# Give a description of Loop Quantum Gravity your grandmother could understand

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Of course, assuming your grandmother is not a theoretical physicist.

I'd like to hear the basics concepts that make LQG tick and the way it relates to the GR. I heard about spin-networks where one assigns Lie groups representations to the edges and intertwining operators to the nodes of the graph but at the moment I have no idea why this concept should be useful (except for a possible similarity with gauge theories and Wilson loops; but I guess this is purely accidental). I also heard that this spin-graph can evolve by means of a spin-foam which, I guess, should be a generalization of a graph to the simplicial complexes but that's where my knowledge ends.

I have also read the wikipedia article but I don't find it very enlightening. It gives some motivation for quantizing gravity and lists some problems of LQG but (unless I am blind) it never says what LQG actually is.

So, my questions:

1. Try to give a simple description of fundamentals of Loop Quantum Gravity.
2. Give some basic results of the theory. Not necessary physical, I just want to know what are implications of the fundamentals I ask for in 1.
3. Why is this theory interesting physically? In particular, what does it tell us about General Relativity (both about the way it is quantized and the way it is recovered from LQG).
This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Marek
I've deleted my answer, since it didn't really address the question and seems to have offended Jerry Schirmer and space_cadet. I do hope, though, that if anyone answers the question they will try to make some sort of contact with the question of what principle determines the low-energy effective action, which any theory that reduces to semiclassical GR must somehow do. "Finiteness" and "background independence" are not good answers for this.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Matt Reece
@matt I was definitely not offended. Your critique was simply a reflection of the lack of communication between highly active sub-fields which is either out of complacency or is limited by ideological factors. So please do contribute a critical commentary in the form of an answer. I look forward to answering your doubts and queries. Your insight is invaluable. Perhaps in this way I might even secretly indoctrinate you into the church of LQG ;)

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user user346
This is an interesting question. I'm tempted to say this subject is naturally far beyond the realm of a layman. I mean, even as someone with an undergraduate physics education, I have no hope of understanding QLG.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Noldorin
@Noldorin: I guess having some knowledge of quantum theory and GR should be enough to understand general concepts of quantum gravity, so that's basically the level I was referring to by grandmother. In other words, I want as simple answers as possible. I'll leave it to experts whether layman explanation is possible or not (Feynman for one was amazing at explaining lots of graduate level concepts in layman terms).

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Marek
@Marek: He was indeed - while he was sometimes criticised for his lack of rigour, he was a good communicator it seems! I'd like to think I have good knowledge of (non-relativistic) QM and a bit of basic GR, but I'm still afraid I'm clueless about QLG, etc.!

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Noldorin
@Marek: Great question!

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Robert Filter
Every single answer here assumes you are very knowledgable in quantum physics. Where's the answer for an aspiring learner!? I read New Scientist and watch Feynman videos and started to watch college physics lectures and probably know much much more about particle physics than anyone else I know and the average American. And yet this is all way over my head. Is there truly an answer that a normal grandmother would understand?

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user user33486

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Here is the way I would try to explain Loop Quantum Gravity to my grand mother. Loop Quantum Gravity is a quantum theory. It has a Hilbert space, observables and transition amplitudes. All these are well defined. Like all quantum theories, it has a classical limit. The conjecture (not proven, but for which there are many elements of evidence), is that the classical limit is standard General Relativity. Therefore the "low energy effective action" is just that of General Relativity.

The main idea of the theory is to build the quantum theory, namely the Hilbert space, operators and transition amplitudes, without expanding the fields around a reference metric (Minkowski or else), but keeping the operator associated to the metric itself. The concrete steps to write the theory are just writing the Hilbert space, the operators and the expression for the transition amplitudes. This takes only a page of math.

The result of the theory are of three kind. First, the operators that describe geometry are well defined and their spectrum can be computed. As always in quantum theory, this can be used to predict the "quantization", namely the discreteness, of certain quantities. The calculation can be done, and area and volume are discrete. therefore the theory predicts a granular space. This is just a straightforward consequence of quantum theory and the kinematics of GR.

Second, it is easy to see that in the transition amplitudes there are never ultraviolet divergences, and this is pretty good.

Then there are more "concrete" results. Two main ones: the application to cosmology, that "predicts" that there was big bang, but only a bounce: And the Black Hole entropy calculation, which is nice, but not entirely satisfactory yet. Does this describe nature? We do not know...

carlo rovelli

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Carlo Rovelli
answered Jan 27, 2011 by (290 points)
You clearly have an awesome grandmother!

