The connection in Loop Quantum Gravity

[marcus]

In the Physics forum, jby started a LQG thread which at one point included some discussion of Connections. It might be helpful to have a thread focused on the connection which is taking over the central role from the metric for some purposes.

GR used to be the study of the metric on a smooth manifold. A metric would give rise to an idea of curvature and an idea of parallel transport of vectors along a path from one point to another. Imagine a sphere and a tangent vector at the equator pointing in some direction and imagine scooting that vector up to the north pole. What direction does it point then?

The simplest way to imagine a connection, I believe, is as a machine to parallel translate vectors along paths from one point to another. If you have a metric and derive a connection from it and then your disk crashes and you have forgotten the metric, then you can RECOVER the metric from the connection (except for scale). So the connection-----the definite proceedure for parallel transport----has the same information in it, including an idea of curvature.

Rovelli begins his brief history of LQG with 1986 when it occurred to someone to quantize the connection instead of the metric

http://www.livingreviews.org/Articles/Volume1/1998-1rovelli/index.html

in the index of Rovelli's review of LQG you see section 3: History.
Just jump to "3 History of Loop Quantum Gravity" and you find the first few sentences are about this 1986 move to connections.

[edit: the 1986 paper was Ashtekar's seminal reformulation
of GR in terms of connections, also called the "new variables", if interested, look at the brief history in section 3 for yourself]

Anyone is welcome to take over the job of telling the story. I am just summarizing from Rovelli's LivingReviews article at this point

 

[chroot]

marcus,

Can you recommend any introductions to LQG? I am decently capable in GR, but cannot seem to bridge the gap to LQG.

- Warren

 

[marcus]

the essential steps in building a quantum theory are
you have a configuration space, say the real line showing, for example, positions of some particle. And you make the space of
(lets say square integrable) functions on R.

That is called L²(R)-----square integrable functions on R.

The functions ψ in L²(R) are called "states" and they form a vectorspace (infinite dimensional but nice) called a Hilbert space and they represent a blend of knowledge and uncertainty about the particle's position.

One can modify L²(R) in various ways to get related Hilbert spaces and construct operators (observables) on whatever Hilbert space you end up using.

the main thing is that when you realize that Nature won't let you pin her down to a particular point in R, then you build a space of functions on R and work with those. And physical observations correspond to operators on that space.

Now in studying gravity we have a manifold and a collection A of all the possible connections A that can be on the manifold.

And the main breakthru will be, how do we construct a Hilbert space out of L²(A)? How do we give that meaning?

All the possible connections is taking the place of the real line R or of any other space of positions or configurations or what have you. All the possible connections includes the idea of all the possible metrics and curvatures and geometries there can be on this underlying manifold (which so far has no particular character except a smooth topology but which you are free to stretch and bend everywhichway.)

A Hilbert space requires an inner product. How can we set up functions f( A ) and h( A ) and arrange to integrate their product to get the usual sort of Hilbertspace inner product?

After that, how do we set up operators on the hilbertspace?

It was so easy in 1920 for Schroedinger and those guys to set up
of L²(R) the hilbertspace of functions defined on the real line and suchlike simple stuff----and so hard to set up L²(A) the hilbertspace of functions defined on connections.

In the real line case the inner product was just ∫ f*(x)h(x) dx, taking the complex conj of one of the two functions if you were using complex numbers but basically just integrate the product of the two functions!

How to do this with f( A ) and h( A ) two functions defined on the space of possible connections?

 

[marcus]

Originally posted by chroot
marcus,

Can you recommend any introductions to LQG? I am decently capable in GR, but cannot seem to bridge the gap to LQG.

- Warren

I am going thru Rovelli's easy beginning parts in this thread.
Have you looked at Rovelli?

We could write an introduction to LQG right here at PF, if a good one does not exist.

Yell and scream for Instanton to return to PF. He was here a few days ago and recommended Thiemann's review of LQG.
arXiv:gr-qc/0110034

Also Baez has one arXiv gr-qc/9504036

I guess "gr-qc" means general relativity and quantum cosmology.

But the short answer chroot is sadly NO I do not know of any
entrylevel intro to LQG.

How familiar is the idea of a hilbert space to you? If you focus on one thing at a time, the problem now is to build a hilbert space---how to define the inner product. I would be happy just to get that down solid, for the moment.

 

[chroot]

Hilbert spaces are fine. I know QM, jeez. :P

- Warren

 

[quantumdude]

Originally posted by chroot
marcus,

Can you recommend any introductions to LQG? I am decently capable in GR, but cannot seem to bridge the gap to LQG.

- Warren

Try this:

http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0110/0110034.pdf

I want to go through it, if I can ever get done with that @$#%*! book on Fields by Siegel.

 

[marcus]

BTW chroot,
you know some GR so you know this

that if you are on a spherical map of the earth, a globe, and you parallel translate a tangent vector from a point on the equator up to the north pole and then down another longitude line to the equator and then back along the equator you end up with a different vector!

If you know this you are way ahead because most people have not realized it.

And so doing that loop of parallel transport generates a shuffling around of the tangent space at the point you come back to.

It is a symmetry on it.

So every loop induces a transformation of the tangent space.

And the curvature enclosed by that loop is reflected in the rotation of tangent vectors that going around the loop causes.

