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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).
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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. –  Matt Reece Jan 1 '11 at 7:03
    
@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 ;) –  user346 Jan 1 '11 at 7:57
    
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. –  Noldorin Jan 1 '11 at 19:32
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@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). –  Marek Jan 1 '11 at 19:48
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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? –  user33486 Nov 10 '13 at 15:06

4 Answers 4

up vote 16 down vote accepted

@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 ;).


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I've just read the first paragraph, so let me address that before I'll delve into your (great, I suppose) answer. I do realize that my question is probably very broad but I don't know how to make it more focused with my current level of knowledge of LQG (read that as non-existent). When I'll understand a little about basics and will find it interesting, be sure that I'll start asking more focused questions. All right, enough blabbering and let me read your answer :-) –  Marek Jan 1 '11 at 14:01
    
Okay, I've read the answer now and it has blown me away completely. First, I always disliked metric and whenever possible I work with connection and vielbeins in GR (in my opinion they are both easier to work with, show the structure of the theory better and are also easily generalized to non Levi-Civita connections; especially to connections with torsion). So this approach is totally to my liking. –  Marek Jan 1 '11 at 14:17
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Second: you state that string theory is not background independent. This is off-topic to my question and also quite outside my area of expertise, but AFAIK you are quite wrong. True, string theory is formulated in some background space but the results are background independent which is the only thing that matters. But I guess this could make a nice question of its own. –  Marek Jan 1 '11 at 14:20
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You did say "correct me if I am wrong", so I will. In your first paragraph you seem to confuse diffeomorphism invariance with background independence, they are two separate issues. String theory is diffeomorphism invariant. String theory is often formulated in 4d Minkowski space times a Calabi-Yau space. The latter is curved. So generically people take the background to be curved. I agree that perturbative string theory is not manifestly background independent, but you more or less need a background to set up perturbation theory, so it is not clear how it could be otherwise. –  pho Jan 2 '11 at 3:56
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@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. –  pho Jan 3 '11 at 23:11

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

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You clearly have an awesome grandmother! –  Nigel Seel Jan 27 '11 at 17:23
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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. –  anna v Feb 2 '11 at 11:33
    
a question: where do the elementary particles come in in this scheme? at least su2xsu3xu1? –  anna v Feb 2 '11 at 11:35
    
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. –  Carlo Rovelli Feb 6 '11 at 8:14
    
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! –  Carlo Rovelli Feb 6 '11 at 8:18

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.

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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. –  Marek Jan 1 '11 at 18:38
    
I've edited my answer slightly to add further comment, that's actually a pretty interesting question... –  user566 Jan 1 '11 at 18:51
    
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. –  Eric Zaslow Jan 1 '11 at 23:55
    
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. –  user566 Jan 2 '11 at 0:11
    
Only found space_cadet's answer later, which addresses Ashtekar variables (though not technically -- s/he could elaborate on her/his #1). –  Eric Zaslow Jan 2 '11 at 0:52

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¢.

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I hope your grandmother understand this type of math. :) –  Rafael Feb 2 '11 at 4:52
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I guess this is where the loops got their name, right? So good point, and also good to see you back :) –  Marek Feb 2 '11 at 8:38
    
@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! :-) –  Daniel Feb 2 '11 at 13:17

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