Loop Quantum Gravity

Study matter and its motion through spacetime...

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Re: Loop Quantum Gravity

#61  Postby campermon » Aug 20, 2010 11:09 pm

Well done Joe!!

:beer:
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Re: Loop Quantum Gravity

#62  Postby twistor59 » Aug 21, 2010 8:07 pm

Post 6 - Spin Networks - Well almost !

The stuff I've described so far (Ashtekar Connection/Densitized Triad on 3 space, Hamiltonian Constraint, Diffeomorphism Constraint and Gauss Constraint) is all machinery for recasting classical (non quantum) GR as a gauge theory. Now we want to quantize it.

The thing that makes the LQG approach to quantization a bit different is that we're no longer allowed to assume the existence of a background spacetime metric anywhere along the way. The metric is part of the dynamical system we're trying to model. In discussions on LQG you will see lots of mention of "background independence". This is what is meant.

We have our space of connections and to quantize, we somehow need to get a Hilbert space of functionals out of it. (A functional is "just" a function defined on a space of functions, which of course will usually be an infinite dimensional space. In other words a functional maps a function to a number). The space of all connections is just too "big", so what we do is the following:

Define a graph Γ to be a join-the-dots picture on the three-space Σ, with a bunch of links which meet at nodes.

If there are L links, then if we have a function f: SU(2)L->ℂ

(SU(2)L just means L copies of SU(2))

we can use the pair (Γ, f) to define a functional ψ(Γ, f) on the space of connections. It maps the connection A to the complex number

ψ(Γ, f)[A] = f(he1[A], he2[A],.....,heL[A])

where hei[A] is the element of SU(2) given by the holonomy of the connection A along the link ei i=1..L

Now to build a Hilbert space we need to get ourselves a scalar product. If I have another function g: SU(2)L->ℂ then I can define the scalar product of the two functionals to be

(Γ, f) | ψ(Γ, g)> = ∫f(he1[[A], he2[A],.....,heL[A])*g(he1[[A], he2[A],.....,heL[A]) dhe1dhe2....dheL

* denotes complex conjugate, and dhe is the Haar measure on SU(2).

The cool thing is that the scalar product defined this way is finite, since SU(2) is a compact group. With this scalar product, we have a Hilbert space 𝓗Γ associated with the graph Γ. Define the Hilbert space

𝓗kin = direct sum over all graphs Γ in Σ of 𝓗Γ

OK so we've got a Hilbert space. What next ? Observables are represented as operators on the Hilbert space, and Poisson brackets become commutators. In elementary QM we have position and momentum, q and p. The analogs here are A, E (connection and densitized triad - not writing the indices out). How do A and E act on the Hilbert space ?

To define this, we start by introducing a basis of the Hilbert space. To do this, we use the Peter-Weyl theorem, which states that a basis of functions on a compact group (like SU(2)) is given by the matrix elements of the unitary irreducible representation of the group. See appendix below for a short discussion of reps Djmn of SU(2).

So we can expand a generic function f on SU(2) as

f(g) = Σ fjmnDjmn(g)

This is extendable in an obvious way to our functionals ψ(Γ, f) (too many indices to write out here though).

Using this basis for functionals on 𝓗kin we can define a representation of the Holonomy/Flux algebra that I introduced at the end of post 5. What we want is something like

Aia ψ[A] = Aia ψ[A]

Eai ψ[A] = (𝛿/𝛿Aia) ψ[A]

where the bold entries are the operators which operate on the wave functionals ψ[A] and 𝛿/𝛿 denotes functional differentiation.

Now that we know that we can represent our wave functionals as expansions of matrix elements of group elements (the D's), this becomes straightforward. For example, if h𝛾[A] is the operator representing the holonomy along a curve 𝛾, then its action on a functional is got just by modifying the functional by multiplying each basis element by its value ( an SU(2) element). So for example if Djmn(he[A]) appears in the expansion of the functional, we replace it by Djmn(h𝛾[A]he[A])

The action of the flux can be defined too, but is a bit messier as it involves the functional derivative, so I won't attempt to type it in.

