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An Alternative Route To A Quantum Theory of Gravity; Quantum Loop Gravity

Producing a quantum theory of gravity is particularly difficult for several reasons. Firstly quantum gravity implies quantised space time!!! Secondly gravity is intrinsically so weak it has not yet been possible to detect classical gravitational waves, let alone its graviton (it is also difficult to combine the two realms, since QT deals with very small scale phenomena, while GR deals with large masses). Thirdly, quantum field theories are written as spinor fields, to which, the Riemannian techniques of GR not applicable Finally regarding unification, the GR group structure is non compact, while QT group is Unitary and there is a theorem which states that there is no finite unitary representation of a non compact group (this is why we have to resort to supernumbers). At first glance GR and QT look very differently mathematically, as one deals with space-time and direct observables while the other with Hilbert space and operators. One approach to synthesizing the two and providing a quantum theory of gravity, involves Topological Quantum Field Theories (TQFT). [A topologist is sometimes defined as a mathematician who cannot tell the difference between a tea cup and a doughnut, since they are diffeomorphic to each other, both having a genus of 1 ]. Quantum states are given topologies and cobordism allows a description of how quantum (gravity) states evolve i.e. TQFT maps structures in differential topology to corresponding structures in quantum theory. The state of the universe can only change when the topology of space itself changes and TQFT does not therefore presume a given fixed topology for space-time. Quantum operators are therefore related to cobordism and n-category theory (i.e. algebra of n-dimensions) is a useful advance in understanding the cobordism theory of TQFT. A topological quantum field theory will be a way to go from general relativity to quantum mechanics, i.e. given a manifold called “space”, it will spit out a Hilbert space, and given a spacetime it will spit out a linear operator. Therefore, we are looking for some kind of map between the world of manifolds and cobordisms and the world of Hilbert spaces and linear operators. This was the approach taken by Atiyah in his axiomatisation of topological quantum field theories. General relativity has taught us not only that space and time share the property of being dynamical with the rest of the physical entities, but also -more crucially- that spacetime location is relational only. Quantum mechanics has taught us that any dynamical entity is subject to Heisenberg's uncertainty at small scale. Therefore, we need a relational notion of a quantum spacetime in order to understand Planck scale physics.

There have been two reactions to the apparent inconsistency of quantum theories with the necessary background-independence of general relativity. The first is that the geometric interpretation of GR is not fundamental, but just an emergent quality of some background-dependent theory. The opposing view is that background-independence is fundamental, and quantum mechanics needs to be generalized to settings where there is no a-priori specified space-time. This geometric point of view is the one expounded in TQFT. In recent years, progress has been rapid on both fronts, leading ultimately to String Theory (which is not background independent) and Loop Quantum Gravity (LQG), which is background independent and also incorporates the diffeomorphic invariance of GR. Topological quantum field theory provided an example of background-independent quantum theory, but usually with no local degrees of freedom, and only finitely many degrees of freedom globally. This is inadequate to describe gravity, since even in the vacuum, the metric has local degrees of freedom according to general relativity (e.g. those due to the propagation of gravity waves in empty space). The main merit of loop quantum gravity, on the other hand, is that it provides a well-defined and mathematically rigorous formulation of a background-independent, non-perturbative generally covariant quantum field theory. So far, the theory has lead to two main sets of physical results. The first is the derivation of the (Planck scale) eigenvalues of geometrical quantities such as areas and volumes. The second is the derivation of black hole entropy for ``normal'' black holes (but only up to the precise numerical factor). In LQG, physical space is a superposition of spin networks, in the same sense in which the electromagnetic field is a quantum superposition of n-photon states; in this manner space has a discrete and combinatorial character .[LQG has also predicts an inflationary effect, that is required to explain the horizon and flatness problem in cosmology]

A spin network state corresponds to a possible geometry of space. The basic picture is that spin network edges represent flux tubes of area: an edge labeled with spin j contributes an area proportional to sqrt(j(j+1)) to any surface it pierces.
Rovelli and Smolin didn't postulate this, they derived it. Remember, in quantum theory, observables are given by operators on the Hilbert space of states of the physical system in question. You typically get these by "quantizing" the formulas for the corresponding classical observables. So Rovelli and Smolin took the usual formula for the area of a surface in a 3-dimensional manifold with a Riemannian metric and quantized it. Applying this operator to a spin network state, they found the picture I just described: the area of a surface is a sum of terms proportional to sqrt(j(j+1)), one for each spin network edge poking through it.

I should note that the spin networks appearing in the loop representation are different from those Penrose considered, in two important ways. First, they can have more than 3 edges meeting at a vertex, and the vertices must be labeled by "intertwining operators", or "intertwiners" for short. This is a concept coming from group representation theory; what we've been calling "spins" are really irreducible representations of SU(2). If we orient the edges of a spin network, we should label each vertex with an intertwiner from the tensor product of representations on the "incoming" edges to the tensor product of representations labelling the "outgoing" edges. When 3 edges labelled by spins j1, j2, j3 meet at a vertex, there is at most one intertwiner

f: j1 tensor j2 -> j3,

at least up to a scalar multiple. [The triangle inequality and so on --- are just the conditions for a nonzero intertwiner of this sort to exist.] That's why Penrose didn't label his vertices with intertwiners: he considered the case where there's essentially just one way to do it! When more edges meet at a vertex, there are more intertwiners, and this extra information is physically very important. One sees this when one works out the "volume operators" in quantum gravity. Just as the spins on edges contribute *area* to surfaces they pierce, the intertwiners at vertices contribute *volume* to regions containing them.

Now Barbieri considers "simplicial spin networks": spin networks living in a fixed 3-dimensional manifold chopped up into tetrahedra. He only considers spin networks dual to the triangulation, that is, spin networks having one vertex in the middle of each tetrahedron and one edge intersecting each triangular face. In such a spin network there are 4 edges meeting at each vertex, and the vertex is labelled with an intertwiner of the form

f: j1 tensor j2 -> j3 tensor j4

where j1,...,j4 are the spins on these edges. If you know about the representation theory of SU(2), you know that j1 tensor j2 is a direct sum of representations of spin j5, where j5 goes from |j1 - j2| up to j1 + j2 in integer steps. So we get a basis of
intertwining operators:

f: j1 tensor j2 -> j3 tensor j4
by picking one factoring through each representation j5:

j1 tensor j2 -> j5 -> j3 tensor j4

where:

a) j1 + j2 + j5 is an integer and |j1 - j2| <= j5 <= j1 + j2

b) j3 + j4 + j5 is an integer and |j3 - j4| <= j5 <= j3 + j4.

Using this, we get a basis of simplicial spin networks by labelling all the edges and vertices by spins satisfying the above conditions. Dually, this amounts to labeling each tetrahedron and each triangle in our manifold with a spin! Let's think of it this way.
Now focus on a particular simplicial spin network and a particular tetrahedron. What do the spins j1,...,j5 say about the geometry of the tetrahedron? By what I said earlier, the spins j1,...,j4 describe the areas of the triangular faces: face number 1 has area proportional to sqrt(j1(j1+1)), and so on. What about j5? It also describes an area. Take the tetrahedron and hold it so that faces 1 and 2 are in front, while faces 3 and 4 are in back. Viewed this way, the outline of the tetrahedron is a figure with four edges. The midpoints of these four edges are the corners of a parallelogram, and the area of this parallelogram is proportional to sqrt(j5(j5+1)). In other words, there is an area operator corresponding to this parallelogram, and our spin network state is an eigenvector with eigenvalue proportional to sqrt(j5(j5+1)). Finally, there is also a *volume operator*
corresponding to the tetrahedron, whose action on our spin network state is given by a more complicated formula involving the spins j1,...,j5. . . The relationship between spin networks and tetrahedra connects the theory of spin networks, with approaches to quantum gravity where one chops up space into tetrahedra --- like the "Regge calculus" and "dynamical triangulations" approaches.

