Loop Quantum Gravity

Study matter and its motion through spacetime...

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Loop Quantum Gravity

#1  Postby twistor59 » Jul 01, 2010 2:00 pm

Anybody know how it works ?
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Re: Loop Quantum Gravity

#2  Postby Sityl » Jul 01, 2010 8:51 pm

it works in a loop
Stephen Colbert wrote:Now, like all great theologies, Bill [O'Reilly]'s can be boiled down to one sentence - 'There must be a god, because I don't know how things work.'


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Re: Loop Quantum Gravity

#3  Postby twistor59 » Jul 10, 2010 11:32 am

LOL - OK I'll take that as a "no" then.

Given that it's held by the physics community to be one of the two currently most promising approaches to quantum gravity, I thought it would be kind of interesting to learn what it was. If people are interested, I'm happy to post my learning process here as I go along. This will certainly take several months due to the combination of not having a vast amount of time and being a slow learner. I will do the learning thing anyway, but I won't bother to post any of it unless people register some sort of interest.


So anybody interested ?
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Re: Loop Quantum Gravity

#4  Postby Mononoke » Jul 10, 2010 1:33 pm

twistor59 wrote:LOL - OK I'll take that as a "no" then.

Given that it's held by the physics community to be one of the two currently most promising approaches to quantum gravity, I thought it would be kind of interesting to learn what it was. If people are interested, I'm happy to post my learning process here as I go along. This will certainly take several months due to the combination of not having a vast amount of time and being a slow learner. I will do the learning thing anyway, but I won't bother to post any of it unless people register some sort of interest.


So anybody interested ?


I'm game I've actually been interested in learning about this.
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Re: Loop Quantum Gravity

#5  Postby Joe09 » Jul 10, 2010 5:37 pm

Im interested, dunno if i wld understand it though as i dont have undergraduate knowledge (yet)
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Re: Loop Quantum Gravity

#6  Postby ashley » Jul 11, 2010 8:51 pm

Go for it.
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Re: Loop Quantum Gravity

#7  Postby twistor59 » Jul 12, 2010 7:43 am

OK I'll start putting some thoughts together. As I said, this will be a slow process ....
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Re: Loop Quantum Gravity

#8  Postby tnjrp » Jul 12, 2010 8:52 am

I've read the popular science version by the head honzo Lee Smolin himself, Three Roads to Quantum Gravity, translated. Which is to say I know about as much about it as WikiPedia does. An independent treatise looking into the current status would be of great interest.
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Re: Loop Quantum Gravity

#9  Postby Mononoke » Jul 12, 2010 9:14 am

twistor59 wrote:OK I'll start putting some thoughts together. As I said, this will be a slow process ....


http://arxiv.org/PS_cache/gr-qc/pdf/0409/0409061v3.pdf

that might be a good place to start. I have to warn you I've only got a bachelor in physics. although I've taken a few graduate classes, stuff like gauge theory is very alien to me. My math is a little bit more advanced so hopefully i can catch up
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Re: Loop Quantum Gravity

#10  Postby Nautilidae » Jul 12, 2010 1:32 pm

:coffee:
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Re: Loop Quantum Gravity

#11  Postby Mazille » Jul 12, 2010 1:33 pm

[youtube]http://www.youtube.com/watch?v=FMSmJCKaaC0[/youtube]

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Re: Loop Quantum Gravity

#12  Postby Nautilidae » Jul 12, 2010 1:37 pm

Mazille wrote:[youtube]http://www.youtube.com/watch?v=FMSmJCKaaC0[/youtube]

(Consider it an on topic bookmarker.)


I love this clip.
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Re: Loop Quantum Gravity

#13  Postby twistor59 » Jul 14, 2010 8:26 am

OK, so as I mentioned previously, I’ve decided to try to learn what loop quantum gravity is, and I thought that it might be useful to post my learning process as I go along, so that people can chip in with extra insights/questions/objections and hopefully we’ll all get to know a bit more than we did before. At this stage I know virtually nothing about what loop quantum gravity, so I can’t plan a coherent and logically ordered set of posts on the subject, so please expect a random mishmash of half understood ideas and concepts. I don’t know who will read this as it’s not exactly mainstream interest – maybe only people who are fairly high on the geekometer spectrum .