Intuitively this is why loops are so valuable for extracting information about geometry from connections (the machines of parallel transport). You run the machine on a loop and it tells you part of what is happening.

A loop is almost an "observable", an operator on connections and also a physical observation defined on the space of connections.

Is any of this intuitive for you or is it a disaster that I am talking like this?

 

[marcus]

Originally posted by chroot
Hilbert spaces are fine. I know QM, jeez. :P

- Warren

Yeah I thought so. So we will have done something if we just see how to define the inner product on A the space of connections on the manifold. Rovelli explains how on one page so let's go ahead and do that.

 

[chroot]

Well, duh, marcus... the failure of parallel transport to return a vector to itself is the very definition of curvature. The Riemann metric is just an expression of what happens to vectors when parallel-transported around infinitesimal loops.

- Warren

 

[marcus]

Originally posted by chroot
Hilbert spaces are fine. I know QM, jeez. :P

- Warren

Rovelli defines the Hilbert space concisely, he says:

We can start ``à la Schrödinger'' by expressing quantum states by means of the amplitude of the connection, namely by means of functionals Ψ(A) of the (smooth) connection. These functionals form a linear space, which we promote to a Hilbert space by defining a inner product. To define the inner product, we choose a particular set of states, which we denote ``cylindrical states'' and begin by defining the scalar product between these.

Pick a graph Γ , say with n links, denoted γ_i, immersed in the manifold M. For technical reasons, we require the links to be analytic. Let U_i((A) be the parallel transport operator of the connection A along γ_i. U_i((A) is an element of SU(2). Pick a function f(...g_i...) on [SU(2)]ⁿ . The graph Γ and the function f determine a functional of the connection as follows

ψ_{Γ, f}(A) = f(... U_i((A)...)


That is enough for one chunk of Rovelli. So concise. He is going to define the inner product on these functionals. We need to pause for intuition to catch up. What is one of these functionals?

The graph is just a collection of n paths (n "links") from one place to another. And parallel transporting along each one gives you a matrix-group element g. And doing all n paths gives you n separate group elements g1, g2, ...gn. And f just happens to be a numerical valued function defined on n-tuples of group elements like that!

So you give me a connection A and I give you back a number.

Which I get from running the graph Γ and using the function f on the n-tuple of things that result.

********
He is going to tell us how to do an inner product of two of these
ψ things.


This is a page labeled 6.2 in Rovelli's review. I may not have said it so clearly but it does not seem fundamentally so hard. Or?

 

[chroot]

Originally posted by marcus
So you give me a connection A and I give you back a number.

Which is the definition of a functional on the connections: a map from the connections to the reals.

- Warren

 

[marcus]

Originally posted by chroot
Well, duh, marcus... the failure of parallel transport to return a vector to itself is the very definition of curvature. The Riemann metric is just an expression of what happens to vectors when parallel-transported around infinitesimal loops.

- Warren

Indeed chroot! It seems to be very elementary. You are the perfect person to discuss this with and it makes me hope that we can bring LQG on board here in a fairly concrete way.

As you know with inner products the job is two fold. You have to define the inner product of any two of a spanning set of functions and you have to verify it is bilinear and satisfies some sensible conditions like an inequality or two. Let's omit that stuff!
Lets just see how he defines it.

I have to go look how to write the integral sign in PF. be back shortly.

The essence is that with any two graphs Γ and Γ' you can merge them into a union where both
f and f' are defined on their respective subgraphs as before and as zero elsewhere. So then you just integrate f* multiplied by f' on the [SU(2)]ⁿ. They do it every time. It is always how the innerproduct on these spaces of functions is defined!

 

[marcus]

Hello again, my computer seems to have slowed down
considerably. By this time you may be happily reading
Rovelli directly. It is a bother copying Rovelli equations into PF
because I have to redo all the subscripts and superscripts.

However selfsufficient you chroot may be we still must think of other readers! So I will write out the integral that defines the innerproduct.

That will create the initial hilbert space and in some sense LQG takes off from there with a process of
refining the hilbertspace and choosing a BASIS for it in a clever way (this brings in "spin networks" as a better basis than just any old loops) and defining operators like the area operator andvolume operator.

It is amusing that in the quantum theory of a particle what one wants immediately is position and momentum operators.
But in the case of geometry what one wants are area and volume operators. they turn out to have discrete eigenvalues---that is, area and volume are quantized in steps of Planck area and Planck volume. wow.

Well now I know how to write the integral so here is the inner product between

the psi of Γ f and the psi of Γ', f'.

The recipe is you merge the graphs into their union, which involves a new larger n. And you extend the definition of f and f' so they are just defined as zero where they weren't defined before. So now you have f and f' defined on n-tuples of group elements.

And any compact group has a natural left and right invariant Haar measure---basically the uniform measure one expects. like on the unit circle for rotations. it is even a probability measure, can take it to sum to one.

So there is a natural uniform measure on the n-tuples
[S(2)]ⁿ and

you just integrate the mothers!

∫ f*(...g_i...)f'(...g_i...) dg₁...dg_n


Integrate with respect to invariant Haar measure on the group!


That gives a number for the two psi's. These psi's are functionals on the connections. The inner product so defined is bilinear etc etc.

