Mononoke wrote:

The Damned wrote:

Joe09 wrote::shock: i cant get to uni quicker

Mostly its post doctorate stuff I think, and you'll have to be damned good to get involved at any level that will get you in on the ground floor. You'll be at it for a while. But hell I'd give my high teeth to know about this stuff at more than a pop science level. Kudos to twistor for his posts. Bookmark.

I don't like the Big Bang but that clip is superb btw.

it's grad school stuff. But as long as you have the math and basic degree of physics knowledge you should be fine

Sorr twistor, I've started reading this stuff yet. maybe over this weekend

No I mean to actually work in it. Although thinking about it you could be doing a PhD under Smolin or someone who is heavily invested in it.

I'd argue to study it it's not any level stuff, well university perhaps. an A level in physics and maths looks like all that is required to me like for a degree in physics.

Post 3 Loops and Stuff

In my last post I said that this time I was going to talk about the generation of spacetime from spin networks. I lied. I want to say a bit about gauge theories and loops first.

Electromagnetism as a Gauge Theory

Maxwells wonderful equations are

divB= 0
∂B/∂t + curl E = 0
div E = rho
∂E/∂t - curl B = -j

If I define a four vector potential

Aa = (ϕ, A)

then the E and B fields in Maxwells equations are given by

B = curl A and E= -grad ϕ

The E and B fields don't change if I make a gauge transformation to Aa

Aa -> Aa - ∂a𝛼(x)

for any old function 𝛼(x)

∂a is partial differentiation with respect to the coordinate xa a=0..3. 0 is time.



Suppose I start to do some QED. The electron field is obtained from the Dirac Lagrangian

L = ψbar(i𝛾a∂a + m)ψ

The Lagrangian is a magic function that, if you do some shit (Euler Lagrange equations) to it, you get the fundamental equations of your theory. Lagrangians appear all the time everywhere and are awesome tools. There is a Lagrangian for QED, QCD, GR, Electroweak, standard model, supergravity. Everything. It encodes the content of your theory.

ψ is the electron field. It's a spinor.

𝛾a is a Dirac gamma matrix.

Whenever you see a repeated index, like "a" in the Lagrangian, where one is superscript and one is subscript, it is understood that there is a summation over that index.

Now wavefunctions, like the electron one, are not supposed to change the physics if you screw around with their phase. So if I multiply

ψ by e-ie𝛼 and ψbar by eie𝛼

(I forgot to mention, the bar is a kind of complex conjugation) where 𝛼 is a constant, the Lagrangian doesn't change and hence the field equations (Dirac equation in this case) don't change.

But since the phase isn't supposed to be a big deal, what happens if I let the phase change be variable, i.e. a function of position ? Unfortunately the partial derivative acts on 𝛼(x) and we get a new Lagrangian and the equations of motion are spoiled.

However, and here is some gauge theory magic, ..... if we, at the same time, replace our partial derivative in the Lagrangian by

∂a -> ∂a - ieAa

then the Lagrangian is now invariant if we do the phase change to ψ

ψ by e-ie𝛼 ; ψbar by eie𝛼

and at the same time a gauge transformation

Aa -> Aa - ie∂a𝛼(x)


So gauge theory somehow ties the Maxwell potential A up with a local invariance under phase changes to the electron field. All gauge theories have this sort of feature. For electromagnetism it's phase changes. For weak interaction it's rotations in weak isospin space. For QCD it's rotations in colour space. For gravity it's changes in local coordinates.


Faraday Lines of Force and Loops in Electromagnetism

Everybody is familiar with the picture of “lines of force” in an electric field – i.e. the curves you get by joining up the little arrows. If there are charges present, these lines originate/terminate on the charges. If there are no charges present, you can still have the fields, but the fields in this case must form closed loops (otherwise Gauss’s law fails). We normally consider electric fields in some region of three dimensional space, which is swept out by a whole bunch of these lines of force, but there is no reason why we cannot consider the case of a single one dimensional line-of-force loop (of course the loop is “infinitely thin”, so writing such a solution of Maxwell’s equations will require a Dirac delta function, but it can be done). Such loops can be considered as the basic excitations of the electric field, and they satisfy Gauss’s law.

There is a view that these loops are more fundamental than just the electric field itself. The electric field describes the situation at each point in space individually, but holonomy along a loop describes a relationship between two points. One motivation for thinking that the field itself is not sufficient is provided by the Aharonov Bohm effect.