Some abbreviations:
LQG = Loop Quantum Gravity
GR = General Relativity
QED = Quantum Electrodynamics
QFT = Quantum Field Theory

So, since I don’t yet know much about LQG and my copy of Rovelli’s monograph hasn’t arrived from Amazon yet, I thought I’d start with:

Post1: WHY DO WE WANT A QUANTUM THEORY OF GRAVITY ANYWAY?

Good question. In fact I remember SkinnyPuppy started a thread over on RDF with precisely this question. Gravity is expressed in terms of the spacetime in which everything lives. The stuff of our universe (fields/particles) consists of objects (particles and fields) which live “in” or “on” this spacetime. Why not just leave the spacetime as it is (“classical” and described by general relativity), and keep the quantum nonsense just for the fields which live on it ?

1 Singularities
Well, the first thing that comes to mind is that even if we treat the spacetime purely classically, as dictated by general relativity, we find, under a set of conditions which are still pretty general, Hawking and Penrose proved several decades ago that the spacetime will contain singularities. These are regions where general relativity is unable to describe what the spacetime is like. At a singularity, a key variable which GR uses to describe the spacetime - the curvature - becomes infinite. Infinities happen in other theories too - for a point charge, the electric field becomes infinite at the origin, but this is due to the pointlike nature of the source. A GR singularity is not like this - there doesn't have to be a source in the region where the curvature diverges. The gravitational field manages to screw itself up without any outside help !

It is hoped that a quantum theory of gravity will somehow avoid this singular behaviour.

2 All the other Force Fields are Quantised, so why not Gravity ?
The other 3 forces of nature - strong, weak and electromagnetic forces all interact with "matter" in a manner described by quantum laws. The forces themselves are described heuristically, in quantum field theoretic terms, by interaction of an "exchange particle" with the matter in question.

Electromagnetic Force = Exchange of (virtual) photons between electrically charged particles
Weak Force = Exchange of (virtual) electroweak gauge bosons between weakly interacting particles
Strong Force = Exchange of (virtual) gluons between quarks

Since we humans don't have much imagination, we expect gravity to be described in a similar way – gravitational force = exchange of gravitons etc.

3 Planck Scale / Cutoffs
Firstly, Quantum Field Theory in a few paragraphs:
QFT is conventionally built up in Minkowski space (the flat spacetime of special relativity): The basis for building up a quantum field theory is a function called the Lagrangian, this tells you which fields (electrons, photons, quarks etc) you’re dealing with, their symmetry properties, which of the fields interact with each other and how strongly they interact etc. You can then take the Fourier transform of the fields. This will split them into “positive frequency” and “negative frequency” components. After introducing (anti)commutation relations, the Fourier coefficients of the positive frequency components are then treated as annihilation operators and the Fourier coefficients of the negative frequency components as creation operators. This process is known as second quantization. Being operators, the operators need something to operate on. They operate on vectors in a potentially infinite dimensional space called a Hilbert space. The starting point for this process is a state vector called the vacuum, usually denoted as |0>.

With this structure you can start to calculate probability amplitudes using perturbation theory. In QFT perturbation theory works like this: in the Lagrangian, the interaction between fields is defined by creating terms with a “product” of the fields which interact with each other. For example in QED, the Lagrangian would contain terms like

eψ-ψA

Here e is the electron charge (you could replace it with the fine structure constant if you want). ψ denotes the electron/positron field, and A denotes the electromagnetic field(ψ with a minus superscript is meant to be “psi bar”, but I don’t know how to type this, also there should be a gamma matrix in the formula and a Lorentz index on the A, but I’m just trying to convey the concept and skip the technicalities).

Suppose I want to compute the probability amplitude for some process – let’s take a trivial process – an electron goes from point A to point B. If it doesn’t interact along the way, there is a simple formula for this probability amplitude ( the electron propagator). However, suppose along the way it emits a virtual photon and then reabsorbs this photon. Drawing this on a diagram, we see the diagram has a loop. Following the rules and computing the probability amplitude in this case, we unfortunately get the answer infinity !

This useless result comes from the rule which says that we have to integrate, from zero to infinity, over the momentum of the virtual photon. We do this since the photon could have any momentum, and we have to add the probability amplitudes for each possibility. There is a very cunning way around this problem, called renormalisation which was far from obvious – QFT had been around for a long time before Feynman/Schwinger/Tomonaga invented the renormalisation procedure in the 50s (?).