Then we take limits of linear combinations of these "cylindrical" psi's upon which the inner product has been defined. But that is a standard proceedure in defining hilbert spaces.

Originally posted by marcus
Indeed chroot! It seems to be very elementary. You are the perfect person to discuss this with and it makes me hope that we can bring LQG on board here in a fairly concrete way.

As you know with inner products the job is two fold. You have to define the inner product of any two of a spanning set of functions and you have to verify it is bilinear and satisfies some sensible conditions like an inequality or two. Let's omit that stuff!
Lets just see how he defines it.

I have to go look how to write the integral sign in PF. be back shortly.

The essence is that with any two graphs Γ and Γ' you can merge them into a union where both
f and f' are defined on their respective subgraphs as before and as zero elsewhere. So then you just integrate f* multiplied by f' on the [SU(2)]ⁿ. They do it every time. It is always how the innerproduct on these spaces of functions is defined!

 

[jeff]

Marcus,

Since you started this thread to teach LQG, it seems odd that given that the point of departure of LQG is the ashtekar reformulation of GR, you haven't said much about it beyond some vague or general reference to metric and connection. For example, in words, describe the correspondence between the degrees of freedom as they appear in the ashtekar versus conventional formulations and in particular explain the physical significance of su(2) in all of this. Again, since you didn't start this thread without pretension, I don't think my request is unfair.

 

[jeff]

Originally posted by chroot
marcus,

Can you recommend any introductions to LQG? I am decently capable in GR, but cannot seem to bridge the gap to LQG.

- Warren

What does "decently capable in GR" mean?

 

[marcus]

Originally posted by steinitz
Marcus,

Since you started this thread to teach LQG, it seems odd that given that the point of departure of LQG is the ashtekar reformulation of GR, you haven't said much about it beyond some vague or general reference to metric and connection. For example, in words, describe the correspondence between the degrees of freedom as they appear in the ashtekar versus conventional formulations and in particular explain the physical significance of su(2) in all of this. Again, since you didn't start this thread without pretension, I don't think my request is unfair.

Last edited by steinitz on 06-04-2003 at 08:07 AM

 

[marcus]

Originally posted by steinitz
What does "decently capable in GR" mean?

 

[jeff]

It means that in this thread I want to tailor my explanations to fit the poster's level of understanding. Just because you've recognized one or two obvious analogies between LQG and what little you know about quantum theory doesn't mean you really understand LQG, after all, the paper you're looking at is at a somewhat technical level. As usual, I answer your questions but you don't answer mine. So what about su(2)?

 

[marcus]

If anybody is displeased that we are going over parts of Rovelli's review of LQG and wants us to stop, just say so----I and chroot (if he is still around) will be happy to stop.

If anybody wants us to interrupt going thru Rovelli in order to
discuss personalities or rank or STATUS or qualifications or manners or whatever-----instead of just Rovelli and LQG, which is sort of the topic of the thread----then, gosh, I don't know what to do. Just leave the thread? Create a new thread for deciding who the authority is and who should test other people's knowledge?

Or if anybody wants to argue about the ORDER in which we pick up bits of subject matter----whether to discuss this before that---I really have no answer except that you can always make your own thread to have a workshop or studyhall on whatever you like and do in whatever order you believe best.

This is said without rancour to anybody reading the thread.
I don't know now whether or not to proceed because I don't want to have to argue or listen to arguments about side issues.

The person who originally asked for help understanding essentials of LQG is jby. He posted a question about it on Physics. I'm actually trying to tune this to what I think
is his receiver-band, whether he is listening or not. If he is around
he can say if he feels like our going ahead on this track or quitting---that might have some bearing.

 

[marcus]

Originally posted by steinitz
It means that in this thread I want to tailor my explanations to fit the poster's level of understanding. Just because you've recognized one or two obvious analogies between LQG and what little you know about quantum theory doesn't mean you really understand LQG, after all, the paper you're looking at is at a somewhat technical level. As usual, I answer your questions but you don't answer mine. So what about su(2)?

 

[jeff]

Originally posted by marcus
the connection...is taking over the central role from the metric for some purposes.

...it occurred to someone to quantize the connection instead of the metric

Which connection?

Originally posted by marcus
GR used to be the study of the metric on a smooth manifold.

Except for exotic applications like LQG, it still pretty much is.

Originally posted by marcus
A metric would give rise to an idea of curvature and an idea of parallel transport of vectors along a path from one point to another.

So the connection has the same information in it [as the metric], including an idea of curvature.

These are true only for metrics chosen to be compatible with the covariant derivative. Otherwise metric and connection are independent.

Originally posted by marcus
Imagine a sphere and a tangent vector at the equator pointing in some direction and imagine scooting that vector up to the north pole. What direction does it point then?

If you meant "scooting" it up a meridian, than the holonomic change would in fact be null. It's only with respect to the 2-spheres embedding in 3-space that the vector changes direction, and that has nothing to do with parallel transport on the sphere which defines the sphere's intrinsic curvature.

 

[jeff]

Originally posted by marcus
is an element of SU(2). Pick a function f(...g_i...) on [SU(2)]ⁿ .

Why su(2)?

The graph is just a collection of n paths (n "links") from one place to another.