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Nigel Seel
Well, I am a grandmother though my eldest grandson is only ten, so could not have explained this.I think I get the gist :), thanks.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user anna v
a question: where do the elementary particles come in in this scheme? at least su2xsu3xu1?

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user anna v
Hi Anna. The elementary particles are there. They have their own Hilbert space and the dynamics is coupled to gravity. Intuitively: they move over the discrete quantum space. They do not cause particular problems.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Carlo Rovelli
On the other hand, they are not "unified" to gravity, in the sense in which electric and magnetic forces are unified in Maxwell theory. Loop quantum gravity does not address the unification problem. It "only" addresses the problem of having a quantum theory of gravity and spacetime. In this, it is, say, like Newton theory, which is a theory of gravity and not other forces, or QCD, which is a theory of the strong force, and not other forces. One problem at the time; quantum gravity is hard enough for not trying to do all at once: We are far from the end of physics!

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Carlo Rovelli
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@Marek your question is very broad. Replace "lqg" with "string theory" and you can imagine that the answer would be too long to fit here ;>). So if this answer seems short on details, I hope you will understand.

The program of Loop Quantum Gravity is as follows:

1. The notion of diffeomorphism invariance background independence, which is central to General Relativity, is considered sacrosanct. In other words this rules out the String Theory based approaches where the target manifold, in which the string is embedded, is generically taken to be flat [Please correct me if I'm wrong.] I'm sure that that is not the only background geometry that has been looked at, but the point is that String Theory is not written in a manifestly background independent manner. LQG aims to fill this gap.

2. The usual quantization of LQG begins with Dirac's recipe for quantizing systems with constraints. This is because General Relativity is a theory whose Hamiltonian density ($\mathcal{H}_{eh}$), obtained after performing a $3+1$ split of the Einstein-Hilbert action via the ADM procedure [1,2], is composed only of constraints, i.e.

$$\mathcal{H}_{eh} = N^a \mathcal{V}_a + N \mathcal{H}$$

where $N^a$ and $N$ are the lapse and shift vectors respectively which determine the choice of foliation for the $3+1$ split. $\mathcal{V}_a$ and $\mathcal{H}$ are referred to as the vector (or diffeomorphism) constraint and the scalar (or "hamiltonian") constraint. In the resulting phase space the configuration and momentum variables are identified with the intrinsic metric ($h_{ab}$) of our 3-manifold $M$ and its extrinsic curvature ($k_{ab}$) w.r.t its embedding in the full $3+1$ spacetime, i.e.

$${p,q} \rightarrow \{\pi_{ab},q^{ab}\} := \{k_{ab},h^{ab}\}$$

This procedure is generally referred to as canonical quantization. It can also be shown that $k_{ab} = \mathcal{L}_t h_{ab}$, where $\mathcal{L}_t$ is the Lie derivative along the time-like vector normal to $M$. This is just a fancy way of saying that $k_{ab} = \dot{h}_{ab}$

This is where, in olden days, our progress would come to a halt, because after applying the ADM procedure to the usual EH form of the action, the resulting constraints are complicated non-polynomial expressions in terms of the co-ordinates and momenta. There was little progress in this line until in 1986 $\sim$ 88, Abhay Ashtekar put forth a form of General Relativity where the phase space variables were a canonically transformed version of $\{k_{ab},h^{ab}\}$ This change is facilitated by writing GR in terms of connection and vielbien (tetrads) $\{A_{a}^i,e^{a}_i\}$ where $a,b,\cdots$ are our usual spacetime indices and $i,j,\cdots$ take values in a Lie Algebra. The resulting connection is referred to as the "Ashtekar" or sometimes "Ashtekar-Barbero" connection. The metric is given in terms of the tetrad by :

$$h_{ab} = e_a^i e_b^j \eta_{ij}$$

where $\eta_{ij}$ is the Minkowski metric $\textrm{diag}(-1,+1,+1,+1)$. After jumping through lots of hoops we obtain a form for the constraints which is polynomial in the co-ordinates and momenta and thus amenable to usual methods of quantization:

$$\mathcal{H}_{eha} = N^a_i \mathcal{V}_a^i + N \mathcal{H} + T^i \mathcal{G_i}$$

where, once again, $\mathcal{V}_a^i$ and $\mathcal{H}$ are the vector and scalar constraints. The explanation of the new, third term is postponed for now.