As shown in the diagram, we have an electron source, two paths to a screen, passing either side of a solenoid. If the current in the solenoid is switched on, a magnetic field is generated, but this field is entirely contained within the solenoid. The experiment goes as follows:
Current in solenoid off. Turn electron source on. We see an interference pattern on the screen.
Current in solenoid switched on. The interference pattern on the screen shifts.

How can this happen, since the paths C1 and C2 traversed by the electrons have no electric or magnetic field from the solenoid in them (the electromagnetic field is completely contained within the solenoid) ?

In fact, the phase difference for electrons travelling the two paths is

∮A.dl integral is over the contour C2-C1

Where (C2-C1) is the path obtained by going along C2, then backwards along C1, ie along a LOOP (woo hoo I knew I’d get to mention loops eventually !!), and A is the magnetic vector potential. Now here’s the thing: by Stoke’s theorem, the integral of A round the loop is equal to the integral of curlA inside the loop. But curlA is just the magnetic field B, which sure isn’t zero inside the loop.

So this tells me that the potential A contains the physics, the fields E and B clearly do not contain all the physics. But we know that A overdescribes the physics, because I can make a gauge transformation to A and get exactly the same result. Moreover, I get the same result if I add 2π to the phase difference. So for the Aharonov Bohm experiment, the physics is contained in the phase factor

exp (i∮A.dl ) again the integral is over C2-C1

Seeing phase factors appearing here should remind you of the phase factors that appeared in the definition of electromagnetism as a gauge theory in the previous section.

So to summarise, for electromagetism, the physics can be described if you specify what happens when things get transported round loops. LQG relies on a clever way of describing gravity in similar terms to this.

Edited for formatting

Carrying on the conversation a few posts back about DSR - some interesting remarks on the falsification of quantum spacetime from section 7 of this paper:

7 Quantum Gravity In Crisis
An important result in cosmology was obtained recently which can elucidate
the nature of spacetime down to the smallest scale. This is the observation
of the highest energy gamma rays from a gamma ray burst GRB 090510 by
the Fermi Gamma-Ray Space Telescope [21]. A single 31-GeV photon was
detected from a source at a redshift of z = 0.903 which corresponds to a
distance of 7.3 billion light years from Earth. It was the last of the seven
pulses in a short burst that lasted for 0.829 s. One of the two postulates of
Einstein’s special relativity is Lorentz invariance in that all observers mea-
sure exactly the same speed of light in vacuum, independent of the motion
of the source and of the photon energy. In certain quantum theories of grav-
ity, there is great interest in the possibility that Lorentz invariance might be
broken near the Planck scale due to quantum fluctuation of spacetime and
the notion of spacetime foam. A variation of photon speed is an indication
that Lorentz invariance is violated. This may be revealed by observing the
sharp features in the gamma ray burst light-curves. If the spread in travel
time of less than 0.9 s between the highest and lowest-energy gamma rays
in the burst GRB 090510 is all attributed to quantum effects, then a thor-
ough analysis shows that any quantum effects in which the speed is linearly
proportional to energy do not show up until the distance is down to about
0.8LPl, which is below the Planck length. This result therefore rules out a
number of quantum gravity models that predict such linear variation with
energy.

The gamma ray burst reported above is significant in that it allows for
the exploration of spacetime near Planck length by using effects accumulated
over cosmological distances since direct access to Planck energy in experi-
ments is not possible. The result indicates that there is no evidence so far
of any quantum nature of spacetime above the Planck length. Spacetime
there is smooth and continuous. The speed of light is constant and special
relativity is right. At the Planck length, quantum black holes would appear
in observation and they effectively provide a natural cutoff to spacetime. For
observable purpose, it is not necessary to consider theories below the Planck
length. Further detections using gamma ray bursts with even higher energy
photons will settle the question of quantum spacetime definitively. It would
be amazing that in effect spacetime is classical and there is no need for a
quantum theory of gravity. There would be an underlying theory for gravity
which is not gravity, just as statistical mechanics is the underlying theory of
thermodynamics. Unification would have a very different meaning from the
current understanding involving quantum gravity as a fundamental premise.

twistor59 wrote:

Quantum Gravity in Crisis wrote:The result indicates that there is no evidence so far
of any quantum nature of spacetime above the Planck length.

And why did they expect there to be any? I thought the difficulty was only with the sub-Planck scale anyway.

iamthereforeithink wrote:

twistor59 wrote:

Quantum Gravity in Crisis wrote:The result indicates that there is no evidence so far
of any quantum nature of spacetime above the Planck length.