Why are these called "graphs" and what are these "places"?

And parallel transporting along each one gives you a matrix-group element g.

No, it just gives a group element g. Then you must select an (irreducible) matrix representation of g.

Originally posted by marcus So you give me a connection A and I give you back a number.

Are these states so defined physical states?

Originally posted by marcus I may not have said it so clearly but it does not seem fundamentally so hard. Or?

Yeah, I advise you to go with "Or?"

 

[chroot]

So steinitz, are you just being an elitist bastard because you like being one of the only people in the world who understand LQG? Does it bother you people like us are actually attempting to understand it too? Do you feel that we are invading the territory which you hold and use to make yourself feel superior? What is your psychological problem?

- Warren

 

[jeff]

Originally posted by chroot
So steinitz, are you just being an elitist bastard because you like being one of the only people in the world who understand LQG?

Like most high energy theorists, only partly.

Originally posted by chroot
Does it bother you people like us are actually attempting to understand it too?

Does it bother you having someone here who really can answer your questions correctly and point out the gaps in you're understanding, because it sure seems to bother marcus.

I'm not breaking any rules here by challenging people to backup their claims, even when some of them do react defensively, and especially when they present with a great deal of pretension erroneous views as fact (have you ever noticed that they're usually the same people?)

 

[chroot]

Originally posted by steinitz
Like most high energy theorists, only partly.

Can you put down your guard then? I'm not challenging your superiority.

Does it bother you having someone here who really can answer your questions correctly and point out the gaps in you're understanding, because it sure seems to bother marcus.

Nope. I, for one, am happy to have you here -- but I wish you would be a little less severe. Perhaps you could help fill the gaps in my knowledge. I looked over Polchinski's String Theory book in the Stanford book store a while back, and am seriously underprepared to begin reading it. Right now, my highest achievement in learning mathematical physics is reading "Gauge Fields, Knots, and Gravity" by Baez. I'm hoping to begin working on quantum field theory soon enough.

Marcus is in the same boat as me -- he doesn't know LQG or string theory, but would like to, and is grabbing up as many papers and tidbits as he can find. He's going through some process of discovery, and is simply trying to share what he's discovering. Yes, he's likely to make many mistakes -- but at the same time, it's commendable. If you have any interest in flexing your pedagogical muscles, we'd love to have your help!

- Warren

 

[chroot]

Originally posted by steinitz
What does "decently capable in GR" mean?

Oh, and this means I've read 90% of Wald's "General Relativity" and understood 75% of it. I'm by no means an expert, but understand all the machinery and can solve simple problems.

- Warren

 

[jeff]

Originally posted by chroot
I looked over Polchinski's String Theory book in the Stanford book store a while back, and am seriously underprepared to begin reading it. Right now, my highest achievement in learning mathematical physics is reading "Gauge Fields, Knots, and Gravity" by Baez. I'm hoping to begin working on quantum field theory soon enough.

Marcus is in the same boat as me -- he doesn't know LQG or string theory, but would like to, and is grabbing up as many papers and tidbits as he can find. He's going through some process of discovery, and is simply trying to share what he's discovering. Yes, he's likely to make many mistakes -- but at the same time, it's commendable. If you have any interest in flexing your pedagogical muscles, we'd love to have your help!

- Warren

Polchinski was my main resource while learning string theory. Without a solid foundation in QFT though, you'll be less able to understand the effective field theories that the various string theories reduce to in the low energy limit and how to leveredge them to gain insight into and verify important results in string theory. In any event, many ideas in string theory have analogs in QFT that are more easily understood in the generally much simpler field theory setting. Also, though you might understand string theory (and LQG for that matter) on a methodological level, without a solid grounding in QFT and GR, you may have more difficulty appreciating what makes these programs of research so intriguing (or recognizing when they're pathological).

On the other hand, if you can't wait to dig into polchinski (or whatever) I'll be happy to answer your questions. I'll try to take it easy on marcus.

 

[Sauron]

It is a pity that you have stopped the exposition that was being made.


To try to feel the gap betwen general relativity and Asthekar variables i link here an exposition i made in another forums of how [SU(2)]ⁿ comes into LQG.

http://www.100cia.com/foros/viewtopic.php?t=847

There is a big problem with it, it is on spanish and now i have not the time to translate it. By no means i am sure it has no mistakes. It is based on what i understood of the T. Thiemann paper.

 

[marcus]

Sauron's short summary of LQG

I am happy to meet someone like yourself who has read Thiemann's LivingReviews article and took the trouble to make a short summary of (background independent) Quantum Gravity for a message board which seems to be a Spanish-language counterpart of PF-----in English: ciencia.com = science.com/forums

Originally posted by Sauron
It is a pity that you have stopped the exposition that was being made.
To try to fill the gap between general relativity and Asthekar variables [here is a link to] an exposition i made in another forums of how [SU(2)]ⁿ comes into LQG.

http://www.100cia.com/foros/viewtopic.php?t=847

There is a big problem with it, it is on spanish and now i have not the time to translate it. By no means i am sure it has no mistakes. It is based on what i understood of the T. Thiemann paper.