We have a Hilbert space and a representation of the operators. To get to the Hilbert space of physical states we need to impose the constraints. This is done in 3 stages, one for each constraint

𝓗kin ->GaussConstraint->𝓗0kin ->DiffeomorphismConstraint->𝓗Diff -> HamiltonianConstraint->𝓗phys

Next post I'll talk a bit about the implementation of the Gauss constraint. States in the resulting Hilbert space are called spin network states.


Appendix - Irreducible reps of SU(2)

Denote by σ1, σ2, σ3 the Pauli spin matrices (familiar to everybody from angular momentum bollox in elementary QM). Remember that you can find wavefunctions which are simultaneous eigenfunctions of the total angular momementum operator L2 and Lz ? In group theoretic terms this is because σ2 = (σ1)2+(σ2)2+(σ3)2 is a Casimir operator.

So, we can find vectors which are simultaneously eigenvectors of σ2 and (say) σ3. Traditionally denote them by

|jm> where j=0,1/2, 1, 3/2.... and m=-j, -j+1,....-1,0,1, .....j-1, j

j labels the total angular momentum and m the z component of the angular momentum.

A generic rotation on 3 space can be written as

R(α,β,γ) = e-iασ1e-iασ2e-iασ3

where α,β,γ are the Euler angles that parametrize the rotation.

(What's happening here is that the Pauli spin matrices are elements of the Lie algebra of SU(2), and to generate elements of the Lie group, you need to exponentiate).

The irreducible representations of SU(2) are given by the matrices

D[j]mn(g) = <jm|R(α,β,γ)|jn>

where the group element g is parametrised by the Euler angles α,β,γ.
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Re: Loop Quantum Gravity

#63  Postby newolder » Aug 21, 2010 10:37 pm

... and this beautiful theory predicts what, for the rest mass of an electron? Is it a fractional dimension theory? :ask:
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Re: Loop Quantum Gravity

#64  Postby twistor59 » Aug 22, 2010 6:45 am

newolder wrote:... and this beautiful theory predicts what, for the rest mass of an electron? Is it a fractional dimension theory? :ask:


Sheesh ! What do you want ? Blood ?

LOL ! no, it doesn't make any predictions about the elementary particle masses. It's not, and doesn't claim to be, a theory of everything. It's purely a way of quantizing the gravitational field at this stage. I think there has been some preliminary thought on how to put "matter" into the theory, but I haven't got to that bit yet. After I get to the end of describing the formalism, I'll summarise what I understand to be the main results so far.

And no, not a fractional dimension theory.
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Re: Loop Quantum Gravity

#65  Postby newolder » Aug 22, 2010 9:18 am

:thumbup: it's still interesting stuff. :)
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Re: Loop Quantum Gravity

#66  Postby klazmon » Aug 22, 2010 11:06 am

twistor59 wrote:
newolder wrote:... and this beautiful theory predicts what, for the rest mass of an electron? Is it a fractional dimension theory? :ask:


Sheesh ! What do you want ? Blood ?

LOL ! no, it doesn't make any predictions about the elementary particle masses. It's not, and doesn't claim to be, a theory of everything. It's purely a way of quantizing the gravitational field at this stage. I think there has been some preliminary thought on how to put "matter" into the theory, but I haven't got to that bit yet. After I get to the end of describing the formalism, I'll summarise what I understand to be the main results so far.

And no, not a fractional dimension theory.


I will be interested to see if you can recover GR at the end as a classical limit. ;)
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Re: Loop Quantum Gravity

#67  Postby twistor59 » Aug 22, 2010 11:45 am

klazmon wrote:
twistor59 wrote:
newolder wrote:... and this beautiful theory predicts what, for the rest mass of an electron? Is it a fractional dimension theory? :ask:


Sheesh ! What do you want ? Blood ?

