Understanding LQG: How Gravitons Interact with Spacetime

[cosmolojosh]

How does LQG work? I know there are gravitons in it but how do they interact with spacetime?
Why is it a theory of quantum gravity but not everything? Thanks, Cosmolojosh

 

[marcus]

How does LQG work? I know there are gravitons in it...

Gravitons not really very much. LQG is mainly about the warping of spacetime. In some restricted nearly flat cases you can talk about gravitons--where the geometry is almost rigid--very special cases though.

Gravitons and a rigid uncurved geometry is not what LQG is mainly about. It is a theory of the dynamic changing geometry of spacetime. In that sense it is very much like ordinary vintage 1915 General Relativity (GR), except it is quantum.

In GR gravity is less of a force and more of a curved geometry thing. Planets move along their orbits because in the 4D geometry (curved by the mass of the sun) the paths they are going feel to them like straight lines----the shortest distance way to go.

So LQG is all about warping of 4D geometry (not about forces). But the difference is that it is a quantum theory so there is uncertainty in the geometry. When you measure an angle or an area or a distance you do not have complete certainty how it will come out. There may be an average or an expectation value, but there is, like, some fuzziness in the geometry. That is mainly how it is different from the pure classical 1915 General Relativity, where there is no fuzziness. Quantum does not always mean grainy or granulated discrete, but it always involves indeterminate fuzziness. Don't think grainy geometry, think uncertain geometry.

Theoretically, in LQG you can describe the present quantum state of the geometry of the universe (or of a small region) with a graph consisting of labeled nodes and links. Practically speaking, to describe the quantum state of the geometry of the whole universe would take an impossibly complicated graph, with a huge huge number of nodes and links.
So what they study and think about are actually small labeled graphs corresponding to a small bit of fuzzy geometry.

There is also LQC (loop quantum cosmology) where they try to deal with the geometry of the whole universe in a quantum way. They do it by making simplifying assumptions. Like that the matter is all uniformly spread out and the curvature is all even all over (no bumps or wrinkles). As long as you assume enough uniformity you can study the quantum geometry of the whole universe, and actually run computer simulations of stuff happening around the big bang. Since 2007 a lot of numerical simulation studies have been done.

Photons play almost no role at all in LQC. It is all dynamic geometry.

Matter tells geometry how to curve.
Geometry tells matter how to flow.

Gravity=geometry.

(Those are some common sayings that the GR and LQG people have.)

 

[cosmolojosh]

Thanks Marcus. So it is for the most part a quantum theory of relativity. So the gravitons would be talked about as the quantum of space time's warping.

 

[marcus]

I'm not quite sure how best to put it. The graviton plays a very marginal role. In most cases it is not a good mathematical tool to describe how gravity behaves.
It's not like one pictures gravitons actually existing all the time running around conveying gravitational force.
The graviton is a math tool of analysis that is useful and appropriate in a certain type of situation.

The situation is called "perturbative" and it is when the geometry is almost perfectly flat and rigid---in other words where gravity is comparatively weak.
Then, on top of this flat geometry you imagine a little ripple, a little perturbation. Then it is appropriate to use the graviton as a tool to analyze what's happening.

In LQG one almost always ignores gravitons. They are just not part of the picture. The theory does not need a perturbative weak-field picture. It was developed to make NON-perturbative calculation possible, to deal with strong fields (like where classic GR breaks down and develops failurepoints called singularities.) It was developed to work where spacetime is NOT approximately flat, like at the core of a black hole or around the big bounce---early universe cosmology---the start of expansion. Those are places where perturbative analysis is worthless and one would never think of using the graviton concept.

For starters probably best to just completely forget about the graviton idea, learn the basics about LQG and LQC, and then much later in special situations find them being used for weak field approx. flat analysis.

 

[cosmolojosh]

What exactly is the loop in LQG? I have heard it is kind of in a way like the string in string theory

 

[tom.stoer]

What exactly is the loop in LQG? I have heard it is kind of in a way like the string in string theory

LQG as of today is a theory based on SU(2) spin networks. The vertices (with an appropriate SU(2) structure, called intertwiner) represent volume, whereas the links (again with an appropriate SU(2) structure), connecting the vertices represent area. There is no spacetime, no manifold, there is just this spin network. Information regarding spacetime (volume, curvature, ...) can be extracted using operators acting on spin network states.

