge" transformations) are described in equation (3). Now we have to look at (3). This is where it gets a bit more challenging and I probably won't address it till tomorrow. Anybody is welcome to take over the exposition.

αβγδεζηθικλμνξοπρσςτυφχψω...ΓΔΘΛΞΠΣΦΨΩ...∏∑∫∂√ ...± ÷...←↓→↑↔~≈≠≡≤≥...½...∞...(⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅)

 

[vacuumcell]

The intuitive picture of combi LQG that I have is that geometry is made of these bits of volume and area flickering into and out of existence.

Please forgive my simplemindedness, but my understanding is that conservation of energy means that energy can change form, but it can`t simply flicker into and out of existence (at least when were not talking about the quantum vacuum). For example, your remark that…

A foam tells the history of the appearance and disappearance of bits of geometry.

...would be better expressed by keeping only the part in boldface. But maybe this is what you meant.

 

[vacuumcell]

.Perhaps I don't get the real drift here in this latest LQG view, but this just doesn't seem as fundamental nor as potentially insightful as Verlinde and Jacobsen's ideas.

Admittedly, I don't know that much about LQG and so won't speculate on whether you get the real drift. I do however know that the idea of quantum spacetime as a "seething foam" didn't originate with LQG. One thing I've noticed in this forum is that many ideas brought up in the context of LQG are discussed as if they first arose in LQG. This may be one of them.

 

[marcus]

...

But what really gets to the heart of my intuitive discomfort, which I tried to expresss in post #4 is in the Rovelli paper, pg 4:

Such geometrical pictures are helps for the intuition, but there is no microscopic geometry at the Planck scale and these pictures should not be taken too literally in my opinion…...These geometrical pictures can play a very useful role in various situations, but what the theory is about is expectation values of physical observables, not mental pictures of the geometry of individual states.​

I don't know exactly what that means in terms of the math, but THATS WHAT I"M TALKING ABOUT...

interesting ideas, and as always, thanks for bringing these to our attention...

Hi Naty, the discomfort goes back to Bohr and is about any quantum theory. He said QM is not about how nature is---it is about MEASUREMENT. It is about preparing an experiment and then making observations and extracting correlations between measurements.

One cannot say what the particle does in between the slit or the detectors. In QM a particle has no trajectory.

Likewise with QG, which is really about quantum geometry. One does not postulate that there is some continuous spacetime geometry (that would be like a particle's trajectory) at a fundamental or microscopic level. There are only geometric measurements (measuring stuff like area angle volume) and any possible theory can only be about the relationships between measurements.

Bohr expressed the essential about any quantum theory when he said "Physics is concerned not with how Nature is, but with what we can say about it." Some words to that effect.

There are always going to be alternative ways to picture the relation between measurements. And there will be some ways to picture that are very much better than others.

BTW the PF setup is great but you should notice one thing about it. If you quote something in your post, then if I quote your post---your quote goes away! So if you want to quote something from Rovelli's paper, and you want me to be able to carry that along and comment, then you could use INDENT, the 10th button from the left
B I U + + + + + *indent* + + etc etc

Quotes self-destruct. So if I want to be stubbornly pedantic (which is nearly always ) I will write a quote with indent, or with some clunky delimiter like this:

==quote Rovelli page 4==
This physical picture admits variants. In the case Γ is the two-skeleton dual to a triangulation of a 3d space, one can view the individual grains as flat tetrahedra. In some cases, namely for some states, these tetrahedra can be viewed as forming a 3d Regge geometry. (For this, matching conditions between the length of triangles must be satisfied [6].) In the general case, one can associate them a “twisted geometry” [6].

Such geometrical pictures are helps for the intuition, but there is no microscopic geometry at the Planck scale and these pictures should not be taken too literally in my opinion. They are choices of classes of continuous geometries interpolating finite sets of geometrical data. It is clear that many such choices are possible. They are analogous to choices of interpolating functions to visualize or describe sets of data points. For instance, we can interpolate a set of data points by means of an interpolating polynomial, or a piecewise linear function, or a piecewise constant functions... As we will better see below, these choices have strict analogs for quantum geometry.

These geometrical pictures can play a very useful role in various situations, but what the theory is about is expectation values of physical observables, not mental pictures of the geometry of individual states.
==endquote==

Essentially just echoing Bohr and reminding us of what has always been the basic quantum philosophy.

 

[marcus]

Have to give this Bohr quote a post of its own:

There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature...​

As quoted in "The philosophy of Niels Bohr" by Aage Petersen, in the Bulletin of the Atomic Scientists Vol. 19, No. 7 (September 1963); The Genius of Science: A Portrait Gallery (2000) by Abraham Pais, p. 24, and Niels Bohr: Reflections on Subject and Object (2001) by Paul. McEvoy, p. 291

From Wikiquotes:
http://en.wikiquote.org/wiki/Niels_Bohr

This page has several other invaluable gems of wisdom. If anybody didn't look at it already I urge doing so.

 

[marcus]

... I don't know that much about LQG ...I do however know that the idea of quantum spacetime as a "seething foam" didn't originate with LQG. ...

"Seething foam" is your phrase. I didn't say that. As far as I know LQG does not have the picture of quantum spacetime as a seething foam.

As far as I know back around 1970 John Wheeler might have been talking about space as a foam (of microscopic wormholes and the like) an active picture of space. That has no direct connection with the LQG concept of a spin foam.

The LQG spin foam does not seethe and it is not a representation of spacetime Too bad they called it a foam---that was John Baez' choice of terminology I think. It is better thought of as a possible history (involving the creation and annihilation of bits of volume and area) that gets you from an initial space-geometry to a final space-geometry.
The initial and final states are not foams in any sense. They are spin networks.

Have to go, back later.

 

[marcus]

In LQG the idea of what you could measure of space geometry---the basis for observable quantitative geometric relationships etc---is represented not by a foam but by a simple graph.

Simply a (labeled) graph with nodes and links. The nodes contribute to volume. The links to area.

The dynamics governing these graphs (called networks) allows for the creation of new nodes (new volume, i.e. expansion) and for the creation of new links (new grist for area measurement operators).

The dynamics permits one to calculate amplitudes for various histories involving the creation and annihilation of nodes/links, that is, of area and volume information.

Actually the SU(2) labels are involved in calculating the amplitude of any given evolution history of geometry, and in calculating observables. It is not as simple as just counting the nodes contained in a given region (to find vol) or just counting the links crossing some given surface.

So you can, if you like, picture a graph as "seething" with the appearance and disappearance of area and volume information. But that is not a foam!

I will try to clear up some more possible confusion later, time permitting. Have to do something else.

Vacuumcell, thanks for your comments! It is good to see what people (especially the highly alert and qualified) might not understand and might be getting confused about.