This is a second question (in what will probably become a series) in my attempt to understand LQG a little. In the previous question I asked about the general concepts behind LQG to which space_cadet provided a very nice answer.

Let me summarize that answer so that I can check that I've understood it properly. In LQG, one works with connections instead of a metric because this greatly simplifies the equations (space_cadet makes an analogy to "taking a square root" of K-G equation to obtain Dirac equation). The connections should determine the geometry of a given 3D manifold which is a space-like slice of our 4D space-time. Then, as is usual in quantizing a system, one proceeds to define "wave-functions" on the configuration space of connections and the space of these functionals on connections should form a Hilbert space.

Note: I guess there is more than one Hilbert space present, depending on precisely what space of connections we work with. This will probably have to do with enforcing the usual Einstein constraints and also diffeomorphism constraints. But I'll postpone these technicalities to a later question.

So that's one picture we have about LQG. But when people actually talk about LQG, one always hears about spin-networks and area and volume operators. So how does these objects connect with the space_cadet's answer? Let's start slowly

  1. What is a spin-network exactly?
  2. What are the main mathematical properties?

Just a reference to the literature will suffice because I realize that the questions (especially the second one) might be quite broad. Although, for once, wikipedia article does a decent job in hinting at answers of both 1. and 2. but it leaves me greatly dissatisfied. In particular, I have no idea what happens at the vertices. Wikipedia says that they should carry intertwining operators. Intertwinors always work on two representations so presumably there is an intertwinor for every pair of edges joining at the vertex? Also, by Schur's lemma, intertwinors of inequivalent irreps are zero, so that usually this notion would be pretty trivial. As you can see, I am really confused about this, so I'd like to hear

 3. What is the significance of vertices and intertwinors for spin-networks?

Okay, having the definitions out of the way, spin-networks should presumably also form a basis of the aforementioned Hilbert space (I am not sure which one Hilbert space it is and what conditions this puts on the spin-networks; if possible, let's postpone this discussion until some later time) so there must be some correspondence between connection functionals and spin-networks.

 4. How does this correspondence look precisely? If I have some spin-network, how do I obtain a functional on connections from that?

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    $\begingroup$ Haha, did space_cadet pay you to write this?! $\endgroup$ – Noldorin Jan 1 '11 at 21:46
  • $\begingroup$ That is a beautifully crafted answer @Marek. Your cheque is in the mail ;) $\endgroup$ – user346 Jan 1 '11 at 21:57
  • $\begingroup$ LOL @Noldorin and @space_cadet. I was just trying to give a little credit :-) $\endgroup$ – Marek Jan 2 '11 at 10:39
  • $\begingroup$ Haha, yes. I was almost expecting to see (space_cadet, 2010) with a bibliography entry with the URL. :) Good job anyway; no need to check for plagiarism hah. $\endgroup$ – Noldorin Jan 2 '11 at 18:32
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    $\begingroup$ @Marek I'm still working on an answer. I will have one shortly. $\endgroup$ – user346 Jan 3 '11 at 2:03

So what are spin-networks? Briefly, they are graphs with representations ("spins") of some gauge group (generally SU(2) or SL(2,C) in LQG) living on each edge. At each non-trivial vertex, one has three or more edges meeting up. What is the simplest purpose of the intertwiner? It is to ensure that angular momentum is conserved at each vertex. For the case of four-valent edge we have four spins: $(j_1,j_2,j_3,j_4)$. There is a simple visual picture of the intertwiner in this case.

Picture a tetrahedron enclosing the given vertex, such that each edge pierces precisely one face of the tetrahedron. Now, the natural prescription for what happens when a surface is punctured by a spin is to associate the Casimir of that spin $ \mathbf{J}^2 $ with the puncture. The Casimir for spin $j$ has eigenvalues $ j (j+1) $. You can also see these as energy eigenvalues for the quantum rotor model. These eigenvalues are identified with the area associated with a puncture.

In order for the said edges and vertices to correspond to a consistent geometry it is important that certain constraints be satisfied. For instance, for a triangle we require that the edge lengths satisfy the triangle inequality $ a + b \lt c $ and the angles should add up to $ \angle a + \angle b + \angle c = \kappa \pi$, with $\kappa = 1$ if the triangle is embedded in a flat space and $\kappa \ne 1$ denoting the deviation of the space from zero curvature (positively or negatively curved).

In a similar manner, for a classical tetrahedron, now it is the sums of the areas of the faces which should satisfy "closure" constraints. For a quantum tetrahedron these constraints translate into relations between the operators $j_i$ which endow the faces with area.

Now for a triangle giving its three edge lengths $(a,b,c)$ completely fixes the angles and there is no more freedom. However, specifying all four areas of a tetrahedron does not fix all the freedom. The tetrahedron can still be bent and distorted in ways that preserve the closure constraints (not so for a triangle!). These are the physical degrees of freedom that an intertwiner possesses - the various shapes that are consistent with a tetrahedron with face areas given by the spins, or more generally a polyhedron for n-valent edges.

Some of the key players in this arena include, among others, Laurent Friedel, Eugenio Bianchi, E. Magliaro, C. Perini, F. Conrady, J. Engle, Rovelli, R. Pereira, K. Krasnov and Etera Livine.