I see no reason not to post your sketch of LQG here in Spanish!
It is interesting to see how "quantum foam" goes over into
" espuma cuántica". It is really interesting to see the same ideas presented in Spanish---with a slightly different flavor to them. So here is your first post:

This was posted 9 May 2003 by Sauron on "100cia.com"
____________________________________________

Espuma cuántica:

La mecáncia cuántica, en concreto el pirncipio de incertidumbre, permite algunos hechos clasicamente prohibidos.

El típico ppio de incertidumbre es el de posición y momento, que indica que no se pueden conocer simultaneamente la posición y velocidad de una partícula con total posición.

Cuando pasamos a considerar la relatividad espacial otro de esoss principios de incertidumbre cobra una relevancia mayor. Hablo de la indeterminación entre tiempo y energí. Es decir las partículas pueden violar la ley de conservación de la energía durante breves instanes de tiempo.

En cuántica no relativista esta indeterminación es poco relevante pero en cuántica relativista esto permite que durante un breve instante de tiempo se creen nuevas partículas. Son las llamadas paartículas virtuales.


Bien, pués pasando por fín a la relatividad general se postula la existencia y relevancia de nuevos principios de indeterminación.

Pero centremos un poco el tema.

La cuantización de la gravedad es harto compleja.

Puesto que la gravedad Einsteniana se basa en el concepto de métrica, basicamente medir distancias, uno podría pensar que la gravitación cuántica sería cuantizar esas distancias, y que existiera una distancia mínima, la longitud de Planck.


Claro, no iba a ser tan fácil, recientemente un Italiano, amelio-Camelia exploró esta posibilidad y sus consecuencias inmediatas. Puedes leer sobre ello en el post del foro de física tituladoo "double special relativity" (o algo similar).


Pero el caso es que esas consideraciones distan basante de llevarnos a una gravitación cuántica.


En realidad el ataque a la gravitación cuántica cambió de rumbo gracias a Feyman, , el gran "papi" de las partículas virtuales. El desarrolló la idea (ya existente) de que la gravedad la media una partícula, el gravitón. Eesta partícula podría interpretarse cómo la portadora de las curvaturas del espacio tiempo que caracterizan la gravedad.


Este ataque de cuantización de la gravedad no ha funcionado bien, es matematicamente inconsistente para partículas puntuales y sólo la consideración de partículas extensas (cuerdas y demás) permiten seguir adelante con este ataque.

Curiosamente este ataque no permite, o al menos no es el más apropiado, para tratar unos cuantos aspectos de la gravitación, en concreto la espuma cuántica.

La idea es que el ppio de indeterminacio de energía-tiempo en RG puede alcanzar un caso extremo, el que esa indeterminación de la energía alcance un valor tan alto que se forma un "agujero negro virtual". y uqe se desintegre rapidamente. Entonces mirado "de cerca" el universo sería un lugar dónde eestan continuamente cereandose y destruyendose agujeros negors.

En realidad podrían formarse y destruirse otros bojetos exótics ligados a agujerson negors, agujeros blancos, agujeros de gusanos, etc.

Entonces a la escala de Planck el universo se vería un poco cómo un queso gruyerere hecho de agujeros que aparecen y desaparecen, Esto es lo que ha dado en llamarse "espuma cuántica".

¿Por que afimro que el gravitón no es bueno para tratar la espuma cuántica?. Pués porque estas rupturas extremas del espacio teimpo no son perturbaciones pequeñas de la métrica del espacio tiempo que es lo que describe bien el gravitón.


De hecho este concepto surge mejor en el marco de otras proximaciones a la gravitación cuántica. La path integral (integral de caminos).


Esta es una forma de descripción de la mecánica cuántica ideadapor Dirac y desarrollada por Feyman. Postula que las parículas no sólo pueden ir de un punto a otro siguiendo las trayectorias clásicas. En realidad pueden ir por cualquier camino. Eso sí, cuanto más alejados de la trayectoria clásica esos caminos son menos influyentes en la probablidad de que un particula vaya de un punto a otro.

En gravitación eso se traduce en que para considerar una probablidad de que un universo en contracción "rebote" y pase a expandirse)hay que sumar sobre todas las métricas posibles, no sólo las clasicamente permitidas.


En esta integral de caminos en principio hab´ria que considerar métricas que no cambiaran la topolo´gia del espacio tiempo (la gravedad clásica no permite estos cambios de topología), pero you puestos a cuantizar ¿por que no permitir estos cambios de topología?.


Es decir, podría considerar que la interacción gravitatoria de dos partículas sería la resultante de considerar que se desplazan por todas las métricas intermedias entre ellas -con las métricas clasicamente compatibles siendo las más importantes- y además podríamos permitr que esas métricas incluyeran "aguejeros" diversos si queremos dar forma a la espuma cuántica.


Por tanto el tema de la espuma cuántica en estas dos aproximaciones es un poco cuestión de fé, es decir, parecería natural permitir su existencia pero no haya nada que lo pruebe.

Y aquí viene a rescate la mejor candidata a una gravedad cuántica que tenemos en la actualidad, la LQG (loop quantum gravity). Esta , a diferencia de las supercuerdas, coje la gravitación de Einstein y los métodos típicos de cuantización (cuantización canónica) e intenta armonizarlos.