The main ideas underlying this approach can thus be summarized as follows. One begins by reformulating general relativity as a dynamical theory of connections, rather than metrics. This shift of view does not change the theory classically (although it suggests extensions of general relativity to situations in which the metric may become degenerate). However, it makes the kinematics of general relativity the same as that of SU(2) Yang-Mills theory, thereby suggesting new non-perturbative routes to quantization. Specifically, as in gauge theories, the configuration variable of general relativity is now an SU(2) connection Aia on a spatial 3-manifold and the canonically conjugate momentum Eai is analogous to the Yang- Mills ‘electric’ field. However, physically, we can now identify this electric field as a triad; it carries all the information about spatial geometry. In quantum theory, it is natural to use the gauge invariant Wilson loop functionals, P exp INT A, i.e., the path ordered exponentials of the connections around closed loops as the basic objects .The resulting framework is often called ‘loop quantum gravity’. To produce a spin network - which is the basic tool of LQG - involves the assignment of a non-zero half integer (‘spin’) –or, more precisely, a non-trivial irreducible representation of SU(2)– to each of the edges. To specify a state in Hilbert space H,~j , one only has to fix an intertwiner at each vertex which maps the incoming representations at that vertex to the outgoing ones. (For a trivalent vertex, i.e., one at which precisely three edges meet, this amounts to specifying a Clebsch-Gordon coeffcient associated with the j’s associated with the three edges.) Each resulting state is referred to as a spin network state

General relativity has taught us not only that space and time share the property of being dynamical with the rest of the physical entities, but also {more crucially{ that spacetime location is relational only Quantum mechanics has taught us that any dynamical entity is subject to Heisenberg's uncertainty at small scale. Therefore, we need a relational notion of a quantum spacetime in order to understand Planck scale physics. Fundamentally, loop quantum gravity is a very conservative approach to quantum gravity. It starts with the equations of general relativity and attempts to apply the time-honored principles of quantization to obtain a Hilbert space of states. There are only two really new ideas in loop quantum gravity. The first is its insistence on a background-free approach. That is, unlike perturbative quantum gravity, it makes no use of a fixed ‘background’ metric on spacetime. The second is that it uses a formulation of Einstein’s equations in which parallel transport, rather than the metric, plays the main role. It is very interesting that starting from such ideas one is naturally led to describe states using spin networks! In ordinary quantum field theory we calculate path integrals using Feynman diagrams. Copying this idea, in loop quantum gravity we may try to calculate path integrals using ‘spin foams’, which are a 2-dimensional analogue of Feynman diagrams. In general, spin networks are graphs with edges labeled by group representations and vertices labeled by intertwining operators. These reduce to Penrose’s original spin networks when the group is SU(2) and the graph is trivalent. [A spin network applies Feynmans method in a combinatorial way, producing the so called 6j symbols of a tetrahedron, which encodes the interaction of spinors in a gauge invariant (and space independent) way ] Similarly, a spin foam is a 2-dimensional complex built from vertices, edges and polygonal faces, with the faces label by (irreducible) group representations and the edges labeled by intertwining operators. When the group is SU(2) and three faces meet at each edge, this looks exactly like a bunch of soap suds with all the faces of the bubbles labeled by spins — hence the name ‘spin foam’. If we take a generic slice of a spin foam, we get a spin network i.e.each of its faces gives a spin network edge, and each of its edges gives a spin network vertex. Thus we can think of a spin foam as describing the geometry of spacetime, and any slice of it as describing the geometry of space at a given time. Ultimately we would like a ‘spin foam model’ of quantum gravity, in which we compute transition amplitudes between states by summing over spin foams going from one spin network to another: Spin networks serve simultaneously as a tool for calculations in representation theory and as a description of the quantum geometry of space. Spin foams extend this idea to the geometry of spacetime. [An alternative approach is to employ Lorentzian 4 dimensional spin network tetraheda i.e. 4 simplex which have 5 vertices]

Loop Quantum Gravity is a nonperturbative quantization of 3-space geometry, with quantized area and volume operators. In LQG, the fabric of space-time is a foamy network of interacting loops mathematically described by spin networks (an evolving spin network is termed a spin foam; spin foams are to operators what spin networks are to states/bases). These loops are about 10E-35 meters in size, called the Planck scale. In previous lattice theories the field is represented by quantised tubes/strings of flux which only exist on the edges of the lattice and the field strength is given by the value of integrating around a closed loop. In LQG space and time are relational! As in GR where there are many ways of slicing a section of space time, there are many ways of slicing an evolving spin network - thus there are no things only processes! [A spin network is a graph with edges labeled by representations of some group and vertices labeled by intertwining operators. Thanks in part to the introduction of spin network techniques, we now have a mathematically rigorous and intuitively compelling picture of the kinematical aspects of loop quantum gravity.] The loops knot together forming edges, surfaces, and vertices, much as do soap bubbles joined together. In other words, space-time itself is quantized. Any attempt to divide a loop would, if successful, cause it to divide into two loops each with the original size. In LQG, spin networks represent the quantum states of the geometry of relative space-time. Looked at another way, Einstein's theory of general relativity is a classical approximation of a quantized geometry. The problem is, how do we understand the quantum geometry of space-time? The problem of time makes it difficult to apply a canonical hamiltonian approach and consequently the path integral Lagrangian is preferred. Unlike most field theories which are set on a manifold with fixed metric, the dynamics of quantum gravity are generated by 'constraints'. Quantum gravity is so hard because whereas there are tricks for quantizing Poisson manifolds (i.e. phase space) by introducing operators obeying Heisenberg's UP, the phase space of GR is infinite dimensional with singularities. From the study of Einstein's equation and QM, we arrive at (utilizing Poison brackets), . . . Gauss constraints (gauge invariance due to Yang-Mills theory, these producing Kinematic states, which do not generate dynamics) .. . .diffeomorphic constraints (due to a GR symmetry) . . . . and Hamiltonian constraints (these do produce a dynamic description but are difficult to define), which have led us to spin bundle spaces L*(A), quotient spaces L*(A/G) and bivectors that are associated with spin networks. We quantise gravity by means of a connection A and its canonical momentum conjugate E, with which we create spin networks for spinor states and from this we can produce area operators which act on space to quantise its geometry!.

So lets take another perspective on the physical meaning of these loop states. Consider taking a spin-1/2 particle and move it around in a path that traces out a knot. When you do this using the Levi-Civita connection, it comes back "rotated" by some SU(2) matrix. If you take the trace of this matrix (sum of diagonal entries) and divide by two, you get a number between -1 and 1. This number is called a "Wilson loop".
This should remind you of the Bohm-Aharonov effect where a split electron beam takes two paths from A to B. Depending on the magnetic flux through the loop, one can have constructive or destructive interference in the split beam experiment.
Mathematically, one can imagine moving the electron around a loop that starts at A, goes to B by one path, and goes back to A by the other path. If this phase corresponding to going round the loop is 1 we get total constructive interference in the split beam experiment, while if it's -1 we get total destructive interference. Just as the Bohm-Aharonov effect measures interference effects due to the magnetic field, the above Wilson loop sort of measures the interference effects due to GRAVITY! Now, in the Rovelli-Smolin loop state corresponding to a particular knot K, the expectation value of a Wilson loop around any knot K' will be 1 if K and K' are isotopic, and 0 otherwise! That's the physical meaning of the loop states: they describe quantum states of geometry in terms of the resulting interference effects on spin-1/2 particles.

Now, there is a more general kind of diffeomorphism-state than the loop states. These are the spin network states! Here one fancies up the Wilson loop idea and imagines a graph embedded in space --- i.e. a bunch of edges and vertices ---where each edge is labelled by a spin that can be 0,1/2,1,3/2, etc. In the simplest flavor of spin network, one only allows 3 edges to meet at each vertex, and requires j3 to be of the form

j3 = |j1-j2|, |j1-j2| + 1, ...., j1+j2-1, j1+j2.

where j1, j2, j3 are the spins labelling the edges adjacent to the given vertex. For example, we can have the the three spins be 1/2,3, and 5/2, because it's possible for a spin-1/2 particle and a spin-3 particle to interact and form a spin-5/2 particle. Here
by "possible" I simply mean that it doesn't violate conservation of angular momentum. Mathematicians would say the spins should be thought of as irreducible representations of SU(2), and the condition above is just the condition that the representation j3 appears as a summand in the tensor product of the representations j1 and j2.
Just as we can compute a kind of "Wilson loop" number from a knot that a spin-1/2 particle goes around, we can compute a number from a spin network.

Given a real-analytic manifold M, a compact connected Lie group G and a principal G-bundle P -> M, there is a canonical generalized measure' on the space A/G of smooth connections on P modulo gauge transformations. This allows one to define
a Hilbert space L^2(A/G). Here we construct a set of vectors spanning L^2(A/G). These vectors are described in terms of `spin networks': graphs phi embedded in M, with oriented edges labelled by irreducible unitary representations of G, and with vertices labelled by intertwining operators from the tensor product of representations labelling the incoming edges to the tensor product of representations labelling the outgoing edges. We can study the operator that corresponds to the measurement of volume, in non-perturbative quantum gravity, and compute its spectrum. The operator is constructed in the loop representation, via a regularization procedure; it is finite, background independent, and diffeomorphism-invariant, and therefore well defined on the space of diffeomorphism invariant states (knot states). The family of eigenstates are in one to one correspondence with the spin networks, which were introduced by Penrose in a different context. It is possible to compute the corresponding component of the spectrum, and exhibit the eigenvalues explicitly. The other eigenstates are related to a generalization of the spin networks, and their eigenvalues can be computed by diagonalizing finite dimensional matrices. Furthermore, we can show that the eigenstates of the volume also diagonalize the area operator. The spectra of volume and area can be considered as predictions of the loop-representation formulation of quantum gravity on the outcomes of (hypothetical) Planck-scale sensitive measurements of the geometry of space.