LOL ! no, it doesn't make any predictions about the elementary particle masses. It's not, and doesn't claim to be, a theory of everything. It's purely a way of quantizing the gravitational field at this stage. I think there has been some preliminary thought on how to put "matter" into the theory, but I haven't got to that bit yet. After I get to the end of describing the formalism, I'll summarise what I understand to be the main results so far.

And no, not a fractional dimension theory.


I will be interested to see if you can recover GR at the end as a classical limit. ;)


Good point. Me too !

I haven't read those bits in detail yet, but I think the way it's approached is to try to construct, in the LQG formalism, a coherent state and extract the classical picture from that. I'm not sure if that work has been completed yet though...
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Re: Loop Quantum Gravity

#68  Postby twistor59 » Aug 24, 2010 12:14 pm

Post7: Spin Networks and Geometric Operators

We have a graph with some nodes and L links joining some of them (for the moment consider this to be embedded in three space. It will eventually become an abstract graph). If we have a function f mapping L copies of SU(2) to the complex numbers then the pair ( Γ, f) defines a functional on the space of connections. To use it to get a complex number from the connection A, just create L SU(2) elements by taking the holonomies of A along each of the L edges, then applying the function f. Given another function f' like f, we showed how to define a scalar product between f and f'.

We defined a basis for functions on SU(2) by expanding in terms of the irreps of SU(2). We used this to define a basis for the functions on SU(2)L.

The next job is to impose the Gauss constraint, i.e. chuck away the states in 𝓗kin that aren't SU(2) gauge invariant. Somewhere a few posts back I said that under gauge transformations, a holonomy he along an link e gets mapped to

gstartheg-1end

where gstart is the group element at the node at the start of the link e, and g-1end is the (inverse of) the group element at node at the termination of link e. Now since typically several links connect to any given node, there are several simultaneous gauge group elements to take into account at that node. We want the overall effect to be such that our functional ψ(Γ, f) is invariant under this gauge transformation.

What we need is some way to connect up, at each node, the collection of SU(2) holonomies consistently into a complex number that's invariant under the SU(2) transformations. The object that enables you to do this is called an "intertwiner", or Clebsch-Gordan coefficient. We thus require an intertwiner at each node.

There is a nice illustration of the operation of the intertwiner given in appendix A1 of Carlo Rovelli's book (a preprint of the book used to be available online, but I can't find this anymore):

SpinNetwork2.jpg
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Suppose we have 3 links meeting at a node, and the links are assigned spin 5/2, 5/2 and 2 representations. Then we can think of a link with spin 5/2 as bringing 5 fundamental representations (each spin 1/2) to the node - this can be represented by 5 lines coming in. Similarly for the other two links. The intertwiner at the node connects up these fundamentals with each other as shown. The "connecting up" is basically contraction of spinor indices - the intertwiner is an object with indices which are contracted with various indices of the "incoming" spinors at the node.

By taking states corresponding to a graph Γ, an assignment of an SU(2) representation jl to each link l, and an intertwiner in to each node n, we have succeeded in imposing the Gauss constraint. The graph with its set of representations assigned to the links and intertwiners assigned to the nodes is called a spin network.

Spin networks were thought up, essentially out of the blue, by Roger Penrose a long time ago (see this 1971 link ). Penrose suspected at the time that spacetime at a fundamental level was combinatorial in character. LQG seems to have given new life to that idea.

The Hilbert space of functionals 𝓗0kin is the direct sum over all graphs Γ of the Hilbert spaces H0Γ of functionals defined by a given graph Γ. Moreover H0Γ itself decomposes as the direct sum over the intertwiner spaces. (This is the analog of the Fock space decomposition of the free field Hilbert space in standard QFT).

Observables

For a given spin network state , an area operator can be defined (in terms of the flux operator which I mentioned – but did not write down – in post 6). For a given two-surface S it has eigenvalues which satisfy

A(S) ψΓ = Σ hSQRT( γ2 jp(jp+1) ) ψΓ

where the sum is over all points p where the links of the spin network intersect the surface S, and jp is the . (There are some technicalities, but this is the general idea, illustrated in the diagram below). Note that spin network states are eigenstates of this operator.