Esto you lo habían hecho anteriormente, sin demasiado éxito. El problema es que surgían uans ecuaciones -llamadas ligaduras- muy dificiles de resolver (La más famosa de estas ligaduras se concoe cómo ecuacion de Wheller-deWitt).


Lo novedoso de la loop quantum gravity es que en vez de utilizar directamente la métrica usa unas variables ligadas a ella, las variables de ASthekar, en las que las ligaduras toman una forma más sencilla. Además estas ligaduras se basan en algo muy conocido por los físicos de partículas, el grupo SU(2).


Para resolver estas ligaduras introducen unos estados, los estados de loop, que son estados que permiten describir cualquier configuración gravitatoria en forma de unaintegral sobre un camino cerrado, un loop.

Sobre estos estados las ecuaciones de las ligaduras toamn una forma particularmente sencilla, y se convierten en ecuacioines algebraicas en vez de ecuaciones diferenciales.

En relaidad esta formulaciónha devenido en las llamadas "spin networks", en la uqe estos estados de loop se caracterizan cómo una red con nodos (a traves de consideracioes típicaas de la teoría de grupos) y las ecuaciones lo que describen es cómo evolucionan estos nodos.

Es decir esta LQG lleva de manera natural a una versión discretizada y no continua de la evolución temporal.


Una versión "covariante" de la LQG, basada en path integral y en lagrangianos en vez de hamiltonianos, esta empezando a desarrollarse, y la "spin network" se trasforma en lo que se ha dado en llamarse "spin foam" , traducible por "espuma de spin".


Cómo indica el nombre esta "spin foam" permite la aparción de la espuma cuántica de una manera bastante natural.


He hecho una descripción bastante superficial de los temas indicados, hora no tengo tiempo para entrar en detalles, peor espero que os sirva de algo.


Cómo miniconclusión indicar que la "espuma cuántica" es algo de lo que se tiene una intuición de que podría, o quizás hasta debería, existir. Sin embargo al no sber cuantizar correctamente la gravedad no podíamos ver cómo esta idea intuitiva toma forma. La LQG nos lleva muy cerca de poder analizar esta idea rigurosamente, cómo tantos otros aspectos de la gravedad cuántica.

En cualuqier caso la LQG es "sólo" una teoría matematicamente consistente de la gravedad cuántica, pero no hay evidencias experimentales uqe la avalen (ni que la nieguen, los fenómenos de gravedad cuántica no son relevantes en la mayoría de los procesos que podemos medir, pero tal vez en un futro no muy lejano...). Por tanto incluso si finalmente se demuestra que alberga en su parato matemático la "espuma cuántica" esot no demostrará la existencia de esta, pero al menos sabremso que si no existe la gravitación einsteniana cuantizada no es la teoría que describe la gravedad para latas energías.

 

[marcus]

Chroot
Stanford bookstore---peninsula techies rule
at least theyre probably the world's most cross-disciplinary

bike riders trade the lead around, pushing the air aside
for the others

you are welcome to make the next onslaught

here BTW is a kind of distant target, goal, something
on the horizon to steer by:
Sauron's followup post (which I didnt bring here) is the
good one and towards the end it mentions the Immirzi
parameter.
As Sauron said, it is a free parameter which different
people get slightly different theories by choosing different Immirzi, and it is "order one". It might for example be around 1/8
or it might...you get the idea!

Last year Olaf Dreyer discovered that (this is totally unbelievable
and, in a strange way, funny) in fact is is about 1/8.
this determines finally the quantum of area and of volume
and distinguishes from among several versions of LQG
and this (actually a bit Earth'shaking) result came out in
a little over 3 pages----
arXiv:gr-qc/0211076

I think we should steer the conversation so that, without a lot of technical jargon and namedropping, we get to where some of us
understand something of the significance of that result.

In 1997 Immirzi said that not knowing that parameter represented a "crisis" in LQG, and now suddenly it is known.
Here is an overview by Immirzi himself, also short (8 page):
arXiv:gr-qc/9701052

I tend to be historically oriented---a major watershed has just been reached (by Olaf Dreyer) so I see things in that light. You may have a different sense of direction so steer things however you see fit, if you want to do a section of the thread.

 

[lethe]

Originally posted by chroot
Can you put down your guard then? I'm not challenging your superiority.
[...] but I wish you would be a little less severe.

wow, chroot, the tables sure have turned over here. time was, all you did was tell people that they are ignorant, and don t understand anything. getting some of your own medicine, eh?

 

[chroot]

lethe,

hehehe, well, I'd rather have people who know more than me call me stupid here than to waste my time looking at Mac's pasta pot over there. And I've been reprimanded a few times for being an arrogant **** on here too. :)

- Warren

 

[thermonuclear]

I'm not sure that placing here post in spanish is a good idea. But, anyway, marcus, you've copied here the wrong post of Sauron. The correct one is an attempt to explain very precisely the emergence of the Ashtekar formulation from GR as basic knowledge to start with LQG discussions, and it is placed two post below in the forum where you found the other one. Since it it contains also some mathematical formulation I'll add it here for completeness.


----Begin----
Desafortunadamente ahora no tengo a mano el libro dónde mejor explicado esta el tema, pero de todas formas voy a intentarlo.


