# One half of LQG's proposed math framework topic

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LQG posted a header for discussion of quantum gravity math framework. I want to restart that with a narrower focus so as not to foster contention between theories but rather to discuss only the mathematical underpinnings of what is known as Loop Quantum Gravity---also as "quantum geometry" and "background independent quantum gravity".

An intuitive overview of the subject is given in a short paper for general audience by Ashtekar
"Quantum Geometry in Action: Big Bang and Black Holes"
http://arxiv.org/math-ph/0202008 [Broken]

Ashtekar is the founder of this field of research because in 1986 he came up with new variables for Einstein's 1916 general relativity model and it is Ashtekar's new variables which enabled the development of LQG. So his perspective on it is especially worth noting.

The basic LQG approach to gravity is to take seriously the idea that "gravity is geometry" that came with GR and to quantize geometry. GR makes the very shape or geometry of space a dynamic evolving thing---guiding the flow of matter and energy and in turn being influenced by the distribution of matter and energy.

There is a standard proceedure for quantizing systems going back to the 1920s called "canonical quantization" which has been successfully applied to one classical system of equations after another---understanding the rules or customs of quantizing can help understand LQG since it is just a continuation (using Ashtekar's 1986 new variables) of a long-standing effort to quantize general relativity.

I will try to sketch this briefly. The first thing one needs to specify in a customary quantization is the "configuration space".
For a single particle this could just be the x-axis---the real line R giving all possible positions of the particle.

Around 1962 when John Wheeler began quantizing GR he made the configuration space the set of all possible distance-functions ("metrics") on a manifold. This choice configuration space led to difficulties. In 1986 Ashtekar made the set of tangent-vector-transport, or parallel transport, functions ("connections") usable as a configuration space.

The idea is to have something analogous to the real line R that describes possible classical states or configurations that the system can be in and then build something like L2(R) on it.

This "Ell-two" is conventional math jargon for the "Hilbert" space of all square integrable functions on whatever, say the real line.
It is your basic customary representation of the quantum states on that set of configuration possibilities. By restricting to functions whose squares can be integrated to give a finite number we sneak in the possibility of defining an "inner product" of any two functions---just multiply the two functions together and integrate.

the inner product is like the dot product of 3D vectors---just multiply them together componentwise and add up.

It is what behavioral psychologists call a "tropism"----rats eat cheese, squirrels run up tree-trunks, physicists make Hilbert spaces. They do it every time. Give one of the little fellows a configuration space like R and he will build the space of square-integrable functions like like L2(R) on it.

And immediately afterwards one always sees OPERATORS defined on the hilbert space. In fact "canonically conjugate pairs" of "self-adjoint" operators. In the case of L2(R) there are the position and momentum operators which correspond to observing the particle's position and momentum.

So this may give you some idea of what to expect the mathematical basics of quantum gravity to look like. There is an underlying continuum---a manifold M----but it has no pre-designated shape. The possible shapes are given by the set A of all the "connections" living on the manifold.
A connection is a nifty gadget that describes how a tangent vector swings around as you move its base point from place to place on the manifold---it's actually a differential form with values in a Lie algebra---and it encodes the shape of the manifold. Different connections encode different shapes or geometries.

And then according to the time-honored tradition, we expect to see something like L2(A) appear (and in fact it does)

At this point there is a titanic struggle to get an orthogonal set of basis vectors for L2(A). The quantum states are an inner product space---a Hilbert space---and one can say when two square-integrable functions are perpendicular to each other: they have inner product equal to zero. A "nice" set of basis vectors----a set of square-integrable functions using which all the rest can be described by taking linear combinations---should be not merely linearly independent but actually orthogonal.
And it is in the struggle to get a nice efficient basis for the Hilbert space that Roger Penrose's "spin networks" appear.

And finally, as one expects, OPERATORS appear on the Hilbert space (which may in the meantime have undergone some factoring down in size). And happily enough the operators that correspond to measuring the area of a surface (in different quantum states of the geometry) and the volume of a region (again in different geometry quantum states) turn out to have discrete definite numbers for their eigenvalues, which people have already succeeded in calculating!

This discreteness of the area and volume in quantum geometry is the root cause of recent sucess removing the big-bang singularity and relating black hole surface area (quantized in discrete steps) to (semi)classical calculations of black hole vibration modes and entropy. These are themes discussed in Ashtekar's not-too-technical paper "Quantum Geometry in Action" which I gave the link to earlier

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another look at Loop's question

The PF poster who was originally wondering about the mathematical framework of Loop Quantum Gravity may not be around, haven't seen him for a while.

Further exploration is possible. I think the mathematical framework of LQG where it is applied to cosmology is especially interesting.

LQG gets rid of the classical singularity at time zero, so instead of a "big bang" there is a transition between the prior contracting phase and the observed expansion. Evolution across the boundary is governed by a quantum operator equation that is actually a difference equation---time gets to be discrete at very small scale.

The approximation to the classical model becomes very close soon after time zero----IIRC on the order of 100 Planck time units. So there is a brief interval right around time zero when the universe would appear not to make everyday classical sense,
but it soon settles down into the classical Friedmann equation track.