AreaOperator.jpg
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Analogously, there is a volume operator. Calculation of its eigenvalues is a bit more complicated, but basically if you have a 3 dimensional sub-region R of three space, the volume of R receives a contribution from each node of the spin network that lies inside R. See diagram:

VolumeOperator.jpg
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Re: Loop Quantum Gravity

#69  Postby twistor59 » Aug 27, 2010 7:59 am

Post8 – Diffeomorphism Invariance
Remember our plan for step by step imposition of the constraints, which will eventually lead us to the Hilbert space of physical states:

𝓗kin ->GaussConstraint->𝓗0kin ->DiffeomorphismConstraint->𝓗Diff -> HamiltonianConstraint->𝓗phys

In the last post, we imposed the Gauss constraint (i.e. SU(2) gauge invariance) which led us to 𝓗0kin, the set of spin-network states. One issue with 𝓗0kin is that it’s non separable. The graphs Γ we have been talking about are embedded in three-space. The non separability of 𝓗0kin can be traced back to the fact that it contains superpositions of states with different graphs. If I distort the embedding of the graph a tiny bit, I get a different graph, and hence generate different states from it.

Post8_1.jpg
Post8_1.jpg (12.06 KiB) Viewed 2823 times


The solution to this is the imposition of the diffeomorphism constraint. Remember, a diffeomorphism is just a smooth map from the manifold into itself

Post8_2.jpg
Post8_2.jpg (44.52 KiB) Viewed 2823 times


Imposition of the diffeomorphism constraint means that we end up with spin networks defined, not upon graphs Γ, but upon equivalence classes of graphs Γ under diffeomorphisms.

So I effectively regard two graphs as being the same if I can distort one into another by a diffeomorphism. For example, a circular loop is the same as a distorted circular loop, but not the same as if the loop has a knot in it.

Post8_3.jpg
Post8_3.jpg (15.96 KiB) Viewed 2823 times



So our Hilbert space 𝓗Diff is spanned by the knotted spin networks. Note that by regarding things as identical under diffeomorphisms, we are moving away from regarding the spin networks as embedded in the three surface, and more towards regarding them as topological entities in their own right.

The final step in getting to the space of physical states is obtained by imposing the Hamiltonian constraint, which will lead us onto the idea of spin foams.
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Re: Loop Quantum Gravity

#70  Postby twistor59 » Aug 30, 2010 5:18 pm

Post 9: Dynamics

The last step in the quantization programme

𝓗Diff -> HamiltonianConstraint->𝓗phys

is the application of the Hamiltonian constraint. In my, admittedly not worth too much, opinion, this step does not go quite as well as the others. It is done by first writing the (classical) Hamiltonian in terms of the Ashtekar variables (connection and densitized triad). The expression is quite messy and looks like it will give some difficulty when replacing the expressions with operators.

This was done by Thiemann who re-expresses the nasty expression in terms of the volume of Σ and some Poisson brackets. This is then re-written in terms of the holonomy and flux operators. The operation of the Hamiltonian ends up looking like adding, to a given vertex, two new vertices and a new link.

NicolaiFig5.jpg
NicolaiFig5.jpg (8.61 KiB) Viewed 2810 times


I haven't tried to closely follow the construction of the Hamiltonian this way, but:

1 Its action is known and finite, but
2 It adds new links to the spin network. The spins of the added links is arbitrary
3 There is some arbitrariness in the order in which things are done.
4 It's not clear how to apply it to a complete spin network

These ambiguities 2-4 dont look too great, so the quantization procedure cannot be considered to be completed yet. It is still under investigation.


Spin Foams


One approach (which has been around for 10 years or so) to get past these (considerable) difficulties with the Hamiltonian in LQG is the development of the theory of spin foams.