I won’t go into renormalisation in detail, but basically one way to do it involves first “taming” the divergent integral by integrating not up to infinity, but to some arbitrary cutoff momentum value Λ. Then the integral is split into a convergent and divergent part and the divergent part subtracted by means of adding “counterterms” to the Lagrangian. After a bit of jiggery pokery replacing the bare coupling constants by their physical values and applying a renormalisation prescription, finite answers which are independent of the cutoff can be extracted from the integral. For a nice not-too-technical presentation of renormalisation, see here.

This whole process seems rather contrived and inelegant (clever yes, but elegant – no). The only reason I mention all this stuff is that one hope for a full quantum gravity theory is that the need for this messy process may vanish. The reason for this hope is that integrals to infinity in the loop momentum will no longer be needed, since in the new theory quantum gravity will provide a natural cut off for the integrals (or whatever the equivalents of the integrals are in the new theory). By combining the fundamental constants G, h and c in various dimensionally consistent ways, “natural” values of energy, mass, length, time etc can be derived. These are the values at which quantum gravity is expected to become relevant. The divergent integrals giving so much trouble in QFT would somehow be cutoff naturally at momenta below the Planck energies.


4 Time in Quantum theory vs Time in Relativity
I mentioned in 3 that a crucial part of QFT in Minkowski space was the separation of the field into positive and negative frequency components. We needed this in order to have the concept of creation and annihilation operators which we needed to describe it as a particle theory. To be able to define positive and negative frequency components, we need a time coordinate. This is fine in Minkowski space - if I make a Lorentz transformation to a new time coordinate I still have the same definition of positive and negative frequencies.

However, in general relativity,
no unique way to choose a time coordinate
=> different choices of time coordinate give different definitions positive/negative frequencies
=> different definitions of creation and annihilation operators
=> in a given state, different definitions of the presence or absence of particles.

(In fact, you don't even need curved spacetime for this - a non inertial frame will do - this is the root of the Unruh effect).

A key aspect of quantum theory is that physical observables are modelled as operators. When the system is in a certain state, the value of the observable is one of the eigenvalues resulting when that operator is applied to the state vector (see http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics). In QFT, time is treated in a different way to the other parameters of the system. It is not an observable. There is no operator I can apply whose eigenvalues are the time. It is merely a parameter. This feels a little unnatural - the fact that there is this classical background clock ticking, driving the quantum system.

Hope has been expressed that in a quantum gravity theory, time will somehow emerge FROM the theory rather than being part of the background on which the theory is formulated.

5 Why Not Stick with Quantum Field Theory on a Curved Background ?
We know how to do QFT on Minkowski space. Much work has been done to investigate what happens to QFT when you replace the Minkowski metric with a general spacetime. I said in (4) that the presence of quantum particles depends upon spacetime curvature (or acceleration, according to the principle of equivalence). There is a now well-developed theory which treats this and many fascinating results have been obtained (Unruh effect, Hawking radiation, black hole entropy). Maybe that's as far as it goes ?

Well if it stops here, we do not yet have a complete theory of gravitons. Classical GR admits gravitational wave solutions, so we would expect there to at least be a quantised way of treating these. Most other wave phenomena in physics seem to have a quantum treatment. Gravitons should be able to interact with each other as well as with “source” matter, so the theory which models them must be able to handle this. If this is tried, using the conventional renormalisation scheme of (3), the prescription fails – renormalisation does not work. What happens is that more and more counter terms are required at successively higher order approximations of the theory.

Well, that’s just a bit of woffle about some of the motivations for quantizing gravity. It’s not an exhaustive list.

LQG is based around the technique of canonical quantization of GR, so in my next post I’ll try to say something about the canonical formulation of GR.
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Re: Loop Quantum Gravity

#14  Postby Joe09 » Jul 14, 2010 5:26 pm

:shock: i cant get to uni quicker
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Re: Loop Quantum Gravity

#15  Postby twistor59 » Jul 19, 2010 8:02 am

Post 2 General Relativity and its Canonical Formulation

In this post I want to say a bit about the canonical formulation of GR, since LQG is based on something similar. To do this I’ll start by saying what the mathematical ingredients of GR are – just a flavour no pretence of completeness or rigour.