Nb: Thus far we have made no modifications to the theoretical structure of GR. The Ashtekar formalism describes the exact same physics as the ADM version. However, the ARS (Ashtekar-Rovelli-Smolin) framework exposes a new symmetry of the metric. The introduction of spinors in quantum mechanics (and the corresponding Dirac equation) allows us to express a scalar field $\phi(x)$ as the "square" of a spinor $\phi = \Psi^i \Psi_i$. In a similar manner the use of the vierbien allows us to write the metric as a square $g_{ab} = e_a^i e_b^j \eta_{ij}$. The transition from the metric to connection variables in GR is analogous to the transition from the Klein-Gordon to the Dirac equation in field theory.

3. The application of the Dirac quantization procedure for constrained systems shows us that the kinematical Hilbert space, consisting of those states which are annihilated by the quantum version of the constraints, has spin-networks as its elements. All of this is very rigorous and several mathematical technicalities have gradually been resolved over the past two decades.

This answer is already pretty long. It only gives you a taste of things to come. The explanation of the Dirac quantization procedure and spin-networks would be separate answers in themselves. One can give an algorithm for this approach:

1. Write GR in connection and tetrad variables (in first order form).

2. Perform $3+1$ decomposition to obtain the Einstein-Hilbert-Ashtekar Hamiltonian $\mathcal{H}_{eha}$ which turns out to be a sum of constraints. Therefore, the action of the quantized version of this Hamiltonian on elements of the physical space of states yields $\mathcal{H}_{eha} \mid \Psi \rangle = 0$. (After a great deal of investigation) we find that these states are represented by graphs whose edges are labeled by representations of the gauge group (for GR this is $SU(2)$).

3. Spin-foams correspond to histories which connect two spin-networks states. On a given spin-network one can perform certain operations on edges and vertices which leave the state in the kinematical Hilbert space. These involve moves which split or join edges and vertices and those which change the connectivity (as in the "star-triangle transformation"). One can formally view a spin-foam as a succession of states $\{ \mid \Psi(t_i) \rangle \}$ obtained by the repeated action of the scalar constraint $\mid \Psi(t_1) \rangle \sim \exp{}^{-i\mathcal{H}_{eha}\delta t} \mid \Psi (t_0); \mid \Psi(t_2) \rangle \sim \exp{}^{-i\mathcal{H}_{eha}\delta t} \mid \Psi (t_1) \cdots \rangle$ [3].

4. The graviton propogator has a robust quantum version in these models. Its long-distance limit yields the $1/r^2$ behavior expected for gravity and an effective coarse-grained action given by the usual one consisting of the Ricci scalar plus terms containing quantum corrections.

There is a great deal of literature to back up everything I've said here, but this is already pretty exhausting so you'll have to take me on my word. Let me know what your Grandma thinks of this answer ;).

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user user346
answered Jan 1, 2011 by (1,985 points)

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Marek
Glad you like it @marek. Because my hands hurt after writing all that! The gauge formulation of GR is definitely much nicer to work with regardless of how you feel about the quantization procedure. The understanding of instantons and topological defects is instantly and dramatically improved through this approach. As for whether or not string theory is "background independent" - that is something I would like to learn from the experts! Anyhow, I hope my answer helps stimulate discussion on what I feel is a very pretty way of looking at gravity, if nothing else :)

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user user346
Hi @matt, I need some time to eat breakfast and gather my energies. I'll tackle that point later today. Perhaps you might want to ask it as a separate question. This answer is already fairly long. But, short answer, yes such a limit exists and is understood ... not thoroughly and there are many conceptual and technical gaps yet to be filled. For now I'll leave you with this one reference: Rovelli - Graviton propagator from LQG

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user user346
@space_cadet: could you explain how does one arrive at the concept of spin-network? If I understand things correctly, they form a basis of Hilbert space of functionals on connections. So presumably there is a one to one correspondence between the said functionals and spin-networks? I'd imagine that given a connection and a spin-network, one could somehow use holonomy of the connection over the spin-network to obtain a number. All of this is just a wild guessing and should more properly form a question of its own, but right now I'd just welcome a reference that explains these matters.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Marek
@marek would you like to ask that as a separate question or should I add to my answer? Its a bit too long for a comment. Also your "wild guesswork" is surprisingly close to being right on target!