If you think of a spin network moving forward in time, a link sweeps out a surface, and a node sweeps out a line (echoes of strings->branes anyone ?). As we said above, the Hamiltonian generates new nodes and links, so in the spacetime picture, a new surface gets added

NicolaiFig7.jpg
NicolaiFig7.jpg (20.47 KiB) Viewed 2810 times


So you can see the motivation for studing the higher dimensional variant of spin networks. A spin foam can be thought of as a path in a Feynman path integral sum to detemine the amplitude for transition between an initial spin network state at time T1 and final spin network state at time T2. Just as in the derivation of the path integral, we can imagine a succession of timeslices and the Hamiltonian to operate on each one, generating a succession of intermediate states.

Although they were originally thought of as time evolutions of spin networks, spinfoams eventually came to be thought of as an independent approach in their own right, and became generalisations of spin networks in the sense that the "group theory stuff" (spins, representations, intertwiners) now gets attached to higher dimensional components (faces etc) instead of just to nodes and edges.

In this approach, the spin foams built with representations of either SO(3,1) (or equivalently SL(2,ℂ)) or SO(4) attached to their faces. In the SO(4) = SU(2)xSU(2) case (tensor product), a subset of representations is used, namely (j, j) j=1/2, 1, 3/2.... This leads to the "Barrett-Crane" spinfoam models.

More recently, a new class of spinfoam models, known as EPRL models (Engle, Livine, Pereira, Rovelli) have been introduced.



SL(2, ℂ) Spin Networks


In the previous set of posts, I outlined the "derivation" of the spin network Hilbert space kinematic formalism from the canonical quantization of gravity (using an ADM type formalism and the Ashetkar variables etc).

There is an alternative derivation, which leads to a slightly different picture which will be quite useful when discussing the dynamics. The derivation in question starts with a discrete GR on a 4d lattice (hence Rovelli refers to it as "covariant lattice quantization"). Look at the boundary of the lattice. This is a 3d manifold, and each tetrahedron in the boundary bounds a single simplex of the 4 manifold and each boundary triangle bounds just two tetrahedra.

Let Γ be the dual of the boundary triangulation. This is a 4-valent graph with nodes dual to the boundary tetrahedra and links dual to the boundary triangles. (See here). The gauge group is SL(2, ℂ) and the Hilbert space is

𝓗ΓSL(2, ℂ) = L2[SL(2, ℂ) L / SL(2, ℂ) N]

(there is something else called the "simplicity constraint" which comes into the definition of this Hilbert space, which I haven't really understood). The simplicity constraints mean that the classical theory corresponds to GR, not just any old url=http://en.wikipedia.org/wiki/BF_model]BF[/url] theory.

Now, there is a map from the old canonical LQG-style Hilbert space 𝓗Γ into 𝓗ΓSL(2, ℂ). To define it, use the Peter-Weyl decomposition (like we did a few posts back)

L2[SL(2, ℂ) L ] = sum over (pl, kl) of tensor product over links l of (𝓗*(pl,kl) tensor 𝓗(pl,kl) )

where SL(2, ℂ) unitary irreps are labelled by (p in ℝ, k in ℤ+).

Now if we pick an SU2 subgroup of SL(2, ℂ), then each SL(2, ℂ) irrep (p, k) decomposes as a sum of SU2 irreps. The sum is over the spins from k up to infinity. Neglecting everything except the lowest (k) spin rep in the sum allows us to define a map

Yγ : 𝓗Γ -> 𝓗ΓSL(2, ℂ)

SU2 spin-j things get mapped to (p=γj, k=k) SL(2, ℂ) things.

Apparently (!) this map respects the simplicity constraints, which means that the quantum theory is still GR.