Manifold
A manifold is a geometric object that looks locally like a piece of Euclidean space RN. RN is just the space of ordered N-tuples of real numbers (x1,...xN). So a (N dimensional) manifold has the property that I can plaster it with patches, and the points in each patch are labelled by the coordinates x1...xN. I may not be able to find one patch to do the job over the whole manifold – for example if I take the 2 dimensional manifold which is the surface of a sphere, I can make much of it look like R2 by using polar coordinates θ and φ. These can’t be used everywhere however, since the poles don’t have a well defined, unique φ coordinate. I can, however, cover the sphere with overlapping patches such that each patch looks like a piece of R2. In GR, the manifold is spacetime, and there are 4 coordinates. Most people number the coordinates from 0 to 3, with 0 being the time coordinate.

Tangent Vector
It’s fairly intuitive what one of these is (but a little more involved to define mathematically). Taking the sphere, I can imagine a tangent vector at a point to be a little arrow stuck onto the sphere at a point P and perpendicular to the line from the origin to P. (If I were to define tangent vector mathematically I wouldn’t need to use the concept of “perpendicular” here).

Metric
The metric (which means “measure”) is a rule which defines how you can take the scalar product of any 2 tangent vectors, so you can use it to compute the angle between the vectors or the length of a vector. Since on an N dimensional manifold I can define N basis tangent vectors, my metric just needs to define how to take the scalar product of any 2 basis vectors, so it has N2 components. However, scalar products are symmetric, so it really only has N(N+1)/2 independent components.

Connection
A connection is a mathematical object which tells you how to parallel transport a vector. Parallel transport is concerned with how the vector changes as you change the point at which it’s attached to the manifold. Suppose it’s attached at point P1 and I want to “move” it to point P2 along a curve C connecting P1 and P2. Conceptually
a) At each intermediate point P along the way, it must become a remain vector at P i.e. it’s not allowed to “dip in” or “dip out” of the manifold – it stays in the tangent space
b) It “preserves the metric”, i.e. keeps the vector the same length
c) It stays “as near as possible to staying parallel to itself whilst respecting (a) and (c)

The mathematical object that tells you how to do this is called a “connection coefficient”, or sometimes “Christoffel symbol”. Γabc

A curve which parallel transports its own tangent vector is called a geodesic and in GR geodesics are the curves traced out in spacetime by particles that are freely falling under gravity.

(For the pedants: When I say "connection" here, I always mean "torsion free Levi Civita connection")

Curvature
Staying with my surface-of-sphere manifold example, if I take a vector at the North pole, parallel transport it along a line of longitude to a point on the equator, parallel transport it a quarter way round the sphere along the equator, and parallel transport it back up to the North pole along a line of longitude. If I do this, it ends up pointing in a different direction from the one it started in, simply because of the transport round the path. The picture of this process is herehttp://en.wikipedia.org/wiki/Connection_%28mathematics%29

If parallel transport round closed circuits like this causes the vector to change relative to its initial state, as it does in the case of the sphere, then the manifold is curved. If I were to do a similar exercise on a cylinder, the vector would end up back where it started from, since the cylinder is not a curved manifold.

Mathematically, I can imagine moving vectors around an infinitesimal quadrilateral circuit bordered by a pair of other vectors and in this way generate the Riemann curvature tensor.The Riemann curvature tensor will have 4 indices – two for the vectors bounding the infinitesimal quadrilateral, and two to make up a matrix with which to transform our test vector after it’s been parallel transported around the infinitesimal quadrilateral. Denote the Riemann tensor by Rabcd . It has several symmetry properties, and you end up with only 20 independent components.

In GR, given the metric, I can compute (by differentiation), the connection coefficients, and by some further differentiation, the Riemann curvature tensor. So the metric lies at the root of the interesting geometrical features – it’s a bit like the potential in electromagnetism – you can get all the physical information you want from it by differentiating. Think of the metric in terms of a potential.

Ricci Tensor and Ricci Scalar
Remember I said that the Riemann curvature tensor had 4 indices Rabcd, .If I sum over the first and third indices, I end up with something with only two indices
Rbd = sum over index a (Rabad)
This is called the Ricci tensor and plays a key part in GR. It has 10 independent components.

Above I mentioned the idea of a geodesic as a curve traced out in spacetime by something freely falling. If I take a small sphere of test particles and let them go in free fall, then the Ricci tensor is associated with the volume change of this sphere.

I can throw away even more information if I sum over the two indices b,d of the Ricci tensor. The object I end up with is the Ricci scalar, or “curvature scalar”, called R.