Veamos; hay tres puntos clave:

1. El formalismo ADM con el slicing habitual del espacio teimpor. Por fijar notación escribiré algunas fórmulas básicas que no dudo conoces:

ds^2 = N^2dt^2 + qij(dx + Ndt)(dx[j] + N[j]dt)

N: Fución de lapso; qij:la 3-métrica de cada slice ; Uso los [] para indicar supraíndices

Si se expresa la acción de Einstein en términos de Nij y la curvatura extrínseca Kij obtenemos a partir del lagrangiano que los momenots canónicos son:

PIij=@L/@(@tqij))= 1/16PiG (q)^1/2 (K[ij] - q[ij]K)

@:derivada parcial; K[ij] curvatura extrínseca PI: letra griega pi, momento asociado (claro)

El resto de la teoría se que la conoces así que me salto escribir la forma explicita del hamiltoniano y las constraints. Más adelante habrá algunas matizaciones.


2. El formalismo del vielbein:

Imagino que también lo conocerás, de todas forma por lo miso, fijar notación hago una pequeña introducción que además tiene algún aspecto relevante.


Este formalismo (tambien conocido cómo el de la tétradad9 fué introducido por Elie Cartán. Entre otras cosas permite introducir fermiones en la relatividad general, algo muy útl, pero aquí no haré hincapié en ese aspecto.

La tétrada Eu, o vielbein, es (así le gusta decir a la gente) una especie de raiz cuadrada la métrica (en realidad es un sistema de referencia ortonormal, bueno algo así), las ecs que cumple son:

g[uv]=EuEu[j]=η[ij] ηij=EuEv[j]=guv


η= métrica de Lorentz.

Asociada al vielbein esta la conexión de spin:

Wuj que cumple Wij=Ev[Eu[v] ;j (Aquí la ; indica derivación parcial en j, es un convenico standard que confio conozcas, el baile del índice elevad entre una y otra fórmula se debe a que estoy usando varias referencias (una básica y otras de apoyo), pero no veo mayor problema en pasar de una a otra convención por ti mismo

Esta conexión juega el mismo papel que la conexión de levi-civitá.

En el formalismo de la tétrada hay una opción iteresante, el considerar la conexión, y no la métrica cómo el campo básico que describe la gravitación, formalemnete esto quiere decir que aplicamos el principio de mínima acción a este campo y no a la métrica. Es este detalle el que permite que en LQG haya la posibilidad de tenr gij=0, es decir, que no haya espacio, mientras que en le formalismo normal la ausencia de gravedad indica que tenemos η cómo vacio.

A partir de W se puede construir una curvatura R (omito ínidces, w tiene 3, R 4). y el correspondiente lagrangiano , análgo al de Einstein.


Ahora you emepzamos a dejar resultados standar y emepzamos a ir a la LQG, para ellos recordar el concepto de dual de un tensor:

F[*]ij=-i/2εij[kl]F[kl] (ε=tensor completamente antisimétrico de Levi-civitá)


(Es importante de que con esta definiciónnos introducimos en el reino de los números complejo)

Bien, pués introducimos el duál de la conexión de spin, lo denotamos Au[ij]. Asociado a este tensor de conexión complejo hay una curvatura compleja Fuv[ij]. Obviamente se puede escribir el lagrangiano de einstein en término d estas cantidades.

No es casualidad que se denoten A y F estas cantidades pués esa es la notación estandard en Yang-Mills.


3. Variables de Ashtekar-Sen:


Aquí viene el punto que peor se aclara en la referencia que estoy usando, pero más o menos creo que se entiende.

Es importante de que no detallé mucho a cómo se introducía la conexión, bien, si no me falla mucho la memoria de cuando estudie esos temas en los libros de mates (fibrados y demás cosas espeluznantes9 esa conexión se puede asociar con un grupo, el grupo de la fibre bundle.

El caso es que aquí pasamos a trabajar en un slice tipo ADM.
por tanto tenemos una 3-variedad, y con esa 3 variedad construimos un fibrado. ¿Que grupo tiene ese fibrado?, pués el grupo de invarianzas locales. Puesto que estamos en el slice uqe estamos ese grupo es el SO(3), Si no trabajaramos con el slice tendríamos una 4-variedady el grupo sería SO(3,1) (Correspondencia al la invarianza lorentz local).


Sé que se puede inrtroducir el tema del grupo sin mencinar los fibrados, pero ahora no recuerdo dónde puedo mirar los detalles. El trabajo original de Asthekar se mete en muchos detalles sin recurrir a fibrados, pero lo he leido muy por encima.


En todo caso simplemente estamos en que la conexión refleja la invarianza local que es SO(3), trasnsformaciones ortogonales galieansas de toda la vida.

Bieeen, esta disgresión es toda mía y hablo de memoria, ahora seguimos con datos más fiables.

SO(3) tiene de grupo recubirdor SU(2), así que you hemos llegado a puerto. Lo que queda es fácil.


Hagamos un pequeño cambio de notación, ahora denoto el viebein con minúscula y reservo la E mayúscula para nuestros nuevos campos.


Ej=(q)1/2ej (q es la 3 métrica contraida qi).

Introducimos un nuevo campo dependiente de la conexión A:

Ai[j]=ε[0ijk]Aijk

La notación es sugerente, E será una especie de campo electrico y A el potencial vector.