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user user346
@inflector-Sorry, but I don't understand what you are saying. I believe I understand the difference between passive and active diffeos, it is explained quite clearly in Wald, Appendix C.1. Neither has anything to do with the issue of background independence as far as I can tell.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user pho
@Jeff-Harvey - Many people mix them up because the one (active diffeomorphism invariance) is evidence of the other (background independence) in the context of general relativity. They are not the same, however, so in this sense you are correct. See the section on Wikipedia here: en.wikipedia.org/wiki/Background_independence especially the section on Background Independence and Diffeomorphism invariance, paragraph beginning with "The resolution to the hole argument."

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user inflector
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Let me make an attempt: LQG is a name for a collection of research programs aimed at quantizing a metric theory of gravity (usually the Einstein-Hilbert action) directly. Those attempts use different variables (starting with Ashtekar's new variables) and different techniques of quantization (e.g path integrals, canonical quantization, loop quantization, etc.) in an attempt to circumvent the problems one encounters when perturbatively quantizing the EH action around flat spacetime.

For a system with finitely many degrees of freedom, unless you are making some preverse choices, quantizing a classical theory usually gives a quantum theory whose classical limit, in turn, is the theory you started from. Also, most the above quantization procedures (with the possible exception of loop quantization) usually give the same quantum theory, perhaps differing in some minor ways.

All of this is far from guaranteed in field theories, because of ultra violet divergences. The questions of whether when you "quantize GR" the resulting theory has a classical limit where it turns into classical GR in flat spacetime, or whether all the different quantizations used in LQG are equivalent to each other, those are open questions as far as I can tell.

The names of different fields tend to be historical, in this case it has to do with the loop variables employed in constructing the so-called kinematical Hilbert space of LQG. Non -perturbative quantum gravity (or simply quantum gravity) usually refers to more than just LQG, for example including things like causal dynamical triangulations (CDT). All of those approaches have in common the belief that some appropriate approach to quantization will enable us to quantize the metric (or some alias thereof) directly.

This is in distinction to the approach titled (for similarly obscure historical reasons) "string theory" in which the fundamental degrees of freedom are not the metric (or the connection, or anything else appearing in the low energy description of gravity), and the low energy degrees of freedom are "emergent" in some appropriate sense of that word.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user user566
answered Jan 1, 2011 by anonymous
Thank you for your insights. It would indeed appear that LQG is a collection of approaches based on certain assumptions (elsewhere this collection is called non-perturbative quantum gravity) rather than one concrete theory. So my current question probably doesn't make much sense. I'll try to ask more focused questions that have a better chance of getting answered later.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Marek
I've edited my answer slightly to add further comment, that's actually a pretty interesting question...

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user user566
The answers seem to point to a way to narrow the question: e.g., "What are Ashtekar variables?" or "What are the Wheeler-de Witt equations?" Technical responses to these might help give shape to the rather broad responses, though they might not be appropriate for our metaphorical grandmothers.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Eric Zaslow
I tried to give an answer that avoids any technical details, and boils things down to what I consider to be the essential points. I think it is always good to have a bird's eyes view before getting into specifics (especially since these specifics are not universal, i.e. some approaches to LQG may not follow the outline of the more details answers). Certainly follow-up questions could be appropriate, especially since the original one is so broad.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user user566
Only found space_cadet's answer later, which addresses Ashtekar variables (though not technically -- s/he could elaborate on her/his #1).

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Eric Zaslow
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The answers above already cover the necessary ground, but they did not mention one 'angle' of it that i find particularly illuminating: holonomy-flux algebras.

The Ashtekar variables describing the metric ($SU(2)$-valued decomposition of the metric) can be understood in terms of its holonomies — and, much like it's done in gauge theories, one can deal with these holonomies' algebras.

I find this interesting because it seems to link to Gauge Theories in a very particular way, à la Wilson Loops. And, if you squint a bit, ;-) , you can also think of 'flux compactification' (string theory; cf. Green-Schwarz-Witten, Vol 2, chapter 14, or http://arxiv.org/abs/hep-ph/0107039) as a much broader 'object' in Physics.

Just my 2¢.

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Daniel
answered Feb 2, 2011 by (735 points)
I hope your grandmother understand this type of math. :)

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Rafael
I guess this is where the loops got their name, right? So good point, and also good to see you back :)

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Marek
@Marek: that's exactly right: the term "loops" come precisely from this decomposition in terms of Ashtekar variables and how they relate to Wilson loops, etc. In this sense, i think it's very fitting that both, String Theory and LQG, hinge on the same object: flux algebras. ;-) (Glad to be back! :-)

This post imported from StackExchange Physics at 2014-04-01 16:50 (UCT), posted by SE-user Daniel

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