I hope this provides some intuition for these structures. Also, I should add, that at present I am working on a review article on LQG for and by "the bewildered". I reserve the right to use any or all of the contents of my answers to this and other questions on physics.se in said work, with proper acknowledgements to all who contribute with questions and comments. This legalese is necessary so nobody comes after me with a bullsh*t plagiarism charge when my article does appear :P

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  • $\begingroup$ Thanks, @space_cadet. Few questions: in a paper by Rovelli and Upadhya (referenced in the first link by @inflector) the vertices carry a vector from the spin-zero subspace of the tensorial product of all the representations of the links meeting at the vertex. This induces the usual Clebsh-Gordon constraints on spins so that total spin-zero is possible and seems to be consistent with what you talk about. But where are the intertwiners? In a paper by Baez (also referenced by inflector) there is a notion an intertwiner between tensorial product of incoming links and outgoing links. (cont.) $\endgroup$ – Marek Jan 3 '11 at 10:07
  • $\begingroup$ So is the Baez's notion of intertwiner what people talk about? If so, how does one choose incoming and outgoing spins? Is that a freedom of a network to choose such directions? Second question: why do people use $SU(2)$? One should presumably start with an $SO(3)$ connection. Lie algebras of course coincide but global properties of these groups differ. Is there a good reason people switched to $SU(2)$? $\endgroup$ – Marek Jan 3 '11 at 10:09
  • $\begingroup$ Third question: what about the 4. part of my question and the correspondence between spin-networks and functionals? I think (after reading the R-U paper) that I could move that to a further question because it relates more to other questions about Hilbert spaces, namely the way the Hilbert is constructed, that it carries actions of gauge and diff. groups and that spin-networks form orthogonal basis. Do you think I should ask these again a separate questions? $\endgroup$ – Marek Jan 3 '11 at 10:12
  • $\begingroup$ @space_cadet: Nice answer again. I think you can increase "impact" (whatever that is here) by just adding a link to an image linked to your answer. And don't worry about the copyright issue, I think you just have to link your real name to your pseudonym used here (and I'm kind of curious, I have to admit :). $\endgroup$ – Robert Filter Jan 3 '11 at 10:34
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    $\begingroup$ @marek - 1. SU(2) is used because spin-networks represent the geometry of a 3-manifold. h extension to 3+1 is done via spin-foams. There the gauge group is SL(2,C) - the universal cover of the Lorentz group. Only recently has it been figured out how to work with SL(2,C) (Magliaro, Perini, Bianchi et al.). 2. It doesn't matter how you choose ingoing or outgoing spins - same logic as in treating electrical circuits - the right signs pop out of Kirchoff's laws. 3. As for the details of functionals and all that check out arXiv:1009.4475 or gr-qc/0404018. It has to do with C* algebras and stuff. $\endgroup$ – user346 Jan 3 '11 at 19:51

Let me answer to the part of the question regarding the relation between spin networks and functionals of the connection. Each spin network defines precisely one functional of the connection.

For instance, take the simplest spin network formed by just one loop. The corresponding functional of the connection is the "trace of the holonomy" of the connection. That is the trace of the group element obtained integrating the connection along the loop. Physically, this is a number that says how much a vector comes back rotated if you parallel transport it along the loop. This of course depends on the connection, so it is a functional of the connection.

A functional of the connection is a quantum state, like functions of x are quantum states psi(x) is usual quantum theory.

It then turns out that these functionals have the property that they diagonalize the area and volume. This like the state psi(x)=exp{ipx} being an eigenstate of the momentum. So, spin networks are functionals of the connection that are eigenstates of the spatial geometry (this is the result of a calculation).

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  • $\begingroup$ In other words, it's just a Wilson loop. But the answer obviously depends on the direction we take along the loop (the reverse direction will produce an inverse element). This seems even more problematic for more complex spin networks. Would you care to elaborate how does one produce a functional there? For a spin-network of genus two (that is, two loops attached to a single vertex), say. Thanks. $\endgroup$ – Marek Feb 6 '11 at 9:11
  • $\begingroup$ Hi @Carlo, perhaps you could merge this with your previous answer to this question ... $\endgroup$ – user346 Feb 6 '11 at 9:13

In LQG there is an Hilbert space, which can be built along the lines you mention. Then we know that in quantum mechanics there are in general several useful basis in the Hilbert space of a system. For instance, for an harmonic oscillator useful bases are the one that diagonalizes the energy, or that which diagonalizes the position. Well, the spin network state are just the state sin a particularly useful basis in the Hilbert space of the theory. It is useful because it diagonalizes certain operators. Area and volume. Therefor it has a nice physical interpretation. Like the discrete basis that diagonalizes the energy of the harmonic oscillator describes states of N quanta of energy, similarly the spin network basis describes "quanta of space", connected to one another...

carlo rovelli

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    $\begingroup$ So what about time? Is it still continuous in this framework while space is quantized? $\endgroup$ – anna v Feb 6 '11 at 12:43

I have found the LQG introduction references here:


to be quite useful. Marcus is a very nice guy, knows most everything going on in the LQG community, has followed all the LQG papers since 2003 or so, and is very helpful. I don't know if he hangs out here or not yet.

The Baez reference marcus lists:

Baez on spinfoams http://arxiv.org/abs/gr-qc/9709052 (see especially pages 1-4 and 33,34)

may be what you want. As marcus points out, it is dated, and the issues Baez points out have been largely covered by more recent research, but the explanations Baez offers are among the easiest to understand.

I should further point out that most of the research in LQG has converged on the idea of using spinfoam not spin networks per se. I'm not sure I completely understand the difference yet, but that's my impression.

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  • $\begingroup$ Thank you. I've read some of the posts at PF and Marcus indeed seems to be very knowledgeable. I missed the post you link to though; I'll let you know whether it helped me. As for the spin-foams: I gathered that there are more approaches to LQG and that indeed people prefer spin-foams nowadays. Also, as space_cadet pointed out, spin-foams are histories of spin-networks. So even if spin-foams can somehow be studied on their own, I think it can't hurt to study spin-networks first. $\endgroup$ – Marek Jan 2 '11 at 22:17

The spin along each each edgelink is also from my understanding a spin-connection. A discrete parallel translation around a lattice gives a curvature, which is a two-form evaluated through the area of the translation.

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