What follows below are a collection of main signposts on the road to the present formulation of Quantum Loop Gravity

Now lets move on to higher dimensions; briefly a"quantum triangle" in 4 dimensions is given by a complex function on the 3-sphere and a "quantum tetrahedron" is a collection of 4 quantum triangles satisfying some constraints. More precisely, let

H = L2(S3)

be the Hilbert space for a quantum triangle in 4 dimensions. Then the Hilbert space for a quantum tetrahedron is a certain subspace T of H x H x H x H, where "x" denotes the tensor product of Hilbert spaces. Concretely, we can think of states in
T as complex functions on the product of 4 copies of S3. (these complex functions do need to satisfy some constraints)
Now let's "second quantize" the Hilbert space T. This is physics jargon for making a Hilbert space out of the algebra of polynomials on T - usually called the "Fock space" on T. As usual, there are two pictures of states in this Fock space: the "field" picture and the "particle" picture. On the one hand, they are states of a quantum field theory on the product of 4 copies of S3. But on the other hand, they are states of an arbitrary collection of quantum tetrahedra in 4 dimensions. In other words, we've got ourselves a quantum field theory whose "elementary particles" are quantum tetrahedra!
The idea of the de Pietri-Freidel-Krasnov-Rovelli paper is to play these two pictures off each other and develop a new way of thinking about the Barrett-Crane model. The main thing these guys do is write down a Lagrangian with some nice properties. Throughout quantum field theory, one of the big ideas is to start with a Lagrangian and use it to compute the amplitudes of Feynman diagrams. A Feynman diagram is a graph with edges corresponding to "particles" and vertices corresponding to "interactions" where a bunch of particles turns into another bunch of particles.
But in the present context, the so-called "particles" are really quantum tetrahedra! Thus the trick is to write down a Lagrangian giving Feynman diagrams with 5-valent vertices. If you do it right, these 5-valent vertices correspond exactly to ways that 5 quantum tetrahedra can fit together as the 5 faces of a 4-simplex! Let's call such a thing a "quantum 4-simplex".

Then your Feynman diagrams correspond exactly to ways of gluing together a bunch of quantum 4-simplices face-to-face. Better yet, if you set things up right, the amplitude for such a Feynman diagram exactly matches the amplitude that you'd compute for a triangulated manifold using the Barrett-Crane model! In short, what we've got here is a quantum field theory whose Feynman diagrams describe "quantum geometries of spacetime" - where spacetime is not just a fixed triangulated manifold, but any possible way of gluing together a bunch of 4-simplices face-to-face.
There remain some major problems however. First, we don't know that the "sum over Feynman diagrams" converges in this theory. In fact, it probably does not - but there are probably ways to deal with this. Second, the model is Riemannian rather than Lorentzian: we are using the rotation group when we should be using the Lorentz group. Luckily this is addressed in a new paper by Barrett and Crane. Third, we aren't very good at computing things with this sort of model - short of massive computer simulations, it's tough to see what it actually says about physics
First of all, we have to remember that Ashtekar reformulated Einstein's equation so that the configuration space for general relativity on the spacetime R x S, instead of being the space of *metrics* on a 3-manifold S, is a space of *connections* on S. A connection is just what a physicist often calls a vector potential, but for any old gauge theory, not just electromagnetism.
Different gauge theories have different gauge groups, so I had better tell you the gauge group of Ashtekar's version of general relativity: it's SL(2,C), the group of 2x2 complex matrices with determinant equal to 1. And I should probably tell you which bundle over S we have an SL(2,C) connection on... but luckily, all SL(2,C) bundles over 3-manifolds are trivial, so we can cut corners by saying it's the trivial bundle. We can think of a connection A on the trivial SL(2,C) bundle over S as 1-forms taking values in the Lie algebra sl(2,C), consisting of 2x2 complex matrices with trace zero.
Okay, so naively you might think a state in the *quantum* version of general relativity a la Ashtekar is just a wavefunction psi(A). But there's one very important catch we can't ignore: general relativity has *constraint* equations, meaning that psi has to satisfy some equations. The first constraint, the Gauss law, just says that we must have

psi(A) = psi(A')

whenever A' is the result of doing a gauge transformation to A. Or at the very least, this should hold up to a phase; the point is that psi is only supposed to record physically significant information about the state of the universe, and two connections are physically equivalent if they differ by a gauge transformation. The second constraint, the diffeomorphism constraint, says we need to have

psi(A) = psi(A')

when A' is the result of applying a diffeomorphism of space, S, to A. Again, the point is that two solutions of general relativity are physically the same if they differ only by a coordinate transformation, or - *roughly* the same thing - a diffeomorphism.
The third constraint is the real killer. It's meaning is that psi(A) doesn't change when we do a diffeomorphism of spaceTIME to the connection A, but it's usually formulated `infinitesimally' as the Wheeler-DeWitt equation

H psi = 0

meaning roughly that the time derivative of psi is zero. Think of it as a screwy quantum gravity version of Schrodinger's equation, where d psi/dt had better be zero!
It's hard to find explicit solutions of these equations. Indeed, it's hard to know what the heck these equations *mean* in a sufficiently precise way to recognize a solution if we found one! However, things were even worse back in the old days. Back in the old days when we thought of states as wavefunctions on the space of metrics, we didn't know ANY solutions of these equations. But nowadays we are very happy, because we know infinitely many times as many solutions! To be precise, we now know ONE solution. This is called the Chern- Simons state, and it was discovered by Kodama:

Now there is a slight catch: the Chern-Simons state is a solution of quantum gravity *with cosmological constant*. This is an extra term that Einstein threw into his equations so that they wouldn't make the obviously ridiculous prediction that the universe is expanding. When Hubble took a look and saw galactic redshifts all over, Einstein called this extra term the biggest blunder in his life. That kind of remark, coming from that kind of person, might make you a little bit reluctant to get too excited about having found a state of quantum gravity with this extra term thrown in! Luckily it turns out that you can take the limit as the cosmological constant goes to zero, and get a state of the theory where the cosmological constant is zero. I like to call this the `flat state', because it's zero except where the connection A is flat. (In fact, if the space S is not simply connected, there are lots of different flat states, because there is what experts call a moduli space of flat connections, i.e., lots of different flat connections modulo gauge transformations.

Now what is this Chern-Simons state? Well, there is a wonderful thing you can compute from a connection A on a (compact oriented) 3-manifold S, called the Chern-Simons action:

CS(A) = integral_S tr(A ^ dA + (2/3)A ^ A ^ A)

which looks weird when you first see it, but gradually starts seeming very sensible and nice. The reason why folks like it is that it doesn't change when you do a small gauge transformation - i.e., one you can get to following a continuous path from the identity - and it changes only by an integral multiple of 8pi^2 if you do a large gauge transformation. Plus, it's diffeomorphism-invariant. It's incredibly hard to write down many functions of A with these properties, so they are precious.
There are deeper reasons why it's so nice, but let's leave it at that for now.
So, the Chern-Simons state is

psi(A) = exp(-6 CS(A)/Lambda)

where Lambda is the cosmological constant. Don't worry about the factor of 6 too much; depending on how you set up various things you might get different numbers, and I can never keep certain factors of 2 straight in this particular calculation. Note however that it looks as if things go completely haywire as Lambda approaches zero, which is why my earlier remark about the `flat state' is a bit nontrivial.
Why does this satisfy the constraints? Well, as mentioned above, the Chern-Simons action was hand-tailored to have the gauge-invariance and diffeomorphism-invariance we want, so the only big surprise is that we *also* have a solution of the Wheeler-DeWitt equation. But clearly nature, or at least the goddess of mathematics, is trying to tell us something if this Chern-Simons state, which has all sorts of wonderful properties relating to *3-dimensional* geometry, is also a solution of the Wheeler-DeWitt equation, which is all about *4-dimensional* geometry, since it expresses the invariance of psi under evolution in TIME. The reason is because Chern-Simons action was *born* as a 3-dimensional spinoff of a 4-dimensional thing called the 2nd Chern class.
What is the physical meaning of the Chern-Simons state? As far as I know Kodama's paper hasn't been vastly surpassed in explaining this. He shows that in the classical limit this state becomes something called the anti-deSitter universe, a solution of Einstein's equation describing a (roughly) exponentially expanding universe. If you are wondering what this has to do with Einstein's introduction of the constant to KEEP the universe from expanding, let me just say this. In our current big bang theory the universe is expanding, but the presence of matter, or any sort of positive energy density, tends to pull it back in,
and if there is enough matter it'll collapse again. Einstein's stuck in a cosmological constant term to give the vacuum some negative energy density, exactly enough to counteract the positive energy density of matter, so things would neither collapse
nor expand, but instead remain in an (unstable, alas) equilibrium. In the deSitter universe there's no matter, just a cosmological constant of the opposite sign, so that the vacuum has positive energy density. In the anti-deSitter universe (invented by deSitter's nemesis anti-deSitter) there's no matter either, but the cosmological constant has the sign giving the vacuum negative energy density, which pushes the universe to keep expanding faster and faster. [SIC]

Now in addition to this physical interpretation, the Chern-Simons state also has some remarkable relationships to knot theory, which were discovered by Witten and, since then, studied intensively by lots of people. Briefly, there should be an invariant of knots and links associated to any state of quantum gravity, and the one associated to the Chern-Simons state is the Kauffman bracket, a close relative of the Jones polynomial, which is distinguished by having a very simple, beautiful definition, and also lots of wonderful relationships to an algebraic structure, the quantum group SU_q(2). I should add that in addition to an invariant of knots and links, a state of quantum gravity should also give an invariant of *spin networks*, and indeed the Kauffman bracket extends to a wonderful invariant of spin networks.