And why did they expect there to be any? I thought the difficulty was only with the sub-Planck scale anyway.

I guess it was this

In certain quantum theories of gravity, there is great interest in the possibility that Lorentz invariance might be
broken near the Planck scale due to quantum fluctuation of spacetime and the notion of spacetime foam.

they mention "near" the Planck scale. This experiment has apparently ruled out "above the Planck scale". I must confess though, I don't understand the details - I haven't got far enough with my study of LQG yet to understand how to estimate this effect.

There is this paper on the gamma ray telescope verification.

ETA and this, which appears to be making some generic statements about quantum gravity predictions

Post 4: Gravity as a Gauge Theory

I talked in the previous post about the description of electromagnetism as a gauge theory and its description in terms of loops. LQG was developed by first reformulating GR as a gauge theory and building up a loop description.

The breakthrough that allowed GR to be usefully described as a gauge theory was provided by Ashtekar in Phys Rev Lett 57 (1986) 2244.

The Ashtekar reformulation makes extensive use of the tetrad formalism. By introducing an appropriate set of basis one-forms, the general metric in general relativity gμν can be made to look locally like a Minkowski metric ηIJ

gμν = eIμ(x)eJν(x)ηIJ

Think of the eIμ(x) as a set of basis vectors at the point x (for the pedants – I am blurring the distinction between covariant and contravariant vectors to keep this explanation simple). Spacetime is 4 dimensional, so I need 4 of these basis vectors (labelled by I), and each one will have 4 components (labelled by μ) in the x coordinate system. These basis vectors are chosen so that the metric components expressed in that basis are just the flat Minkowski space components ηIJ.

In post number 2, I talked about the idea of a connection coefficient. This is used to allow the definition of a parallel transport operation. The infinitesimal form of this parallel transport is a covariant differentiation operation defined on tensors which produces new tensors. Basically, the derivative of a (co)vector aμ is

∂μaν-Γρνμaρ

When we have tetrads, we not only have the general tensor indices (Greek) which are handled by the connection coefficient, but also the Minkowski indices (capital Latin). To handle the latter, we introduce a "spin connection" ω and the covariant derivative of something with both general tensor indices and Minkowski space indices becomes

DμeIν ≝ ∂μeIν + ωIμJeJν - ΓρνμeIρ

The tetrad field encodes the information in the metric gμν, thus contains all the usual GR information (curvature etc). The well known relationships of GR can now be re-written in terms of the e and Γ fields.

To get to the Ashtekar variables, we need to use the ADM formalism of GR I described in post number 2. Basically I described there how spacetime is foliated by a bunch of spacelike hypersurfaces and time evolution is described in terms of the lapse and shift vectors.

The physics is contained in the three-metric gij induced on the hypersurfaces. Recall I said that there were also a set of variables πij “ conjugate” to the
gij. The set of g’s together with the π’s form “phase space”. (The easiest way to picture a phase space is to imagine the phase space of the simple harmonic oscillator. There we have a position coordinate q and a momentum coordinate p. The motion of the oscillator is an ellipse traced out in phase space. The actual ellipse depends upon the initial conditions.)

There are also 4 constraint equations which the πij and gij have to satisfy. These are split into two types:

The "Hamiltonian Constraint" - a single scalar constraint equation
The "Diffeomorphism Constraints" - a three vector's worth of constraints

Think of the constraints as defining a subset of the full space of values of { πij , gij }. The physical configurations are restricted to this subset.

These are first class constraints (Their Poisson brackets with each other preserve the constaint hypersurface). The diffeomorphism constraint generates diffeomorphisms of the 3 surfaces. Combined with the Hamiltonian constraint, we get the generators of spacetime diffeomorphisms on the full spacetime.

Now we can choose a tetrad formulation of GR appropriate for our ADM decomposition (note this is sometimes called “3+1 decomposition” since the time variability is split off from the 3 space dimensions). Note from now on I’m going to use Greek indices for 4 dimensional spacetime indices and latin lower case for three-space indices.

To do this, we can keep the spatial part of the tetrad the same and redefine the time part to be a combination of the lapse function and shift vectors:

eI0 = NnI + NieIi
𝜹kleki elj = gij

Now, as with ADM, we identify our canonical variables and their conjugates, construct an action integral from them, which is then used to derive the physics. Constraints between the set of canonical variables and their conjugates become Lagrange Multipliers. This time, with the tetrad picture, it’s going to be a bit more complex, because we have, instead of the metric gij, the tetrad field and the connection as independent variables. Also, we have an extra symmetry – the invariance under local Lorentz transformations – which we would expect to give rise to new constraints.

