Coexistence of LQG and String Theory?

[marcus]

Does one need to introduce matter to measure these observables?

My answer is YES.
I should say that I'm just an interested onlooker from the sidelines. You should really have someone who is actively doing Loop research to answer your questions.

If you want to measure the area of a surface, that surface has to be defined somehow. a black hole horizon, the top of your work table. Space has no meaning apart from relationship with matter or with features of the gravitational field (like a horizon) defined and observed through matter. I think this is a really important issue and wish I were more able to give an informed answer.

How does the notion of 'spin' come into this in this very intuitive and clear picture you have described? (Am I to imagine that this spin network is "real", i.e. made up from some kind of distribution of energy, or is it just a mathematical construction?)

Just a partial answer for now. The way I think of it, Loop is not about what Nature is made of, it's about how she responds to measurement. A state summarizes past (geometric + ideally matter) measurements. You want a model of how the state evolves so you can predict other measurements. First we have to say how states are going to be presented in math terms.

A GRAPH Γ (made of nodes and links) is not a physical thing, it is just a way of truncating to get a finite number of geometric degrees of freedom. Only such and such measurements are under consideration (a finite web of measurements).

If you look at the first few pages of 1102.3660 you will see how H_Γ the graph Hilbert space is defined.

Think of ordinary quantum mechanics in the simplest case of a particle in a onedimensional "box" which is just the interval [0,1] of the real line. The configurations are just position numbers from 0 to 1, and the state is just a complex valued function defined on that interval. States are square-integrable functions: L₂[0,1].

You will see in 1102.3660 that they do the analogous thing. CONFIGURATIONS are technically "connections" (a diffy geom. term) that assign a GROUP ELEMENT to every leg of the graph.*
A connection tells you how a parallel transport vector swings and sways and rolls around as you move along that leg. So the config. space of (finitized) connections is just the cartesian product G^L where L is the number of links in the graph. And by the way G is SU(2), we are focusing on rotations in 3D.
In diffy geom. a connection describes a configuration of geometry at the classical level, and in our case we are talking about SU(2) connections.

You want a SQUARE INTEGRABLE FUNCTION ON THE SPACE OF CONNECTIONS in direct analogy with the particle in the box. That means L₂[G^L ]

Now how do you get a basis for the vectorspace of complex valued functions on G? or the cartesian product? The Peter-Weyl theorem say to use the REPRESENTATIONS of the group. And you know the representations of SU(2) are labeled by half-integers.

So you label the graph with half-integers and presto you have a machine which can eat "configurations" and give you back a complex number. It can eat an L-tuple of group elements belonging to G^L (which is our finitized or truncated "connection") and chew it up and give a number.
And functions of that form constitute a BASIS of the whole vectorspace L₂[G^L]

You might want to look at 1102.3660, or somebody might be able to suggest something better as an introduction. This what I've supplied here is just a start.

*In classical differential geometry a connection tells you about parallel transport along all possible paths in the manifold. That's too much information. That's why we truncate the information we are dealing with and just consider moving along the legs of a finite graph. And then we recover by considering more and more complicated graphs. But first the theory is constructed on one particular finite graph.

 

[w4k4b4lool4]

Lets say an observer wants to measure one of these observables ... Does one need to introduce matter to measure these observables?

My answer is YES.
I should say that I'm just an interested onlooker from the sidelines. You should really have someone who is actively doing Loop research to answer your questions.

If you want to measure the area of a surface, that surface has to be defined somehow. a black hole horizon, the top of your work table. Space has no meaning apart from relationship with matter or with features of the gravitational field (like a horizon) defined and observed through matter. I think this is a really important issue and wish I were more able to give an informed answer.

I see. This is very interesting, but it is also puzzling at the same time. If LQG is a quantum theory of gravity, which thus far has not incorporated matter, but nevertheless requires matter in order for one to access and measure its observables, then in what sense can we address physical questions in this context? (And I suppose I'm referring to the truly quantum degrees of freedom of the theory, as opposed to the effective classical limit.)

Also, if the quantum gravity theory is defined in this very precise manner (which you've described in a great way by the way Marcus!), then, intuitively, wouldn't we expect to have to carry out a similar procedure for the matter contribution? (Of course, I know the same techniques don't work for matter .. unless, one thinks along the lines of asymptotic safety, maybe); but what I mean is that, although Einstein's equation is classical, it seems to be hinting towards a strong correlation between matter and gravity .. so that I would interpret that as a strong hint that we should be treating both matter and gravity on an equal footing. This elaborate setup that you have described should somehow be, I don't know, let's say a "mirror image" of the matter theory! I suppose AdS/CFT (or maybe I should say gauge/gravity duality) is kind-of thinking along these lines, and what an indirect way to do so, what an indirect way to seal this strong correlation between gravity and matter at the quantum level! Wow! (Of course, the latter is also speculative at present, but, having passed numerous checks, there is a large community that has been convinced that it is true.)

I am a string theorist at heart, but I'm continuously trying to learn about new ideas. After all, the largest breakthroughs have been made by bringing tools from one area of science to another (spontaneous symmetry breaking being a good example I suppose).