The big difference to string theory (where the string lives in a spacetime manifold) is that the spin network does not life in spacetime, it is spacetime. So the vacuum state w/o any vertices is not empty or flat space, it is no space at all.

LQG was never "designed" for unification. That means that you can add matter on top of it, but that there is no unified description of matter and spacetime; they are two distinct entities (but this is the same in string theory; here you describe matter and gravitons on equal footing, but they both life in classical spacetime, so the unification is not complete). What you get in LQG is a unification on a mathematical level: spacetime is (prior to quantization which results in spin networks) described by an SU(2) gauge theory which is rather similar to well-known theories like (e.g.) QCD

 

[diazona]

What exactly is the loop in LQG? I have heard it is kind of in a way like the string in string theory

The name is actually a little misleading, it'd be better to call it "graph quantum gravity" because it's really a theory based on graphs (in the graph theory sense - that is, sets of vertices connected by edges). Of course, in these graphs, you can have loops where edges lead from one vertex to another to another to ... back to the original vertex, and (at least from what I understand) those are the loops of loop quantum gravity. There isn't really anything all that special about the loops themselves.

 

[tom.stoer]

What exactly is the loop in LQG? I have heard it is kind of in a way like the string in string theory

Sorry. Instead of answering your question I was telling you something different.

The loop in LQG is a well defined mathematical object which has been used to derive some mathematical structiures that are used tody. As far as I can see in the current firmulation the loop is not so relevant.

The idea was to transform gravity in a theory that is not based on a metric (this was Einstein's and Hilbert's firmulation) but on variables that are rather closed to gauge fields. Look at electromagnetism: Think about a closed curve C in space, a ring or something like tat, but the detailed geometry doesn't matter. Now look at the vector potential A(x) and define an integral of the potntial along this closed curve. The integral depends on both A(x) and C. In addition you can define a set of surfaces S spanned by the curve. Now look at the electric field E(x) and define an integral over the surface, that means the electric flux through the surface. The flux depends on both E(x) and the surface S.

In LQG the situation is ore complex as A(x) and E(x) are SU(2) valued matrices instad of single functions. Nevertheless one can define similar integrals along the curve C and the surface S, respectively. The integrals are called loop variables and can be used to re-formulate the theory.

Something similar has been used in standard gauge theory, namely QCD. In the continuum formulation this does not work because in QCD one crucial symmetry which allows to reduce the "space of loops" to a tractable mathematical entity is missing. This symmetry is the so-called diffeomorphism invariance of GR, the symmetry under arbitrary coordinate transformations. In lattice-gauge QCD the loops are called Wilson-loops and are fine as the are regulated by the lattice. They are used to study the confinement properties of the theory.

 

[Fra]

I'm just interjecting here a personal projection I've always made on LQG logic, which aims to project some of the frameworks of the somewhat structural realist view of LQG to an subjective view.

This is not stadard though, but I personally find it the most promising angle of LQG (if I have visions of what LQG could become, including the unification that was not originally part of design)

If we think of generalized background independence in the sense of seeing any baggade structure (not just a background space time metric) as part of a generalized "background", then the following interpretation of what Tom said is possible:

The big difference to string theory (where the string lives in a spacetime manifold) is that the spin network does not life in spacetime, it is spacetime. So the vacuum state w/o any vertices is not empty or flat space, it is no space at all.

Lets say that background choice in string theory can be interpreted to come from the "choice of the observer" - in the generalized sense.

Then the wording about could be rephrased in the generalized sense as "the spin network" does not "live" or depend on an "background/observer", it IS the observer. Or maybe put a bit differently, it IS the "image" or information state of the system, that observer encodes.

This is partly similar to Zureks picture, that what the observer IS and what the observers knows (think quantum state) is indistinguishable

So, I would prefer to somewhat oddly interpret the "spin network" with the observer or say the system on the other side of "the screen".

But this would be a modification of LQG, but it's the most constructive projection of LQG that I can make.