*************************

The discreteness of area and volume is derived as follows. Consider the area A of a surface . The physical area A of  depends on the metric, namely on the gravitational field. In a quantum theory of gravity, the gravitational field is a quantum field operator, and therefore we must describe the area of  in terms of a quantum observable, described by an operator ^ A. We now ask what the quantum operator ^ A in nonperturbative quantum gravity is. The result can easily be worked out by writing the standard formula for the area of a surface, and replacing the metric with the appropriate function of the loop variables. Promoting these loop variables to operators, we obtain the area operator ^ A. The actual construction of this operator requires regularizing the classical expression and then taking the limit of a sequence of operators, in a suitable operator topology. The resulting area operator ^ A acts on a spin network state jSi (assuming here for simplicity that S is a spin network without nodes on ): A similar result can be obtained for the volume in terms of the number of nodes that intersect a given surface.

For simplicity and ease of visualization, you can pretend S is a ball, so its boundary is a sphere. Think of a spin network state as a graph embedded in this ball, possibly with some edges ending on the boundary, with all the edges labeled by spins j = 0,1/2,1,3/2, etc., and with the vertices label by some extra numbers that we won't worry about here. Let's call the points where edges end on the boundary `punctures', because the idea is that they really poke through the boundary and keep on going. Physically, these edges graph represent `flux tubes of area': if we measure the area of some surface in this state (or at least a surface that doesn't intersect the vertices), the area is just the quantity

L^2 sqrt(j(j+1))

summed over all edges that poke through the surface, where L is the Planck length and j is the spin labeling that edge. Gauge theories often have "flux tube" solutions when you quantize them: for example, type II superconductors admit flux tubes of the magnetic field, while superfluids admit flux tubes of angular momentum (vortices). The idea behind spin networks in quantum gravity, physically speaking, is that gravity is a gauge field which at the Planck scale is organized into branching flux tubes of area.
But we want to understand, not the kinematical states in general, but the actual physical states, which satisfy the diffeomorphism constraint and the Wheeler-DeWitt equation. We can start by measuring everything we measure by doing experiments right at the boundary of S. More precisely, we can try to find a maximal set of commuting observables that `live on the boundary' in this sense. For example, the area of any patch of S counts as one of these observables, and all these `surface patch area' observables commute. If we measure all of them, we know everything there is to know about the area of all regions on the boundary of S. Thanks to spin network technology, as described above, specifying all their eigenvalues amounts to specifying the location of a bunch of punctures on the boundary of S, together with the spins labeling the edges ending there.
Now Chern-Simons theory gives an obvious candidate for the space of physical states of quantum gravity for which these`surface patch area' observables have specified eigenvalues. In fact, if you hand Chern-Simons theory a surface like the boundary of S, together with a bunch of punctures labelled by spins, it gives you a FINITE-DIMENSIONAL state space.
Let's not explain just now how it gives you this state space; let's simply mumble that it gives you this space by virtue of being an `extended topological quantum field theory.' If you want, you can think of these states as being the `relative states' I discussed in last week's Finds, but not all of them: only those for which the `surface patch area' observables have specified eigenvalues. There is a wonderfully simple combinatorial recipe for describing all these states in terms of spin networks living in S, having edges that end at the punctures, with the right spins at these ends.

Smolin's hypothesis is that this finite-dimensional space of states coming from Chern-Simons theory *is* the space of all physical states of quantum gravity on S that 1) satisfy the self-dual boundary conditions and 2) have the specified values of the surface patch area observables. Now if this hypothesis is true, it means we have a wonderfully simple description of all the physical states on S satisfying the self-dual boundary conditions!

Thanks to work by Hawking, Bekenstein and others, there is a lot of evidence that if one takes quantum gravity into account, the maximal entropy of any system contained in a region with surface area A should be proportional to A. The basic idea is this. For various reasons, one expects that the entropy of a black hole is proportional to the area of its event horizon. For example, when you smash some black holes together it turns out that the total area of the event horizons goes up --- this is called the `second law of black hole thermodynamics'. This and many more fancy thought experiments suggest that when you have some black holes around the right notion of entropy should include a term proportional to the total area of their event horizons. Now suppose you had some other system which had even MORE entropy than this, but the same surface area. Then you could dump in extra matter until it became a black hole, which would therefore have less entropy, violating the second law.

All this suggest an intriguing connection between the vast literature on black hole thermodynamics and the more mathematical problem of relating quantum gravity and Chern-Simons theory. Now the maximum entropy of a system is proportional to the logarithm of the total number of states it can assume. So if the `Bekenstein bound' holds, the dimension of the space of states of a system contained in a region with surface area A is proportional to exp(A/c) for some constant c (which should be about the Planck length squared). Now the remarkable thing about Smolin's hypothesis is that if it's true, this is what one gets, because the dimension of the space given by Chern-Simons theory does grow like this. There is another approach leading to this conclusion that the space of states of a bounded region should have dimensional proportional to exp(A/c), called the 't Hooft-Susskind holographic hypothesis.

**************

So: it seems we can't determine the constant of proportionality in the entropy-area relation, because of this arbitrariness in the Immirzi parameter. But we can, of course, use the Bekenstein-Hawking formula together with our formula for black hole
entropy to determine gamma, obtaining

gamma = ln(2) / sqrt(3) pi

This may seem like cheating, but right now it's the best we can do. All we can say is this: we have a theory of the microstates of a black hole, which predicts that entropy is proportional to area for largish black holes, and which taken together with the
Bekenstein-Hawking calculation allows us to determine the Immirzi parameter.

What do the funny constants in the formula S = (ln 2 / 4 pi gamma sqrt(3)) A mean? It's actually simple. The states that contribute most to the entropy of a black hole are those where nearly all spin network edges puncturing its surface are labelled by spin 1/2. Each spin-1/2 puncture can have either jz = 1/2 or jz = -1/2, so it contributes ln(2) to the entropy. On the other hand, each spin-1/2 edge contributes 4 pi gamma sqrt(3) to the area of the black hole. Just to be dramatic, we can call ln 2 the "quantum of entropy" since it's the entropy (or information) contained in a single bit. Similarly, we can call 4 pi gamma sqrt(3) the "quantum of area" since it's the area contributed by a spin-1/2 edge.
These terms are a bit misleading since neither entropy nor area need come in *integral* multiples of this minimal amount. But anyway, we have

S = (quantum of entropy / quantum of area) A

*****************

Recall that general relativity is usually thought of as a theory about a metric on spacetime - more precisely, a Lorentzian metric. Here spacetime is a 4-dimensional manifold, and a Lorentzian metric allows you to calculate the "dot product" of any two tangent vectors at a point. This is in quotes because, while a normal dot product might look like

(v_0,v_1,v_2,v_3).(w_0,w_1,w_2,w_3) = v_0w_0 + v_1w_1 + v_2w_2 + v_3w_3

relative to some basis, for a Lorentzian metric we can always find a basis of the tangent space such that

(v_0,v_1,v_2,v_3).(w_0,w_1,w_2,w_3) = v_0w_0 - v_1w_1 - v_2w_2 - v_3w_3

Now the metric in general relativity defines a "connection," which tells you a tangent vector might "twist around" as you parallel translate it, that is, move it along while trying to keep it from rotating unnecessarily. Here "twist around" is in quotes because, since you are parallel translating the vector, it's not really "twisting around" in the usual sense, but it might seem that way relative to some coordinate system. For example, if you used polar coordinates to describe parallel translation on the plane, it might seem that the unit vector in the r direction "twisted around" towards the theta direction as you dragged it along.

However in another coordinate system - say the usual x-y system - it would not appear to be "twisting around". This fact isexpressed by saying "the connection is not a tensor". But from the connection we can cook up a big fat tensor, the "Riemann tensor" R^i_{jkl}, which says how much the vector in the lth direction (here the indices range from 0 to 3) twists towards the ith direction when you move it around a teeny little square in the j-k plane. The Lagrangian in ordinary GR is just the integral of the "Ricci scalar curvature," R, which is gotten from the Riemann tensor by "contraction", i.e. summing over the indices in a certain way:

R = R^i_{ji}^j

where we are raising indices using the metric in a manner beloved by physicists and feared by many mathematicians. If you integrate the Lagrangian over a region of spacetime you get the "action", and in classical general relativity (in a vacuum, for
simplicity) one can formulate the laws of motion simply by saying: any teeny change in the metric that vanishes on the boundary of the region should leave the action constant to first order. In other words, the solutions of the equations of general relativity are the *stationary points* of the action. If you know how to do variational calculus you can derive Einstein's equations from this variational principle, as it's called. Mathematicians will be pleased to know that Hilbert beat Einstein to the punch here, so the integral of R is called the "Einstein-Hilbert" action for general relativity.