If we compose this map Yγ with a projection onto the SL(2, ℂ) invariant states (i.e. we do the Gauss constraint for SL(2, ℂ) ), we get a very important map

fγ : 𝓗Γ -> 𝓗ΓSL(2, ℂ)

from SU(2) spin networks to SL(2, ℂ) spin networks.
NicolaiFig7.jpg
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This is used in the "new dynamics" (next post).
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Re: Loop Quantum Gravity

#71  Postby twistor59 » Aug 30, 2010 5:25 pm

Post 9a The New Dynamics

There is a large volume of work on LQG now following a new line of approach. It goes roughly like this:

"We've come a long way down the road of quantizing the classical picture of gravity. It has given us a very promising model of the quantum three geometry, but as yet there is no unique approach to the dynamics. Instead, we will take as a starting point the spin-network Hilbert space picture, and try to construct a "natural" dynamics on it and see where it leads us. The hope is that it will give something that has classical GR as its low energy limit".

So we now forget about the fact that the spin network graphs were once thought of as embedded in a three space Σ. They're now abstract graphs, describing purely combinatorial relations (incidently this is exactly what Penrose was thinking in his original invention of spin networks). We still have the mechanisms for constructing the Hilbert space from them - this only makes reference to the spin network, not to its embedding. We can still construct the area and volume operators on them.

The nodes represent "grains" of space. Nodes connected by a link represent grains which are adjacent. The area of the boundary surface between adjacent grains is determined by the spin on the link.

Elements of the Hilbert space are of course superpositions of basis states which look nothing like classical spacetime. Two grains of space in an element of the Hilbert space will in general be in a superposition of being adjacent and not adjacent ! To select states which resemble classical three geometries, use is made of the idea of coherent states.

Mechanisms have been given for constructing such states, which are in some sense "peaked" around a discretized classical 3 geometry. Discrete versions of differential geometry have been around for some time.



In conventional quantum theory, dynamics represents an evolution from a state on a spacelike three surface at time t1 to a state on a spacelike three surface at time t2. There is a complex number (an "amplitude") which, when squared, represents the probability of this evolution. This amplitude may be computed by a Feynman path integral.

In the background-independent quantum gravity that we're trying to formulate, there isn't really such a time evolution. The "time evolution" moves points around the three space along integral curves of the Hamiltonian flow. It's just another constraint. So what do we do ?

The answer is that we define an amplitude for each state on Σ where Σ is a boundary to some region R of spacetime. This amplitude map

H->ℂ

is the (quantum) dynamics. This seems to be the appropriate generalisation of the conventional picture, where the amplitude is

<φ|ψ>

and |φ> is the initial state on Σ1 and |ψ> is the final state on Σ2. Here the boundary consists of two disconnected components and things are assumed to die away at spatial infinity. Obviously we can't make assumptions like this in the background independent case, because since the whole thing is diffeomorphism invariant, we don't have a notion of spatial infinity any more. If you need your intuition on this renormalizing (as I did), I recommend this discussion.

So the dynamics will be defined by a linear functional W on H, the Hilbert space of states| ψ> on the boundary Σ of the spacetime region R. The associated probability is

P(ψ) = |<W|ψ>|2

In Rovelli's words, we will attempt to describe this (motivated by the Feynman path integral approach) as a sum over "histories of states"

|<W|ψ>| = Sum over σ W(σ)

where σ is a sequence of states (think spinfoam) with ψ as boundary state, and W(σ) is an amplitude for a given state. The task is therefore to build the map W.

The answer turns out to be the spinfoam sum:

<W|ψ> = Σ Π d(jf) Π Wv(σ)

The sum is over spinfoams σ which have ψ as boundary
The first product is over the faces f of the spinfoam
The second product is over the vertices of the spinfoam
d(jf) is the dimension of the representation assigned to face f

Wv(σ) = <Wv| ψv>

where ψv is a natural local spin network surrounding the vertex v (how to construct this local spin network is discussed here).

Finally, <Wv| ψv> = (fγ ψv) (𝟙)

𝟙 just denotes the identity SL(2, ℂ) element on the rep attached to each of the edges of the vertex v.

fγ is the function that I introduced at the end of the last post.