The constraint algebra (the set of Poisson bracket relations amongst the constraints) is now second class. However – (this was the breakthrough that enabled this whole field to develop ) – there is a specific set of variables which simplifies things. These were discovered by Abhay Ashektar and Amitaba Sen.

The variables are, firstly the "densitized triad"

Eai = eeai

(a,b,c are the three space indices, and i,j,k are the "group" indices labelling the basis vectors. e = det(eai)

and secondly the Ashtekar-Barbero connection

Aia = γω0ia + (1/2)εijkωjka


Eai and Aia transform like an SU(2) vector and SU(2) connection respectively.

In addition to the Hamiltonian and Diffeomorphism constraint I talked about previously, in the tetrad formulation 3+1 decomposition we now have an extra constaint called the Gauss constraint. This comes from the extra symmetry we've introduced - since we've defined these tetrads at each point, they can be transformed by local Lorentz transformations into other just-as-good tetrads.

Now that we have a classical phase space, consisting of dynamical variables and their conjugates, plus constraint conditions, what would normally be done is to apply Dirac Quantization to this system. (Note this mechanism is not what you’ll find in most books on quantum field theory. They describe covariant quantization, whereas here we need a quantization to deal with a system on spacetime foliated by time slices and subject to constraints).

The original attempt (using γ=+/-i) at quantization used a wave functional Ψ[A] on the space of connections A (modulo gauge). This attempt was not very successful – it was not possible to establish semiclassical states and thus see how smooth spacetime could emerge (see here).

The solution to this was to move away from defining wave functionals on the space of connections and instead to use holonomies as the basic variables. I’ll describe this in the next post.

Edited to fix error in equation

And then there's this ... which might just alter all sorts of things

http://www.newscientist.com/article/mg20727721.200-rethinking-einstein-the-end-of-spacetime.html

chairman bill wrote:And then there's this ... which might just alter all sorts of things

http://www.newscientist.com/article/mg20727721.200-rethinking-einstein-the-end-of-spacetime.html

Ah yeah that's Horava gravity. He introduces some higher spatial derivative powers into the action (but keeps the time derivatives the same). This causes the propagators to have higher powers on the denominators, and hence be less prone to diverging. Hope is it's renormalisable.

I don't know much about it, but I think it needs some fine tuning of the parameters which go into it, but I guess it's early days yet....

twistor59 wrote:

chairman bill wrote:And then there's this ... which might just alter all sorts of things

http://www.newscientist.com/article/mg20727721.200-rethinking-einstein-the-end-of-spacetime.html

Ah yeah that's Horava gravity. He introduces some higher spatial derivative powers into the action (but keeps the time derivatives the same). This causes the propagators to have higher powers on the denominators, and hence be less prone to diverging. Hope is it's renormalisable.

I don't know much about it, but I think it needs some fine tuning of the parameters which go into it, but I guess it's early days yet....

doesn't it give all sorts of quack results for a slightly non-spherical sun

Post 5 Holonomies

In the last post I mentioned the description of general relativity in terms of a set of canonical variables consisting of the "densitized triad" and the Ashtekar-Barbero connection

Eai = eeai and

Aia = γω0ia + (1/2)εijkγjka

I mentioned that the “obvious” application of Dirac’s constraint based quantization scheme failed and people started to look at the “loop representation” (I talked about describing the Maxwell field in terms of loops in post number 3. Now we want to describe gravity in these terms – using the Ashtekar variables).



Firstly I need to describe the way Aia is used for parallel transport. Recall I said Aia was an SU(2) connection. SU(2) is the special unitary goup of 2x2 complex matrices. Think of the group elements as “acting” by matrix multiplication on two – element complex vectors. The vectors are elements of a vector space representation of G. Consider the case where we have a curve in a manifold M connecting points P and Q. Imagine that we have a copy of the representation space over each point of the manifold. (This object is in general called a vector bundle over M). There is in general no way to compare the representation spaces over the points P and Q – they’re different vector spaces and there is no natural map between them. This map is provided by the connection Aia. It works like this:

From Aia, construct the quantity Aiaσi where σi i=1,2,3 are the Pauli spin matrices. The Pauli spin matrices form a basis for the Lie algebra of SU(2). You can think of Lie algebras as “infinitesimal forms” of the groups – a Lie group is a manifold (for example SU(2) is actually a 3 dimensional sphere), and elements of a Lie algebra are tangent vectors.