Now there's another formulation of general relativity in terms of an action principle -- called the "Palatini" action .The Palatini approach turns out to be more elegant and is a nice stepping-stone to the Ashtekar approach. In the Palatini approach one thinks of general relativity not as being a theory of a metric, but of a "tetrad" and an "so(3,1) connection". To explain what these are, I will cut corners and assume all the fiber bundles lurking around are trivial; the experts will easily be able to figure out the general case. So: an (orthonormal) tetrad, or "vierbein," is a just a kind of field on spacetime which at each point consists of an (ordered) orthonormal basis of the tangent space. If we express the metric in terms of a tetrad, it looks just like the formula for the standard "inner product"

(v_0,v_1,v_2,v_3).(w_0,w_1,w_2,w_3) = v_0w_0 - v_1w_1 - v_2w_2 - v_3w_3

This allows us to identify the group of linear transformations of the tangent space that preserve the metric with the group of linear transfomations preserving the standard "inner product," which is called SO(1,3) since there's one plus sign and three minuses. And from the connection mentioned above one gets an SO(1,3) connection, or, what's more or less the same thing, an so(1,3)-valued 1-form, that is, a kind of field that can eat a tangent vector at any point and spits out element of the Lie algebra so(1,3).

What's so(1,3)? Well, elements of so(1,3) include "infinitesimal" rotations and Lorentz transformations, since SO(1,3) is generated by rotations and Lorentz transformations. More precisely, so(1,3) is a 6-dimensional Lie algebra having as a basis the three infinitesimal rotations J_1, J_2, and J_3 around the three axes, and the three infinitesimal Lorentz transformations or "boosts" K_1, K_2, K_3. The bracket in this most important Lie algebra is given by

[J_i,J_j] = J_k
[K_i,K_j] = -J_k
[J_i,K_j] = K_k

where (i,j,k) is a cyclic permutation of (1,2,3). (I hope I haven't screwed up the signs.) Note that the J's by themselves form a Lie subalgebra called so(3), the Lie algebra of the rotation group SO(3). Note that so(3) is isomorphic to the the cute little Lie algebra su(2) I described in my post "week5"; J_1, J_2, and J_3 correspond to the guys I, J, and K divided by two.

The so(1,3) connection has a curvature, and using the tetrads again we can identify this with the Riemann curvature tensor. So the Palatini trick is to rewrite the Einstein-Hilbert action in terms of the curvature of the so(1,3) connection and the tetrad field. This is called the Palatini action. Charmingly, even though the tetrad field is utterly unphysical, we can treat it and the so(1,3) connection as independent fields and, doing calculus of variations to find stationary points of the action, we get equations equivalent to Einstein's equations. Ashtekar's "new variables" - from this point of view - rely on a curious and profound fact about so(1,3). Note that so(1,3) is a Lie algebra over the real numbers. But if we allow ourselves to form *complex* linear combinations of the J's and K's, thus:

M_i = (J_i + iK_i)/2
N_i = (J_i - iK_i)/2

(please don't mix up the subscript i = 1,2,3 with the other i, the square root of minus one) we get the following brackets:

[M_i,M_j] = M_k
[N_i,N_j] = N_k
[M_i,N_j] = 0

I think the signs all work but I wouldn't trust me if I were you. The wonderful thing here is that the M's and N's commute with each other, and each set has commutation relations just like the J's! The J's, recall, are infinitesimal rotations, and the Lie algebra they span is so(3). So in a sense the Lie algebra of the Lorentz group can be "split" into "left-handed" and "right-handed" copies of so(3), also known as "self-dual" and "anti-self-dual" copies. This is, in fact, what lies behind the handedness of neutrinos, and many other wonderful things.

But let me phrase this result more precisely. Since we allowed ourselves complex linear combinations of the J's and K's, we are now working in the "complexification" of the Lie algebra so(3,1), and we have shown that this Lie algebra over the complex numbers splits into two copies of so(3,C), the complexification of so(3).
Ashtekar came up with some "new variables" for general relativity in the context of the Hamiltonian approach. Here we are working in the Lagrangian approach, where things are simpler because they are "generally covariant," not requiring a split of spacetime into space and time. The Lagrangian approach to the new variables is due to Samuel, Jacobson and Smolin, and in this approach all they amount to is this: so(1,3) connection of the Palatini approach, think of the so(1,3) as sitting inside the complexification thereof, and consider only the "right-handed" part! Thus, from an so(1,3) connection, we get a so(3,C) connection. The "new variables" are just the tetrad field and this so(3,C) connection.
Lagrangian, without explaining exactly what they mean, just to show how amazingly similar-looking they are. In the Palatini approach we have a tetrad field, which now we write in its full glory as e_I^i, and the curvature of the so(1,3) connection,
which now we write as Omega_{ij}^{IJ}. The Lagrangian is then

e_I^i e_J^j Omega_{ij}^{IJ}

(which we integrate against the usual volume form to get the action). In the new variables approach we have a tetrad field again, and we write the curvature of the so(3,C) connection as F_{ij}^{IJ}. (This turns out to be just the "right-handed" part
of Omega_{ij}^{IJ}.) The Lagrangian is

e_I^i e_J^j F_{ij}^{IJ} !

Miraculously, this also gives Einstein's equations.

* * * * * * * * * * * * * * *

In the older Palatini approach to general relativity, the idea was to view general relativity as something like a gauge theory with gauge group given by the Lorentz group, SO(3,1). But to do this one actually uses two different fields: a "frame field", also called a "tetrad", "vierbein" or "soldering form" depending on who you're talking to, and the gauge field itself, usually called a "Lorentz connection" or "SO(3,1) connection". Technically, the frame field is an isomorphism between the tangent bundle of spacetime and some other bundle having a fixed metric of signature +---, usually called the "internal space", and the Lorentz connection is a metric-preserving connection on the internal space.

The "new variables" trick is to use the fact that SO(3,1) has as a double cover the group SL(2,C) of two-by-two complex matrices with determinant one. (The Lie algebra of SL(2,C) is called sl(2,C) and is the same as the complexification of the Lie algebra so(3), which allows one to introduce the new variables in a different but equivalent way) Ignoring topological niceties for now, this lets one reformulate *complex* general relativity (that is, general relativity where the metric can be complex-valued) in terms of a *complex-valued* frame field and an SL(2,C) connection that is just the Lorentz connection in disguise. The latter is called either the "Sen connection", the "Ashtekar connection", or the "chiral spin connection" depending on who you're talking to.
The advantage of this shows up when one tries to canonically quantize the theory in terms of initial data. Here we assume our 4-dimensional spacetime can be split up into "space" and "time", so that space is a 3-dimensional manifold, and we take as our canonically conjugate fields the restriction of the chiral spin connection to space, call it A, and something like the restriction of the complex frame field to a complex frame field E on space. (Restricting the complex frame field to one on space is a bit subtle, especially because one doesn't really want a frame field or "triad field", but really a "densitized cotriad field" - but let's not worry about this here). The point is, first, that the A and E fields are mathematically very analogous to the vector potential and electric field in electromagnetism - or really in SL(2,C) Yang-Mills theory - and secondly, that if you compute their Poisson brackets, you really do see that they're canonically conjugate. Third and best of all, the constraint equations in general relativity can be written down very simply in terms of A and E. Recall that in general relativity, 6 of Einstein's 10 equations act as *constraints* that the metric and its time derivative must satisfy at t = 0 in order to get a solution at later times. In quantum gravity, these constraints are a big technical problem one has to deal with, and the point of Ashtekar's new variables is precisely that the constraints simplify in terms of these variables.