Just a partial answer for now. The way I think of it, Loop is not about what Nature is made of, it's about how she responds to measurement. A state summarizes past (geometric + ideally matter) measurements. You want a model of how the state evolves so you can predict other measurements. First we have to say how states are going to be presented in math terms.

A GRAPH Γ (made of nodes and links) is not a physical thing, it is just a way of truncating to get a finite number of geometric degrees of freedom. Only such and such measurements are under consideration (a finite web of measurements).

If you look at the first few pages of 1102.3660 you will see how H_Γ the graph Hilbert space is defined.

Think of ordinary quantum mechanics in the simplest case of a particle in a onedimensional "box" which is just the interval [0,1] of the real line. The configurations are just position numbers from 0 to 1, and the state is just a complex valued function defined on that interval. States are square-integrable functions: L₂[0,1].

You will see in 1102.3660 that they do the analogous thing. CONFIGURATIONS are technically "connections" (a diffy geom. term) that assign a GROUP ELEMENT to every leg of the graph.*
A connection tells you how a parallel transport vector swings and sways and rolls around as you move along that leg. So the config. space of (finitized) connections is just the cartesian product G^L where L is the number of links in the graph. And by the way G is SU(2), we are focusing on rotations in 3D.
In diffy geom. a connection describes a configuration of geometry at the classical level, and in our case we are talking about SU(2) connections.

I see. So I suppose the analogy with the QM above is:

connections (or g\in G) ~ x\in[0,1] interval

and

SQUARE INTEGRABLE FUNCTION ON THE SPACE OF CONNECTIONS ~ ψ(x) of particle in box

By the way, is the full 4D diffeomorphism group still intact?
And why are you assuming D=4 dimensions? If you start with a higher dimensionality do interesting things happen (I don't know .. like enhanced symmetries or something of the like) for specific values of D? I suppose, ideally, one would like to PREDICT D=4 ...

You want a SQUARE INTEGRABLE FUNCTION ON THE SPACE OF CONNECTIONS in direct analogy with the particle in the box. That means L₂[G^L ]

Now how do you get a basis for the vectorspace of complex valued functions on G? or the cartesian product? The Peter-Weyl theorem say to use the REPRESENTATIONS of the group. And you know the representations of SU(2) are labeled by half-integers.

So you label the graph with half-integers and presto you have a machine which can eat "configurations" and give you back a complex number. It can eat an L-tuple of group elements belonging to G^L (which is our finitized or truncated "connection") and chew it up and give a number.
And functions of that form constitute a BASIS of the whole vectorspace L₂[G^L]

That's very clear and enlightening!

You might want to look at 1102.3660, or somebody might be able to suggest something better as an introduction. This what I've supplied here is just a start.

*In classical differential geometry a connection tells you about parallel transport along all possible paths in the manifold. That's too much information. That's why we truncate the information we are dealing with and just consider moving along the legs of a finite graph. And then we recover by considering more and more complicated graphs. But first the theory is constructed on one particular finite graph.

I see. Have any alternatives (that lack the discreteness you are referring to) been suggested?
Might this discreteness be physical after all?

Thank you very much again Marcus!
It's been a great please reading slowly through your every sentence, laying back, and thinking about what you are saying.

Wakabaloola

 

[marcus]

...
Also, if the quantum gravity theory is defined in this very precise manner (which you've described in a great way by the way Marcus!), then, intuitively, wouldn't we expect to have to carry out a similar procedure for the matter contribution? (Of course, I know the same techniques don't work for matter .. unless, one thinks along the lines of asymptotic safety, maybe); but what I mean is that, although Einstein's equation is classical, it seems to be hinting towards a strong correlation between matter and gravity .. so that I would interpret that as a strong hint that we should be treating both matter and gravity on an equal footing. This elaborate setup that you have described should somehow be, I don't know, let's say a "mirror image" of the matter theory! ...
..

Thanks for this interesting and stimulating comment (and for the encouraging words.) I will try to say how LQG at present includes matter. I don't know much about it and will have to check with the current definitive source 1102.3660. As I recall fermion labels go on nodes of the graph and YM field labels go on links. You have to enlarge the graph Hilbert space H_Γ. It is work in progress.
http://arxiv.org/abs/1102.3660

Also in Loop cosmology they use matter a lot, simple generic scalar fields and such. They do it differently. LQG and LQC are both evolving. By this time next year there could be another 30page paper that replaces the 2011 one I'm going to consult

I misremembered, on page 27 the "current standard source" says
==quote==
...Also, I have not covered several recent development, such as the manifest Lorentz invariant formulation of the theory [131], the coupling to fermions and Yang-Mills fields [14, 132], and to a cosmological constant [11, 133], using a quantum group.
==endquote==
So I must look up [14, 132]
[14] http://arxiv.org/abs/1012.4719
Spinfoam fermions
Eugenio Bianchi, Muxin Han, Elena Magliaro, Claudio Perini, Carlo Rovelli, Wolfgang Wieland
(Submitted on 21 Dec 2010)
We describe a minimal coupling of fermions and Yang Mills fields to the loop quantum gravity dynamics. The coupling takes a very simple form.
8 pages