Now given an element of a Lie algebra, we can generate an element of the Lie group by a process called the exponential map. The connection thus allows us to construct our map between the vector spaces over P and Q given the curve C:

U[A,C] = exp ∫dxa/ds Aia σids
(where the integral is taken along the curve xa(s) ).

This map between the representation spaces at different points, given the connecting curve C is referred to as a holonomy. (Note in most of the mathematical literature the term “holonomy” is used for closed curves only. In LQG papers, it’s conventional to abuse the term).

Under a local gauge transformation g(x) (x⋿ M; g⋿ SU(2)), the gauge transformation law for the connection implies that the holonomy must transform as:

U[A,C] -> g(P) U[A,C] g-1(Q)

where P and Q are the endpoints of the curve C. We’ll be looking at quantization by trying to define a wave functional on the space of holonomies, so of course we will require some conjugate variables. What could be considered as conjugate to a holonomy ?

The answer is a “flux” defined as follows:

A two dimensional area element on our 3-manifold M is
dFi = εijkθj‸θk where θi = dxaeai

We can write this area element in terms of our densitized triad from post number 4:

dFi = = εabcEiadxb‸ dxc

Given a two surface S, and a densitized triad field E, the flux vector is defined as

FSi[E] = ∫dFi (integral is over S)

Given a curve C and two surface S, we can define the Poisson bracket between the holonomy and flux. I’m not going to write it because it would involve an astronomical amount of Unicode !.

We now have defined the basic quantities used in LQG’s quantization program. In the next post, I'll describe how spin networks are constructed and used to generate the Hilbert space for LQG.

I've got a bit confused (again) here:

twistor59 wrote:(Note in most of the mathematical literature the term “holonomy” is used for closed curves only. In LQG papers, it’s conventional to abuse the term). … where P and Q are the endpoints of the curve C.

So, if P<>Q it's not strictly a holonomy? Also, if P<>Q how is it describing a 'loop'?

newolder wrote:I've got a bit confused (again) here:

twistor59 wrote:(Note in most of the mathematical literature the term “holonomy” is used for closed curves only. In LQG papers, it’s conventional to abuse the term). … where P and Q are the endpoints of the curve C.

So, if P<>Q it's not strictly a holonomy? Also, if P<>Q how is it describing a 'loop'?

Nope, it's not strictly a holonomy as mathematicians use the term, but it seems to have crept into the papers on LQG. And no, it's not describing a loop.

The quantization (which hopefully I'll get to when I come back from my holidays Wheeeeeee!) is based on spin networks, which are graphs of curves, each of which has holonomy information. These graphs, of course, contain lots of closed loops.

The very first attempts at doing this stuff I believe used a "loop transform" to convert from the connection representation to the loop representation (like Wilson loops in gauge theories), but in the modern approach, it's all built up from spin networks. (Unfortunately the original papers aren't on the arxiv and I don't have copies).

Enjoy you holiday's twistor. I will be looking forward to your next post when you return

twistor59 wrote:...
Nope, it's not strictly a holonomy as mathematicians use the term, but it seems to have crept into the papers on LQG. And no, it's not describing a loop.

k

The quantization (which hopefully I'll get to when I come back from my holidays Wheeeeeee!) is based on spin networks, which are graphs of curves, each of which has holonomy information. These graphs, of course, contain lots of closed loops.

Do I need to get to grips with Hopf & other fibrations, first?
[youtube]http://www.youtube.com/watch?v=1yJ5EvV0K70[/youtube]

The very first attempts at doing this stuff I believe used a "loop transform" to convert from the connection representation to the loop representation (like Wilson loops in gauge theories), but in the modern approach, it's all built up from spin networks. (Unfortunately the original papers aren't on the arxiv and I don't have copies).

I think you are doing a good thing off your own bat. c u on t'uther side...

well ive just been accepted to do Bsc Physics with Astrophysics at Leicester so gimmie a few and ill come back with my thoughts on LQG

Joe09 wrote:well ive just been accepted to do Bsc Physics with Astrophysics at Leicester so gimmie a few and ill come back with my thoughts on LQG

Well done mate !
Keep us up to date on how you get on...

will do