The price one has paid, however, is that one now seems to be talking about *complex-valued* general relativity, which isn't what one had started out being interested in. One needs to get back to reality, as it were - and this is the problem of the so-called "reality conditions". One approach is to write down extra constraints on the E field that say that it comes from a *real* frame field. These are a little messy. Ashtekar, however, has proposed another approach especially suited to the quantum version of the theory, and in his talk he filled in some of the crucial details.
Here, to save time, I will allow myself to become a bit more technical. In the quantum version of the theory one expects the space of wavefunctions to be something like L^2 functions on the space of connections modulo gauge transformations - actually this is the "kinematical state space" one gets before writing the constraints as operators and looking for wavefunctions annihilated by these constraints. The problem had always been that this space of L^2 functions is ill-defined, since there is no "Lebesgue measure" on the space of connections. This problem is addressed (it's premature to say "solved") by developing a theory of generalized measures on the space of connections and proving the existence of a canonical generalized measure that deserves the name "Lebesgue measure" if anything does. One can then define L^2 functions and work with them. In the case of SU(2), Wilson loops act as self-adjoint multiplication operators on the resulting L^2 space. But in quantum gravity we really want to use gauge group SL(2,C), which is not compact, and we want the adjoints of Wilson loop operators to reflect that fact that the SL(2,C) connection A in quantum gravity is really equal to Gamma + iK, where Gamma is the Levi-Civita connection on space, and K is the extrinsic curvature. Both Gamma and K are real in the classical theory, so the adjoint of the quantum version of A should be Gamma - iK, and this should reflect itself in the adjoints of Wilson loop operators.
. The point is that SL(2,C) is the complexification of SU(2), and can also be viewed as the cotangent bundle of SU(2). This allows one to copy a trick people use for the quantum mechanics of a point particle on R^n - a trick called the Bargmann-Segal-Fock representation. Recall that in the ordinary Schrodinger representation of a quantum
particle on R^n, one takes as the space of states L^2(R^n). However, the phase space for a particle in R^n, which is the cotangent bundle of R^n, can be identified with C^n, and in the Bargmann representation one takes as the space of states HL^2(C^n), by which I mean the *holomorphic* functions on C^n that are in L^2 with respect to a Gaussian measure on C^n. In the Bargmann representation for a particle on the line, for example, the creation operator is represented simply as multiplication by the complex coordinate z, while the annihilation operator is d/dz. Similarly, there is an isomorphism between
L^2(SU(2)) and a certain space HL^2(SL(2,C)). Using this, one can obtain an isomorphism between the space of L^2 functions on the space of SU(2) connections modulo gauge transformations, and the space of holomorphic L^2 functions on
the space of SL(2,C) connections modulo gauge transformations. Applying this to the loop representation, Ashtekar has found a very natural way to take into account the fact that the chiral spin connection A is really Gamma + iK, basically analogous to the fact that in the Bargmann multiplication by z is really q + ip.

&*&*&*&*&*&*&*&*&*&*&*&*

Unification Theory Slides

HISTORY OF LOOP QUANTUM GRAVITY by Carlos Rovelei

MAIN STEPS

1986 . . Connection formulation of classical general relativity Sen, Ashtekar.
Loop quantum gravity is based on the formulation of classical general relativity, which goes under the name of “new variables”, or “Ashtekar variables”, or “connectio-dynamics” (in contrast to Wheeler’s “geometro-dynamics”). In this formulation, the
field variable is a self-dual connection, instead of the metric, and the canonical constraints are simpler than in the old metric formulation. The idea of using a self-dual connection as field variable and the simple constraints it yields were discovered by
Amitaba Sen. Abhay Ashtekar realized that in the SU(2) extended phase space a self-dual connection and a densitized triad field form a canonical pair and set up the canonical formalism based on such pair, which is the Ashtekar formalism.
Recent works on the loop representation are not based on the original Sen-Ashtekar connection, but on a real variant of it, whose use has been introduced into Lorentzian general relativity by Barbero.

1986 . . Wilson loop solutions of the hamiltonian constraint Jacobson, Smolin. Soon after the introduction of the classical Ashtekar variables, Ted Jacobson and Lee Smolin realized in that the Wheeler-DeWitt equation reformulated in terms of the new variables admits a simple class of exact solutions: the traces of the holonomies of the Ashtekar connection around smooth non-selfintersecting loops. In other words: the Wilson loops of the Ashtekar connection solve the Wheeler-DeWitt equation if the loops are smooth and non self-intersecting.

1987 . . The Loop Representation Rovelli, Smolin.
The discovery of the Jacobson-Smolin Wilson loop solutions prompted Carlo Rovelli and Lee Smolin to “change basis in the Hilbert space of the theory”, choosing the Wilson loops as the new basis states for quantum gravity. Quantum states can be represented in terms of their expansion on the loop basis, namely as functions on a space of loops. This idea is well known in
the context of canonical lattice Yang-Mills theory and its application to continuous Yang-Mills theory had been explored by Gambini and Trias, who developed a continuous “loop representation” much before the Rovelli-Smolin one. The difficulties of the loop representation in the context of Yang-Mills theory are cured by the diffeomorphism invariance of GR The loop representation was introduced by Rovelli and Smolin as a representation of a classical Poisson algebra of “loop observables”. The relation with the connection representation was originally derived in the form of an integral transform (an in- finite dimensional analog of a Fourier transform) from functionals of the connection to loop functionals.
Several years later, this loop transform was shown to be mathematically rigorously defined. The immediate results of the loop representation were two: the diffeomorphism constraint is completely solved by knot states (loop functionals that depend only on the knotting of the loops), making earlier suggestions by Smolin on the role of knot theory in quantum gravity concrete; and (suitable extensions of) the knot states with support on non-selfintersecting loops were proven to be solutions of all quantum constraints, namely exact physical states of quantum gravity.

1988 - Exact states of quantum gravity Husain, Br¨ugmann, Pullin, Gambini, Kodama. The investigation of exact solutions of the quantum constraint equations, and their relation with knot theory (in particular with the Jones polynomial and other knot invariants) has started soon after the formulation of the theory and continued since

.
1989 - . Model theories Ashtekar, Husain, Loll, Marolf, Rovelli, Samuel, Smolin, Lewandowski, Marolf, Thiemann.
The years immediately following the discovery of the loop formalism were mostly dedicated to understanding
the loop representation by studying it in simpler contexts, such as 2+1 general relativity, Maxwell , linearized gravity , and, much later, 2d Yang-Mills theory

1992. . Classical limit: weaves Ashtekar, Rovelli, Smolin. The first indication that the theory predicts Planck scale discreteness came from studying the states that approximate geometries flat on large scale These states, denoted “weaves”, have a “polymer” like structure at short scale, and can be viewed as a formalization of Wheeler’s “spacetime foam”.

1992 . . C* algebraic framework Ashtekar, Isham. Abhay Ashtekar and Chris Isham showed that the loop transform introduced in gravity by Rovelli and Smolin could be given a rigorous mathematical foundation, and set the basis for a mathematical systematization of the loop ideas, based on C* algebra ideas.

1993 . . Gravitons as embroideries over the weave Iwasaki, Rovelli. In Junichi Iwasaki and Rovelli studied the representation of gravitons in loop quantum gravity. These appear as topological modifications of the fabric of the spacetime weave.

1993 - Alternative versions Di Bartolo, Gambini, Griego, Pullin. Some versions of the loop quantum gravity alternative to the “orthodox” version have been developed. In particular, these authors above have developed the so called “extended” loop representation.

1994 - . Fermions, Morales-Tecotl, Rovelli. Matter coupling were beginning to be explored in detail. Later, matter’s kinematics was studied by Baez and Krasnov, while Thiemann has extended his results on the dynamics to the coupled Einstein Yang-Mills system..

.
1994 . . The dµ0 measure and the scalar product Ashtekar, Lewandowski, Baez. In Ashtekar and Lewandowski set the basis
of the dierential formulation of loop quantum gravity by constructing its two key ingredients: a diffeomorphism invariant measure on the space of (generalized) connections, and the projective family of Hilbert spaces associated to graphs. Using
these techniques, they were able to give a mathematically rigorous construction of the state space of the theory, solving long standing problems deriving from the lack of a basis (the insufficient control on the algebraic identities between loop states). Using this, they defined a consistent scalar product and proved that the quantum operators in the theory were consistent with all identities. John Baez showed how the measure can be used in the context of conventional connections, extended it to the
non-gauge invariant states (allowing the E operator to be defined) and developed the use of the graph techniques. Important contributions to the understanding of the measure were also given by Marolf and Mour˜ao. .

1994 . . Discreteness of area and volume eigenvalues Rovelli, Smolin. In my opinion, the most significative result of loop
quantum gravity is the discovery that certain geometrical quantities, in particular area and volume, are represented by operators that have discrete eigenvalues. This was found by Rovelli and Smolin in, where the first set of these eigenvalues
were computed. Shortly after, this result was confirmed and extended by a number of authors, using very diverse techniques. In particular, Renate Loll used lattice techniques to analyze the volume operator and corrected a numerical error in . Ashtekar and Lewandowski recovered and completed the computation of the spectrum of the area using the connection representation,
and new regularization techniques. Frittelli, Lehner and Rovelli recovered the Ashtekar- Lewandowski terms of the spectrum of the area, using the loop representation. DePietri and Rovelli computed general eigenvalues of the volume.
Complete understanding of the precise relation between dierent versions of the volume operator came from the work of Lewandowski.

1995 . . Spin networks - solution of the overcom-pleteness problem Rovelli, Smolin, Baez. A long standing problem with the loop basis was its overcompleteness. A technical, but crucial step in understanding the theory has been the discovery of the spin-network basis, which solves this overcompleteness. This step was taken by Rovelli and Smolin in and was motivated by the work of Roger Penrose, by analogous bases used in lattice gauge theory and by ideas of Lewandowski. Shortly after, the spin network formalism was cleaned up and clarified by John Baez. After the introduction of the spin network basis, all problems deriving from the incompleteness of the loop basis are trivially solved, and the scalar product could be defined also algebraically

.
1995 . . Lattice Loll, Reisenberger, Gambini, Pullin. Various lattice versions of the theory have appeared..
1995 Algebraic formalism / Dfferential formalism DePietri Rovelli / Ashtekar, Lewandowski, Marolf, Mour˜ao, Thiemann.
The cleaning and definitive setting of the two main versions of the formalisms was completed in for the algebraic formalism (the direct descendent of the old loop representation); and in for the differential formalism (based on the Ashtekar-Isham C* algebraic construction, on the Ashtekar-Lewandowski measure, on Don Marolf’s work on the use of formal group integration for solving theconstraints, and on several mathematical ideas by Jos´e Mour˜ao).