Next post I'll try to summarise in English rather than mathematics some of this wibble that I've been accumulating.
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Re: Loop Quantum Gravity

#72  Postby twistor59 » Sep 06, 2010 7:54 am

Post 10 Summary
I thought I’d try to summarise what I’ve gathered about this subject so far:

There are well known methods for quantizing classical field theories. The archetypical example is the quantization of classical electromagnetism, giving rise to quantum electrodynamics. Applying these methods to the gravitational field produces a theory which is non-renormalizable. (M.H. Goroff and A. Sagnotti, Nucl.Phys.B266, 709 (1986)). This means that the usual process you apply (separating integrals into divergent and convergent parts) doesn’t work for gravity. Loop quantum gravity is one line of approach to try to produce a model which doesn’t have these problems.

The breakthrough that allowed LQG to start was Abhay Ashetkar’s reformulation of general relativity in a new set of variables. These variables are defined for the case where we “chop up” spacetime into a bunch of three-space slices. These variables allow GR to be formulated as a gauge theory. There are three sets of constraints which the theory must implement:

“Gauss Constraint” – to ensure it’s invariant under local Lorentz transformations
“Diffeomorphism Constraint” – to ensure it’s invariant under arbitrary coordinate changes on the three-space slices
“Hamiltonian Constraint” – to implement the time evolution
Applying the rules of quantization for constrained systems to the Ashtekar variables did not produce a theory for which it was possible to extract a satisfactory classical limit.

This problem was addressed by using holonomies as the basic variable to be quantized. The holonomy (in this context) is the map from the gauge group into itself generated by parallel transport along a given curve. The conjugate variable to the holonomy is a “flux”.

The Hilbert space defined this way would be astronomically huge (even for an infinite dimensional space, there’s infinite and there’s INFINITE !), so we restrict the Hilbert space to the holonomies generated along the links of a graph with nodes and edges. Think of the graph as defining a sort of “skeleton” in the manifold along which we “feel out” the connection. Diffeomorphism invariance is ensured by considering graphs as identical if they can be smoothly deformed into one another.

The Gauss constraint and diffeomorphism constraints are nicely respected by this construction, and they allow area and volume operators to be defined. Areas and volumes, as eignenvalues of the respective operators, turn out to be quantized. A graph node represents a “grain of space” in Rovelli’s terminology and two grains of space are adjacent if there is a link in the graph between the corresponding nodes. The spin number assigned to the link between a pair of nodes represents the area of the boundary between the grains.

The spin network states form a basis for a Hilbert space and they can be thought of as quantum versions of three-geometries (i.e. the geometries of three-spaces). Area and volume operators can be defined on the Hilbert space, and their eigenvalues are physical areas and volumes. These areas and volumes turn out to be quantised. A generic state in the Hilbert space is a quantum superposition of three-geometries.

There have been several approaches to "deriving" spin networks and their rules from classical general relativity, and all have ended up at the same place. The focus in the LQG community seems to be now shifting from trying to derive LQG to starting from the LQG rules and seeing where they lead us. This is the important task, as it is this which will (hopefully) lead to predicitions from the theory. Along the way to this goal, it is necessary to set up a framework within which we can do explicit calculations. In particular, calculations involving the dynamics:

For the dynamics, attempts have been made to define a Hamiltonian operator, which generates time evolution. It achieves this by changing a given spin network by changing the nodes and links. The Hamiltonian operator is not altogether satisfactory – it contains a number of arbitrary parametrisations. There is a parallel approach to describing the dynamics of quantum gravity, known as the spinfoam approach. Spinfoams are a higher dimensional version of spin networks, where spin quantum numbers are assigned to faces rather than just nodes and edges.