I just fail to make sense out of an objective quantum state of the entire universe. What does make snse to me is a quantum state of the observable universe (within our horizon) but then that observable universe and horizon are observer dependent.

/Fredrik

 

[Finbar]

One should recover the graviton description of quantum gravity as a low energy description of any viable theory of quantum gravity. This is one of the challenges LQG faces. On the other hand string theory has this low energy limit already.

Is that fair?

 

[Finbar]

Something similar has been used in standard gauge theory, namely QCD. In the continuum formulation this does not work because in QCD one crucial symmetry which allows to reduce the "space of loops" to a tractable mathematical entity is missing. This symmetry is the so-called diffeomorphism invariance of GR, the symmetry under arbitrary coordinate transformations. In lattice-gauge QCD the loops are called Wilson-loops and are fine as the are regulated by the lattice. They are used to study the confinement properties of the theory.

In what sense is diffeomorphism invariance "missing" in QCD? QCD doesn't break diffeomorphism invariance. The point is that the geometry is non-dynamical in QCD.
Maybe I'm wrong here but are you confusing diffeomorphism invariance with back ground
independence? QCD is diffeomorphism invariance(you can still make arbitrary coordinate transformation) but not background independent (the geometry is fixed, usually to flat space-time) whereas GR is both.

 

[Fra]

In what sense is diffeomorphism invariance "missing" in QCD? QCD doesn't break diffeomorphism invariance. The point is that the geometry is non-dynamical in QCD.
Maybe I'm wrong here but are you confusing diffeomorphism invariance with back ground
independence? QCD is diffeomorphism invariance(you can still make arbitrary coordinate transformation) but not background independent (the geometry is fixed, usually to flat space-time) whereas GR is both.

I think it's just terminology issue, and Tom means the general covariance or "active" diffeomorphism invariance.

Somehow the "passive diff invariance" are somewhat trivial, and is void of physical meaning.

Active is what I also meant implicitly when I used the word here in posts, since the "coordinate change" doesn't say anything as a physics statement.

/Fredrik

 

[tom.stoer]

I think it's just terminology issue, and Tom means the general covariance or "active" diffeomorphism invariance./Fredrik

That's exactly what I mean.

The problem in QCD is that a Wilson loop W_C is labelled by C, which means we use a continuum of points to express a single "variable". In GR - due to active diff.-inv. - most of these degrees of freedome are "gauge". It doesn't matter exactly where you place the loop in spacetime, because due to diff.-inv you can factor out most of these "degrees of freedom". In QCD this is not possible, therefore using Wilson loops in QCD enlarges the number of degrees of freedom you have to quantize.

If you look at the derivation of the kinematical Hilbert space in LQG there is an intermediate step with a non-separable Hilbert space. This is exactly due to the loops. In LQG you can factor away this redundancy, whereas in QCD you can't and you stay with the non-separable space. People tried for some tome to use loops in QCD but it didn't work (except in lattice QCD where this problem does not appear as the lattice is fixed, disceret and finite)

The reason is the coordinate trfs. in QCD are unphysical. They don't do anything with the gluon field. There is no relation between different loops.

 

[Finbar]

Ok I think I get you now. When I make a diffeomorphism this is essentially a gauge transformation for which the gravitational field is the relevant gauge field. So i GR i have to actively change the gravitational field where as the gluons are passive to this gauge transformation.

So the point about the loops in LQG is that they are gauge invariant and don't have the same gauge redundancies that the metric does? In particular they explicitly remove the infinities which crop up at arbitrary small distances.

 

[tom.stoer]

So the point about the loops in LQG is that they are gauge invariant and don't have the same gauge redundancies that the metric does? In particular they explicitly remove the infinities which crop up at arbitrary small distances.

The loops are both gauge invariant and "diff. covariant".

It's not the loop space itself but the class of representatives of diff.-inv states (cylinder functions) that are used to quantize the theory on "physical states", namely the spin networks. Instead of the loops only representatives for a whole class of each diffeomorphic loops is used.

Regarding the infinities at small distances: yes, it's something like this, but you can't (at least I can't) compare one-to-one as the math is totally different.

Have you studied any LQG papers explaining the approach in more detail?