1996. . Equivalence of the algebraic and differential formalisms DePietri. In , Roberto DePietri proved the equivalence
of the two formalisms, using ideas from Thiemann and Lewandowski.

.
1996 . . Hamiltonian constraint Thiemann The first version of the loop hamiltonian constraint is in . The definition of the
constraint has then been studied and modified repeatedly, in a long sequence of works, by Br¨ugmann, Pullin, Blencowe, Borissov and others. An important step was made by Rovelli and Smolin in with the realization that certain regularized loop operators have finite limits on knot states. The search culminated with the work of Thomas Thiemann, who was able to construct a rather welldefined hamiltonian operator whose constraint algebra closes. Variants of this constraint have been suggested in and elsewhere.

1996 . . Real theory: solution of the reality conditions problem Barbero, Thiemann. As often stressed by Karel Kucha.r, implementing the complicated reality condition of the complex connection into the quantum theory was, until 1996, the main open problem in the loop approach.‡ Following the directions advocated by Fernando Barbero, namely to use the real connection in the Lorentzian theory, Thiemann found an elegant elegant way to completely bypass the problem.

1996 . . .Black hole entropy Krasnov, Rovelli. A derivation of the Bekenstein-Hawking formula for the entropy of a black hole from loop quantum gravity was obtained in , on the basis of the ideas of Kirill Krasnov . Recently, Ashtekar, Baez,
Corichi and Krasnov have announced an alternative derivation.

1997 . . Anomalies Lewandowski, Marolf, Pullin, Gambini. ‡“The loop people have a credit card called reality condi-
tions, and whenever they solve a problem, they charge the card, but one day the bill comes and the whole thing breaks
down like a card house” These authors have recently completed an extensive analysis of the issue of the closure of the quantum
constraint algebra and its departures from the corresponding classical Poisson algebra following earlier pioneering work in this direction by Br¨ugmann, Pullin, Borissov and others. This analysis has raised worries that the classical limit of Thiemann’s hamiltonian operator might fail to yield classical general relativity, but the matter is still controversial

.
1997 . . Sum over surfaces Reisenberger Rovelli. A “sum over histories” spacetime formulation of loop quantum gravity was derived in from the canonical theory. The resulting covariant theory turns out to be a sum over topologically inequivalent surfaces, realizing earlier suggestions by Baez , Reisenberger and Iwasaki that a covariant version of loop gravity should look like a theory of surfaces. Baez has studied the general structure of theories defined in this manner. Smolin and Markoupolou have explored the extension of the construction to the Lorentzian case, and the possibility of altering the spin network evolution rules .

SUMMARY OF THE TWO MAIN APPROACHES TO QUANTUM GRAVITY

A spin network applies Feynmans method in a combinatorial way, producing the so called 6j symbols of a tetrahedron, which encodes the interaction of spinors in a gauge invariant (and space independent) way. Unlike Penrose's original formulation, they can have more than 3 edges meeting at a vertex, and the vertices must be labelled by "intertwining operators", or "intertwiners" for short. This is a concept coming from group representation theory; what we've been calling "spins" are really irreducible representations of SU(2).
A 'connection' on the lattice is simply an assignment of an element of a gauge group to each edge of the graph, representing the effect of a parallel transport along the edge. [The holonomy around such a gauge field, is a measure of the field strength, which in turn determines the value of the Feynman path integral]. Each edge of this lattice is assigned a gauge group element that represents the (holonomic) connection and the vertex is also assigned a group element that represents a gauge transformation.
Thus, the confguration variables are now not metrics as in Wheeler's geometrodynamics, but certain spin-connections; the emphasis is shifted from distances and geodesics to holonomies and Wilson loops. [Consider taking a spin-1/2 particle and move it around in a path that traces out a knot. When you do this using the Levi-Civita connection, it comes back "rotated" by some SU(2) matrix. If you take the trace of this matrix (sum of diagonal entries) and divide by two, you get a number between -1 and 1. This number is called a "Wilson loop"]. In its original formulation GR is all about a metric on spacetime, while gauge theories are all about a connection on some bundle over spacetime. While there is a (Leci-Civita) connection involved in GR too, this is traditionally regarded as a subsidiary entity, since it can be computed starting from the metric. To emphasise the gauge theoretic aspect of GR we need to rewrite it so a connection has the starring role and the metric appears as more of a minor character
In ordinary quantum field theory we calculate path integrals using Feynman diagrams. [In Feynman diagrams all vector spaces are representations of some symmetry group G (e.g. Poincare group), also vertices represent morphisms between representations i.e. intertwiners. Penrose took G=SU(2), which has one irreducible representation of each dimension i.e. spin 0 (=1D) spin1/2 (=2D) spin 1 (=3D). [A Feynman diagram is a graph with edges corresponding to "particles" and vertices corresponding to "interactions" where a collection of particles turns into another bunch of particles.]
A finite linear combination of loop states are defined precisely as the spin network states of a lattice YM theory. In LQG the physical space is a quantum superposition of spin networks, which predict a quantisation of area and volume, in the same sense in which, an electromagnetic field is a quantum superposition of an n-photon state where we have quantitative predictions of energy and momentum. Remember, in quantum theory, observables are given by operators on the Hilbert space of states of the physical system in question. You typically get these by "quantizing" the formulas for the corresponding classical observables. So we take the usual formula for the area of a surface in a 3-dimensional manifold with a Riemannian metric and quantized it. Applying this operator to a spin network state, they found the picture just described: the area of a surface is a sum of terms proportional to sqrt(j(j+1)), one for each spin network edge poking through it. Also just as the spins on edges contribute *area* to surfaces they pierce, the intertwiners at vertices contribute *volume* to regions containing them!
Hence in QLG we employ the holonomic (gauge) technique of (Wilson) loop theory and the combinotorial technique of (Penrose) spin networks (which guarantees a relational aspect to space and time that is inherent in GR due to its diffeomorphic invariance) In a 3 particle (spinor) interaction there is an intertwiner between j1, j2 and j3 and we can normalise this invariant tensor, producing what is known as the 3j Wigner symbols. Now when considering more spinor interactions we can spilt these into other trivalent interactions which then defines the Wigner 6j symbols (formed by contracting four 3j symbols). These can then be topologically identified by the sides of a tetrahedron (a 3-symplex), whose invariant properties form the basis for a spin network.[There is one 3j-symbol for each vertex of the tetrahedron and one representation for each edge].
Historically Regge calculus was the first attempt to quantize Riemannian gravity, producing a lattice formulation of GR by dividing up space into small (flat) 3D tetrahedral simplexes, in which the curvature is concentrated along their boundaries. In effect he applied spin networks before Penrose invented them, by analysing space in terms of tetrahedron, in which the amplitude for tetrahedra are given by 6j symbols. Once the space has been chopped up into such tetrahedra, we can find the amplitude for space-time to have a given shape by taking the products of the 6j symbols over all the tetrahedra and multiplying this by a suitable factor for each tetrahedra.
From this Hilbert space of quantum tetrahedron, it was intended to produce 'Feynman propagators' for gravity and recover Einstein's field equation in the macroscopic domain of space-time. Each of the edges is associated a spin j and in one type of approach, the exponential of the action in such a configuration, is a suitable product of the the 6j symbols associated to each of the 6 edges of the 3-simplexes and a partition function is obtained by taking the sum of this products over the possible associations of the spin to the edges. Remember that angular momentum is a quantum (bi)vector and is therefore subject to Heisenberg' Uncertainty Principle. [In considering a Lorentzian (as opposed to a 3-D) spin network we employ a 4-simplex in which there are 4 tetrahedra, 5 vertices, 10 triangles and 10J symbols].
In retrospect the very first spin foam model was the Ponzano--Regge model in 3-D Riemannian quantum gravity, in which we triangulate a given 3-manifold and expresses the partition function . . .INTEGRAL Exp iS . . .as a sum over spin foams lying in the dual 2-skelaton of the triangulation. Subsequent improvements were made by invoking so called quantum groups SUq(2), which are not actually groups but a structure whose representations are j=0, 1/2, . . .k/2 where q=2pi/(k+2) - - - in other words we do not actually need to know the group. [Basically, the partition function will not converge unless we replace SU(2) by the quantum group]
A strict connection exists between quantum tetrahedra and 4-valent vertices of SU(2) spin networks. The relationship of spin networks to Regge-Ponzano method is thus evident, by which we have a combinotorial approach of describing space in terms of tetrahedra which are quantised, by identifying each face with an angular momentum operator. The 4 faces of a tetrahedron are associated with 4 irreducible representations of SU(2), which are represented by a perpendicular line, the 4 of which meet at a central node of the tetrahedron -- the Poincare dual skeleton. [There are actually bivectors associated with each face, in keeping with constraints of GR and quantizing the bivectors/tetrahedra amounts to labelling each face with a pair of spinors.]
We therefore obtain a 4-valent (colour coded) spin network (each line of which represents a quantized unit of action, while the nodes behave as area operators), which can exhibit properties that are gauge invariant. The quantum bivectors allow us to construct area and volume operators which act upon the spin network basis to produce a discrete spectrum which has units of Planck length squared and cubed respectively (in the case of the area operator, the eigenvalues are 1/2sqrt (j(j+1) - - spin network edges are quantized flux tubes of area). Here E( ) measures the area of the surface , area is quantized, and spin network edges give area to the surfaces they intersect! [Since the E operator vanishes unless it grasps an edge, it only acts where spin networks intersects the surface -hence the area eigenvalue depends on the number of times the spin network intersects the surface].This is particularly intriguing given the Bekenstein-Hawking formula saying that the entropy of a black hole is proportional to its area. In keeping with quantum theory we would expect these to correspond to physical observable i.e. we have a quantized space! .
Einstein had originally experimented with the connection approach (so called distant parallelism) in his attempt at a unified theory of electromagnetism and gravity. However, now these were 'spin-connections', required to parallel propagate spinors, and they turn out to simplify Einstein's equations considerably. For example, with the dynamical evolution dictated by Einstein's equations can now be visualized simply as a geodesic motion on the space of spin-connections (with respect to a natural metric extracted from the constraint equations). Since general relativity is now regarded as a dynamical theory of connections, this reincarnation of the canonical approach is called 'connection- dynamics'. We quantise gravity by means of a connection A and its canonical momentum conjugate E, with which we create spin networks for spinor states and from this we can produce area operators which act on space to quantise its geometry!.
Several Lagrangian actions have heuristically been considered (e.g. Chern theory, Yang-Mills, BF theory and the Chern-Simon action), in an attempt to find a connection which is compatible with GR. Ashtekar reformulated Einstein's equation so that the configuration space for general relativity on the spacetime R x S, instead of being the space of *metrics* on a 3-manifold S, is a space of *connections* on S. [A connection is just what a physicist often calls a vector potential, but for any old gauge theory, not just electromagnetism] The gauge group of Ashtekar's version of general relativity is SL(2,C), the group of 2x2 complex matrices with determinant equal to 1.
Another approach to formulating a quntum version of GR is the Chern-Simons state, which is a solution of quantum gravity with cosmological constant. The Chern-Simons action was hand-tailored to have the gauge-invariance and diffeomorphism-invariance we want, so the only big surprise is that we also have a solution of the Wheeler-DeWitt equation.
Now there's another formulation of general relativity in terms of an action principle -- called the "Palatini" action .The Palatini approach turns out to be more elegant and is a nice stepping-stone to the Ashtekar approach. In the Palatini approach one thinks of general relativity not as being a theory of a metric, but of a "tetrad" and an "so(3,1) connection". Charmingly, even though the tetrad field is utterly unphysical, we can treat it and the so(1,3) connection as independent fields and, doing calculus of variations to find stationary points of the action, we get equations equivalent to Einstein's equations.
In the older Palatini approach to general relativity, the idea was to view general relativity as something like a gauge theory with gauge group given by the Lorentz group, SO(3,1). But to do this one actually uses two different fields: a "frame field", also called a "tetrad", "vierbein" or "soldering form" depending on who you're talking to, and the gauge field itself, usually called a "Lorentz connection" or "SO(3,1) connection"
The "new variables" trick is to use the fact that SO(3,1) has as a double cover the group SL(2,C) of two-by-two complex matrices with determinant one. (The Lie algebra of SL(2,C) is called sl(2,C) and is the same as the complexification of the Lie algebra so(3), which allows one to introduce the new variables in a different but equivalent way.) Ignoring topological niceties for now, this lets one reformulate *complex* general relativity (that is, general relativity where the metric can be complex-valued) in terms of a *complex-valued* frame field and an SL(2,C) connection that is just the Lorentz connection in disguise (TheAshtekar-Sen connection). The advantage of this shows up when one tries to canonically quantize the theory in terms of initial data. Here we assume our 4-dimensional spacetime can be split up into "space" and "time", so that space is a 3-dimensional manifold, and we take as our canonically conjugate fields the restriction of the chiral spin connection to space, call it A, and something like the restriction of the complex frame field to a complex frame field E on space. (Restricting the complex frame field to one on space is a bit subtle, especially because one doesn't really want a frame field or "triad field", but really a "densitized cotriad field") The point is, first, that the A and E fields are mathematically very analogous to the vector potential and electric field in electromagnetism - or really in SL(2,C) Yang-Mills theory - and secondly, that if you compute their Poisson brackets, you really do see that they're canonically conjugate. Third and best of all, the constraint equations in general relativity can be written down very simply in terms of A and E. Recall that in general relativity, 6 of Einstein's 10 equations act as constraints that the metric and its time derivative must satisfy at t = 0 in order to get a solution at later times.
Hence we quantise gravity by means of a connection A and its canonical momentum conjugate E, with which we create spin networks for spinor states and from this we can produce area operators which act on space to quantise its geometry!. Such approaches are encouraging in that they agree (when one introduces the `Barbero-Immirzi parameter'), with the Bekenstein-Hawking black hole entropy formula.however, as yet they are unable to produce realistic vacuum states.