The latest approach to dynamics uses spinfoam vertices to compute transition amplitudes. In traditional quantum theory, the goal of the dynamics is to compute a complex amplitude given an initial and a final Hilbert space state. This is done (in the path integral picture) by summing over all the possibilities “in between” which are compatible with those initial and final states. In background-independent quantum gravity, we can’t really adopt this approach because we have no a-priori spacetime to give us a notion of what “initial” and “final” means – i.e. we have no “time” with which to define t=+∞ and t=-∞. Instead, the problem is phrased as “how do we compute an amplitude for any given quantum 3-geometry state Ψ ? Ψ is going to be thought of as a state representing the three geometry bounding “spacetime”, and we compute the amplitude by summing over all the spacetime possibilities which are compatible with it.

Being a three-geometry state, Ψ is decomposed in terms of spin networks. The desired amplitude is computed by summing over all spin foams that have these spin networks as their boundary. For a given spin foam in the sum, each vertex of the spin foam contributes to the amplitude. Each vertex is evaluated by building a small private spin network around it and applying some simple computation rules, based on group theory and the spin foam parameters. Rules for extracting an amplitude from a vertex bring to mind Feynman rules for vertices in standard quantum field theory.

This sum of products of spin foam vertices is generally referred to as the “spinfoam sum”. As with a Feynman perturbation expansion, the full sum is intractable, and approximations are needed to extract numbers. There are several ways to do this, but one way is to choose a boundary state which is peaked around a three-geometry which is “large” compared to the Planck length. If this is done, then the vertex amplitude comes out as the exponential of the Regge action. The Regge action is precisely that which we obtain from a direct discretization of the classical Einstein equations. The significance of this is that it appears possible to encode the Einstein equations in a simple combinatorial model using some straightforward group theory constructs.
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Re: Loop Quantum Gravity

#73  Postby twistor59 » Sep 06, 2010 11:13 am

For anybody who wants to look at this stuff in more detail, I found a link to an online version of Rovelli's book. It doesn't have the very latest stuff it it, but has a lot of the arguments leading up to it:

http://www.cpt.univ-mrs.fr/~rovelli/book.pdf
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Re: Loop Quantum Gravity

#74  Postby newolder » Sep 06, 2010 3:52 pm

hmmm, brain-pop. :cheers:
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Re: Loop Quantum Gravity

#75  Postby hackenslash » Nov 19, 2010 11:53 pm

Donwloaded for later.

Incidentally, I have asked for this topic to be stickied. It would be a shame to lose such brilliant work. I am reading and re-reading until it sinks in.
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Re: Loop Quantum Gravity

#76  Postby Mazille » Nov 19, 2010 11:59 pm

Indeed. :thumbup: Stickied.
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Re: Loop Quantum Gravity

#77  Postby hackenslash » Nov 20, 2010 12:04 am

Wicked. Cheers, Maz.
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Re: Loop Quantum Gravity

#78  Postby twistor59 » Nov 20, 2010 6:19 pm

thanks guys. I should really tart it up a bit if we want to preserve it. I was just writing down stuff as I read about it, so it may not be as coherent/clear an exposition as it could be. I could maybe do with giving some more references in certain places for the group theory/differential geometry stuff since that's not universally known unless people have worked in similar fields before.
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Re: Loop Quantum Gravity

#79  Postby The Damned » Dec 14, 2010 8:37 pm

hackenslash wrote:Donwloaded for later.

Incidentally, I have asked for this topic to be stickied. It would be a shame to lose such brilliant work. I am reading and re-reading until it sinks in.


I don't think I have the maths yet to really get to grips with it, but hell as you say its worth some serious effort. Kudos guys.

Is LQG more testable than string theory, ie by experiment, if so can anyone explain how?
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Re: Loop Quantum Gravity

#80  Postby Mazille » Dec 14, 2010 8:47 pm

twistor59 wrote:thanks guys. I should really tart it up a bit if we want to preserve it. I was just writing down stuff as I read about it, so it may not be as coherent/clear an exposition as it could be. I could maybe do with giving some more references in certain places for the group theory/differential geometry stuff since that's not universally known unless people have worked in similar fields before.

You know, I was thinking that maybe hack could use that for that book of his somehow? :scratch:
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