[132] http://arxiv.org/abs/1101.3264
Spinfoam Fermions: PCT Symmetry, Dirac Determinant, and Correlation Functions
Muxin Han, Carlo Rovelli
(Submitted on 17 Jan 2011)
We discuss fermion coupling in the framework of spinfoam quantum gravity. We analyze the gravity-fermion spinfoam model and its fermion correlation functions. We show that there is a spinfoam analog of PCT symmetry for the fermion fields on spinfoam model, where a PCT theorem is proved for spinfoam fermion correlation functions. We compute the determinant of the Dirac operator for the fermions, where two presentations of the Dirac determinant are given in terms of diagram expansions. We compute the fermion correlation functions and show that they can be given by Feynman diagrams on the spinfoams, where the Feynman propagators can be represented by a discretized path integral of a world-line action along the edges of the underlying 2-complex.
26 pages, 9 figures

 

[marcus]

My memory of how matter is included, and my understanding of details were not so good. I am still struggling and will take a break, but first let's look on page 5 of reference [14] where it says:
==quote 1012.4719 ==
Now combine this fermionic Fock space with the kinematics of quantum gravity. Fermions must reside on the nodes n of a graph, like in lattice gauge theory. Thus we assign a copy of the Fock space F to each node of the graph. Therefore the states of the gravity+fermion theory live on the space (⊗_lL₂[SU(2)])⊗(⊗_nF), divided by the gauge action of SU(2) at each node. We can write states as Ψ(h_l,ψ_n), where l labels the links of the graph and n the nodes. The spin networks that form a basis of this state are a simple generalization of the pure gravity spin networks.
As before, it is convenient to choose an intertwiner basis at each node n that diagonalizes the volume of the node n, and label it with the volume eigenvalue v_n. That is
|j_l,v_n⟩. In the presence of fermions, spin networks carry an extra quantum number c_n at each node, which labels the basis |c⟩ in the Fock space at the node. That is: |j_l,v_n,c_n⟩. At each v-valent node n bounded by links with spins j₁ , ..., j_v , the intertwiner v_n is an invariant tensor in the tensor product of the v representations j₁, ..., j_v
if c_n =∅ or c_n =2. But it is an invariant tensor in the tensor product of the v + 1 representations j₁, ..., j_v, 1/2 if c_n = ±. In this case, the intertwiner couples the spinor to the gravitational magnetic indices.

==endquote==

So the enlarged graph Hilbert space basis is the usual L₂ of L copies of the group, tensored with N copies of Fock space where N is the number of nodes.

 

[w4k4b4lool4]

Thanks for that Marcus!
Unfortunately, I won't have time to delve deeper into this in the next few days.
I've certainly learned a lot from our conversation,
Wakabaloola

 

[marcus]

OK so Waka is busy for the time being and unlikely to respond, but I want to comment on something or some things he said

...
I am a string theorist at heart, but I'm continuously trying to learn about new ideas. After all, the largest breakthroughs have been made by bringing tools from one area of science to another (spontaneous symmetry breaking being a good example I suppose).
...

This thread is about productive interchange between String and Loop (and I think one could ask more broadly about the possibility of carryover to and from other QG lines.)
How much and what is possible? How much is likely.

Most of us (if we hang around BtSM forum) know the work of Kirill Krasnov. He has co-authored with Rovelli and with Freidel. The currently prevalent Loop dynamics could be called EPRL-FK spinfoam dynamics, and he is the K there. He has very interesting ideas which he works on intently often seeming out of touch with the other Loop people.

So what is Krasnov doing tomorrow? He will be in Munich attending the Strings 2012. This is an example of what I think has to happen. What Waka called "bringing tools" or taking away tools. Redistributing useful pieces of flint chipped and shaped in various ways.

From my personal perspective I think it will be especially productive for String people to learn how Loopsters and Cosmologists think. Also maybe how the Asymptotic Safesters think, if Shaposhnikov can be included with them.

Loop is relatively advanced in cosmology. Two recent ones I think of (not perhaps the most important, just ones that come to mind):
Agullo Ashtekar Nelson http://arxiv.org/abs/1204.1288
Artymowski Dapor Pawlowski http://arxiv.org/abs/1207.4353

 

[atyy]

Actually, FK are the loop people I most associate with taking a covariant point of view that need not coennct with the canonical viewpoint - not Rovelli. So I am pleasantly surprised to hear francesca espouse that viewpoint!

Actually, maybe it'd be more accurate to say FC "A priori, a spin foam model of gravity need not be related to canonical loop quantum gravity (LQG). That is, a given model could be a viable quantization of gravity, and nevertheless do not have the kinematical boundary variables of canonical LQG. Such a thing is, at least, conceivable, since we have an analogous example at the classical level: Hilbert-Palatini gravity, which after the Hamiltonian analysis, does not lead to the connection formulation by Ashtekar and Barbero."