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In the alternative approach, M theory is a recent revolution in our understanding of sting theory*! and has demonstrated a symmetry/duality between the 5 competing formulations, namely Type I, IIa, IIb, heterotic-O and heterotic-E as well as supergravity. This is achieved by increasing the number of dimensions to 11 and which also allows for the existence of higher dimensional 'branes' as well as the more usual strings, which represent our universe. Under this scheme it is found that the competing string theories can paired off under what is known as T-duality. For example when the extra dimension is curled up into a circle, M-theory yields the type IIA string, which when observed at low energy, can behave like type IIB theory under high energy. Similarly, if the extra dimension shrinks to a line segment, M-theory reduces to a heteroic-E string, which is conected to the heterotic-O string by dualities! [T- duality can itself also be shown to be a duality of S-duality which relates a gauge charge e, such as that of an electron with its topological soliton counterpart with charge1/e --- the magnetic monopole!]. Here again it becomes important to study the symmetries (and hence topologies) of the types of p-dimensional branes that can exist in such an exotic manifold.
Some of the most recent discoveries include the fact that the weakness of gravity can be shown to be a result of it leaking from a parallel membrane into our own (it is consequently diluted by the extra dimensions that we do not directly observe). Also it suggests that the big bang can be formulated in terms of an initial violent collision between 2 such membranes. (The Ekpyrotic theory)
Now General Relativity also exhibits a diffeomorphic symmetry (reparameterisation invariance), which does not exist in Special Relativity. A natural generalization of the work of Kaluza and Klein, shows that diffeomorphism symmetry of a higher than 4 dimensional space-time manifold, can give rise to the gauge symmetry of a Yang-Mills theory. [This effect can also be obtained from special diffeomorphic symmetries associated with the 2 dimensional string world sheet. Also there are some special cases of diffeomorphic symmetry, which are transformed into Poincare symmetries.] So by using the Kaluza-Klein idea we may 'derive' gauge symmetries from the geometrical symmetries of a higher dimensional space-time. Part of the diffeomorphism symmetry of gravity in higher dimensions becomes a gauge symmetry in four dimensions. In some sense, this just means that one assumed symmetry is interpreted as transformed into another one. Five-dimensional anti-de Sitter space (AdS), has a boundary which is four-dimensional, and in a certain limit looks like flat space-time with one time and three space directions. The AdS/CFT correspondence ( known as the Maldacina conjecture), states that the physics of gravity in this five dimensional anti-de Sitter space, is equivalent to a certain supersymmetric Yang-Mills theory (a Conformal Field Theory without gravity), which is defined on the boundary of AdS. [CFT relates to Weyl group, a gauge invariance which is an angle preserving transformation which is also an inherent symmetry in string theory.] The AdS/CFT correspondence is a type of duality, which states that two apparently distinct physical theories are actually equivalent. On one side of this duality is the physics of gravity in a spacetime known as anti-de Sitter space (AdS). while on the other is a supersymmetric Yang-Mills gauge theory that lives on the boundary of the AdS.
This Yang-Mills theory is thus a `hologram' of the physics which is happening in five dimensions. The Yang-Mills theory has gauge group SU(N), where N is very large, and it is said to be `supersymmetric' because it has a symmetry which allows you to exchange bosons and fermions. In recent studies of M-Theory, AdS black hole solutions of gauged extended supergravities can be considered as rotating, higher dimensional membranes. We therefore embed these known lower-dimensional black hole solutions into ten or eleven dimensions, thus allowing a higher dimensional interpretation in terms of rotating M2, D3 and M5-branes.