Energy-Momentum Tensor
The final ingredient for GR, the energy momentum tensor Tab has two indices. We’ll use the letters a, b, c , d for spacetime indices, i.e. they each run from 0 to 3, and the letters i, j, k, l for space indices, i.e. they run from 1 to 3. Given some matter, component Tij is the i’th component of the momentum flux in the j’th direction. What this means is, imagine a surface whose normal is the j’th direction, then the “flux in the jth direction” is the momentum flowing across this surface. T00 is the energy density etc.

Einstein’s Equations
Ignoring the cosmological constant and the occasional π Einstein’s equations are just

Rab – 1/2Rgab = Tab

Written like this, they look fairly simple, but remember that Rab is a complicated expression involving the Γabc and Γabc is a complicated expression involving the gab. So given a matter distribution Tab, solving for the gab is not an easy matter. Only a handful of exact solutions are known.

In a vacuum, Tab = 0 and Einstein’s equation reduces to Rab = 0. That means 10 curvature components have vanished – however remember that the Riemann tensor has 20 components, the remaining 10 are encapsulated in the Weyl tensor – this is basically the content of the Riemann tensor which isn’t in the Ricci tensor. We said above that the Ricci tensor causes volume change in a sphere of freely falling test particles, well, the effect of the Weyl tensor is to deform the sphere away from its initial spherical shape – it’s a shearing effect, or a tidal force.

Initial Value Formulation
I will hopefully get around to talking about LQG eventually (but not in this post!). LQG theory is constructed in a similar way to an old treatment of GR called the canonical formalism.

The development of the canonical formalism was motivated by a desire to isolate the independent “physical degrees of freedom” of GR. Quantization relies on writing down commutation (or anticommutation for fermions) relations between operators representing these physical degrees of freedom and their conjugates. So essentially by "physical degrees of freedom" I mean a minimal set of functions I can write down, which contain all the physics.

The Riemann curvature tensor has 20 components, but I can get all these from the metric tensor components by differentiating (just like in Maxwell’s theory I can get E and B fields from the electromagnetic potential vector Ab by differentiating). So surely, these metric components are the physical degrees of freedom ? To see why it may not be so simple, we’ll start by taking a look at the easier case of electromagnetism.

In classical Maxwell theory, the electric field vector E has 3 components Ei (i=1,2,3) and the magnetic vector B has 3 components Bi. In trying to find the true physical degrees of freedom, we can do a bit better though, we can use the four-potential Aa = (φ, A), where E=-grad(φ), and B=curl(A). φ is a scalar and A is a three dimensional vector. In fact it's even better than this - I can make changes to the potential of the form Aa ->Aa+daω without altering the physics (ω is just a scalar function). Such changes to Aa are called gauge transformations (another note to pendants - I am leaving out global gauge issues, such as the Aharonov Bohm effect here). In the absence of sources (charges and currents), I can reduce everything to two physical degrees of freedom (which I can express, for example as left and right circularly polarised waves).

That was just the Maxwell case. As you might guess, the same sort of thing applies for the GR case only it's a lot more complicated. People started a long time ago to try to isolate the physical degrees of freedom in GR by asking the question how many functions (and their first time derivatives) would I have to specify at t=0 in order that Einstein's equations would then determine the rest of the spacetime ?

Now in GR what does t=0 mean ? It means that I've introduced a coordinate system and I'm looking at the subset of spacetime I get if I set the time coordinate to zero. Such a subset with a fixed time coordinate is called a spacelike hypersurface, and it might be wiggly because spacetime is wiggly, or even if spacetime is flat it might be wiggly because I happen to have chosen a wiggly coordinate system. Separating these reasons for wiggliness is a large part of the problem. The first systematic attempt to do this that I know of was done by Arnowitt, Deser and Misner, and is called the ADM formalism (although I'm pretty sure I read somewhere that Dirac did something along these lines previously). The ADM reference is a classic.

I've tried to show their approach in this diagram:

ADM.jpg
ADM.jpg (26.54 KiB) Viewed 5906 times


Spacetime is 4 dimensional, but I will not draw 2 of the space dimensions (because I can't draw them) and just draw 1 space and 1 time dimension. I have a spacelike hypersurface at time x0 (which will be a 3 dimensional surface, represented as a wiggly line in my diagram), and another one at a slightly later time x0+dx0. What happens if you look at a fixed space coordinate, x (as in the Maxwell discussion above, bold symbols are 3 dimensional, and so have 3 components) ? You get a wiggly vertical line (labelled by what I call a "3-coordinate" in the diagram because it coordinatizes the 3 space dimensions). I've also shown the nearby fixed space coordinate x+dx.