Hemos usado SO(3), es posible erfectamente usar SU(2) y representaciones espinoriales, entonces las definiciones de E y A varian un pelín, pero en el fondo es lo mismo.

Es interesante darse cuenta de que A es deual de la conexión de la conexión original, y que por tanto estamos en el C y no en R, por tanto nuestro SO(3) sería una versión compleja del habitual. Pero las cantidades físicas son reales, nocomplejas y luego hay que imponer condicones de realidad a posteriori.


Bueno, you tenemos nuestros campos, la acción en función de ellos es.


I=Int{ iAi[j]@tEj - iA0jG[j] + iNVi -1/2(N/q^1/2)S}

Int{..} denota integración y tal. G,V y S son las constraints (ligaduras).


a) G[j]=DiE[ij]
b) Vi=E[j]lFij[l]
c) S=... (Una larga expresión que omito)

Aquí Di denota derivación covariante respecto a la conexión que tenemos.

G es análoga a la ley de gauss y V y S son análogas a las ligaduras de momenot y hamiltonianas del formalismo ADM.


Lo interesante es que ahora todas las ligaduras son polinómicas en los camos, en contraste con le caso ADm en que no lo on. Por tanto son más sencillas de resolver.


Existe la poibilidad de expresar estas ligaduras usando la curvatura extrínseca, la conexión W será proporcinal a esta curvatura extrínseca, ahí tambi´ne se introduce una conexión para el slice y se forma un campo A a partir de estas dos cantidades (hay un parámetro libre, el parmámetro de iramizzi).


Bueno, hasta aquí la conexión de Asthekar-Sen.

Unos comentarios de despedidad.

Con esa conexión se construyen los "wilson loops" correspondientes.

Por construcción esto deriva en campos que autmoáticamente satisfacen la "ley de Gauss".

Además poseen estos campos construidos a traves de los wilson loops algunas invarianzas gauge extra.

Es factible calcular el efecto de los campos E y A (cuantizados, y por tanto transformados en operadores) sobre estos campos.

Por tanto es posible calcular el efecto de estas ligaduras sobre los campos que forman la representción. Las ligaduras se convierten en ecuaciones algebraicas en vez de ecs. diferenciales y los métodos de algebras de Lie ayudan a resolverlas.


En concreto la V se puede resolver más o menos razonablemente.

Otro aspecto, you aprovecho, es que el área clásica se puede definir en términos de esstos campos, por tanto al cuantizar se convierte en un operador y se puede demostrar que ese operador tiene un espectro discreto. De ahí sale el famoso resultado que conceta la LQG con la teoría de DSR de Amelio Camelia.


También del operador área sale la obtención del contaje de microestados de un agujero negro compatibles con el área clásica. El problema es que no es un resultado que se obtenga de un hamiltoniano, sí que no es un resultado deducido de 1os ppios.


En fín, la LQg no es un tema demasiado sencilo y no afirmo no haber cometido algún error, máxime cuando you ha pasado un mes desde que hice mi primer ataque a la teoría. Ahora estoy en una fase dereposo, y dedicado a otras actividades, volve´re a hacer un ataque a la LQG cuando haya ido meidtanod poco a poco los que he visto.


Espero que te haya servido de algo todo esto y que más o menos responda el aspecto que te ineresaba.
----End----

 

[marcus]

reply to thermonuclear

I am glad you pasted that post in. I believe I missed it. it seems to agree closely with papers I have been reading and to be
clear and efficient.

A couple of paragraphs from the end he uses an abbreviation for "from first principles" which I never saw before namely "de 1os ppios."

"También del operador área sale la obtención del contaje de microestados de un agujero negro compatibles con el área clásica. El problema es que no es un resultado que se obtenga de un hamiltoniano, sí que no es un resultado deducido de 1os ppios. "

It is fun to figure out what some technical expressions in Spanish mean----like "agujero negro" is black hole.
this would be a generally valuable summary for everybody here if it were in English (and easier for me to read too! my Spanish just barely suffices to let me get thru it)

Sauron has frequently mentioned DSR (doubly special relativity) and the relation to LQR. I wish someone would summarize that in an equally clear concise way---but hopefully in english.

Here is what he says about Amelio Camelia this time.
He mentions the discrete spectrum of the area operator in LQG
which is sort of a landmark and apparently forms the connection to DSR. Is this correct?


"Otro aspecto, you aprovecho, es que el área clásica se puede definir en términos de esstos campos, por tanto al cuantizar se convierte en un operador y se puede demostrar que ese operador tiene un espectro discreto. De ahí sale el famoso resultado que conceta la LQG con la teoría de DSR de Amelio Camelia."

yeah, thanks for posting this of Sauron!

 

[thermonuclear]

I'm afraid I can give only some hints about that, which you might already know. The inherent discretness of spacetime in LQG leads to a deformation of the dispersion relations, e.g. E^2 = p^2 + m^2 + f(E). How these arise exactly is obscure for me and I believe there are several posibilities which were already postulated. The rest of the stuff is trivial if you know something about DSR: A deformation of the dispersion relation does not (necessarily) imply a violation of the relativity principle if one postulates an additional universal constant for the Plank regime (e.g. the Plank lenght). So, new Lorentz transformations might be found.