DA is the infinitesimal vector, at D, normal (according to the spacetime metric gab) to the surface, with its tip on the second surface at A.

DA = Ndx0 for some scalar function N
If, instead of going in the normal direction along DA, I were to keep the space coordinate x constant, I would have travelled up the integral curve of x0 and ended up at point B. If I initially went up DA I can get back to B by going along a 3-vector AB, tangent to the 3 surface. The coordinates of this tangent vector AB are traditionally denoted Ni, i=1..3.

N and Ni are called respectively the lapse function and shift vector (or as Paul Tod used to say, lips and shaft). If I want to change the point in the 3 surface a bit as well, I get to point C. Working like this I can decompose my metric as

gab = [ -N2+sum over i,j(NiNjgij).......Nj ]
.......[ Ni..................................gij ]

Here gij is just the metric "induced" in the 3 surface from the metric gab on spacetime.

N and Ni aren't physical degrees of freedom - they're just objects which tell you how to evolve in time. The physical degrees of freedom are contained in gij.

In order to attempt to quantize the system, a set of conjugate variables πij are required (think of the p's in elementary quantum mechanics, being conjugate to the q's). There are four equations that the gij and πij components have to satisfy among themselves which can be derived from the ADM formalism, called the constraint equations.

So you can see that in GR, the question of what is fundamentally physical and what is not is quite difficult.

LQG does not actually use the ADM decomposition, it uses a slightly less obvious description of GR, invented by Abhay Ashtekar, which I might describe in a later post ( when I've sorted out how it works !). I just mentioned the ADM formalism to give a flavour of what’s involved in trying to isolate the physical degrees of freedom.

In the next post, I’ll give some overview of how spacetime arises in LQG from spin networks.

Edited for formatting
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Re: Loop Quantum Gravity

#16  Postby Joe09 » Jul 20, 2010 2:42 pm

i think i will fav this thread, as im not going to understand any of it for couple years :( lol
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Re: Loop Quantum Gravity

#17  Postby twistor59 » Jul 20, 2010 4:24 pm

Joe09 wrote:i think i will fav this thread, as im not going to understand any of it for couple years :( lol


Yeah, I'm sorry I haven't been writing it as a systematic pedagogical introduction. I was just intending to slap down my thoughts as I went along reading about LQG. It was never going to be a work of art and it's made slightly more difficult by the limitations on equation formatting available. I promise to try to keep the equations to a minimum !

Are you doing a Uni degree in physics and/or maths ?
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Re: Loop Quantum Gravity

#18  Postby Joe09 » Jul 20, 2010 9:48 pm

No its not your fault, i can follow disorganised trains of thought if i understand them, i just dont have the knowledge yet with what your writing.

Soon, ill be doing Foundation Physics Degree this september (i didnt do well at college) and then after ill be doing Bsc Physics with Astrophysics at Leicester

I understand more about star evolution (than would be expected with my physics knowledge) but thats probably because it fascinates me alot more. For my A2 Physics research paper i did 'What is a neutron star?' got an A in that 8-)
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Re: Loop Quantum Gravity

#19  Postby newolder » Jul 20, 2010 9:51 pm

How is a topic 'stickied', I wonder?

In a recent video-google lecture, Lee Smolin uses LQG to predict the speed of light increases with increasing energy but he gave no mathematical symbols in that talk showing how. Fractional dimensions show such features clearly with bit-maps and paint. The separation between 'Mandelbrot islands' (if real) is likely to remain unengineerable for a while yet... Unless someone has drawings of a vacuum energy pump in action. :ask:
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Re: Loop Quantum Gravity

#20  Postby tnjrp » Jul 21, 2010 6:33 am

newolder wrote:How is a topic 'stickied', I wonder?
At this point, by asking the staff to do it I reckon. One can do this by reporting the starting post.

Unless someone has drawings of a vacuum energy pump in action. :ask:
I have some but I've been handsomely bribed by the nuclear power industry not to show them to anyone :plot:
The dog, the dog, he's at it again!
tnjrp
 
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