(100 Planck time units is hardly any time at all, so the unintuitive quantum state of reality does not last very long)

It seems to me that the mathematical framework that allows getting thru the singularity and making this transition might be worth sketching out. Any interest?

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MathematicalPhysicist
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Originally posted by marcus
The PF poster who was originally wondering about the mathematical framework of Loop Quantum Gravity may not be around, haven't seen him for a while.

Further exploration is possible. I think the mathematical framework of LQG where it is applied to cosmology is especially interesting.

LQG gets rid of the classical singularity at time zero, so instead of a "big bang" there is a transition between the prior contracting phase and the observed expansion. Evolution across the boundary is governed by a quantum operator equation that is actually a difference equation---time gets to be discrete at very small scale.

when you mean classical singularity do you mean the one derived from the equations of Gr?

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Originally posted by loop quantum gravity
when you mean classical singularity do you mean the one derived from the equations of Gr?

Yes! Maybe I should say the "classical GR singularity"
but other people, in articles Im reading, just call it the
"classical singularity"

time marches on and Einstein's 1916 equation (which breaks down at t = 0 with curvature going off the chart) is already "classical".

to me this is just about the most exciting thing in physics right now: the time-zero singularity seems to disappear when GR is quantized by the loop method

so there is emerging a field or subfield of scholarship called "Loop Quantum Cosmology" (should we abbreviate this LQC?)

and it works with quantized Friedmann equations or stuff like that

John Archibald Wheeler started doing that in 1962 in a book called Geometrodynamics

a really good sign is that the LQC equation gives a good approximation to the Wheeler-DeWitt equation if you stay
a little bit away from zero-----and the Wheeler-DeWitt equation was already not bad as a quantized version of the basic cosmology equation that Friedmann derived from Einstein.

So it goes in a kind of line of development:
Einstein---Friedmann---Wheeler-DeWitt----LQC
and finally, at the end, the trouble at time zero is resolved
I keep looking for a paper by Ashtekar and Lewandowski that
has been promised and referred to but is not yet online

jcsd
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I'm not a big LQG fan (though I would have to learn more about it to give a more objective judgement) as astronmer's recent observations would seem to rule out quantized space.

GR is a classical theory and always has been, as classical just means non-quantum.

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Originally posted by jcsd

GR is a classical theory and always has been, as classical just means non-quantum.

I think you are right. That is what it means in a modern physics context. But to other people there may be ambiguity----classical may suggest "newtonian"---so we can throw in a few extra words occasionally for clarity.

Originally posted by jcsd
I'm not a big LQG fan (though I would have to learn more about it to give a more objective judgement) as astronmer's recent observations would seem to rule out quantized space.

jcsd,

I've looked at some articles bearing on this and they do not seem to rule out quantized space, so I would be glad to read what you have seen that gives you that impression!

It is pretty interesting that observations seem able to assist the development of LQG, already in the near term, by establishing bounds on various parameters. It helps for a theory to be testable and to have a narrowed range of choice.

If a theory proliferates in an unconstrained way then it is apt to predict everything and nothing. So one hopes for increasingly stringent tests.

So far, to my knowledge, the "astronomer's recent observations" (of Gamma Ray Bursts in particular) have not narrowed things down decisively. But can be expected to do so in the future---by 2006 according to one writer. I would be glad to have links to whatever it is you are referring to.

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jcsd
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I haven't got a link, but I defintely rmember reading an article a few months ago in which an astronmer calculated that if space was quantized then it would effect the observation of very far away objects (IIRC the resolution, or some other optical property) he then tested this and concluded thatif space is quantized it must be on a scale much shorter than the Planck length.

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Originally posted by jcsd
I haven't got a link, but I defintely rmember reading an article a few months ago in which an astronmer calculated that if space was quantized then it would effect the observation of very far away objects (IIRC the resolution, or some other optical property) he then tested this and concluded thatif space is quantized it must be on a scale much shorter than the Planck length.

jcsd,

Well perhaps some other reader will supply the link. I've seen quite a few articles discussing observational tests of LQG. The people developing the theory are keen to get astronomers to test the theory to whatever extent is possible and this summer's conference at Ashtekar's Institute at Penn State had a bunch of papers by astronomers.

We should strive towards a common understanding of what is meant by "quantized space". In the LQG theory geometry is quantized---it is a quantum theory of shape, curvature, area, volume----but space is not partitioned into a mosaic of bits or steps of uniform size. It is a quantum theory but does not model space with a fixed lattice. I'm wondering what your astronomer meant by "space being quantized".

In LQG the underlying space is a continuum (I sketched this out in this thread). The area and volume operators turn out to have discrete spectra---but the eigenvalues dont go in uniform steps and for macroscopic objects the separation of adjacent values of the area is less than 10-10 of the Planck area.

Ashtekar makes that point emphatically in his general audience paper "Quantum Geometry in Action"
http://arxiv.org/math-ph/0202008 [Broken]
This applies to length as well. In discussing a length operator he says on page 8: "For example the gap between an eigenvalue of a^ of about 1 centimeter and the next one is less than 10-30 times the Planck length."

I've written a^ for the a-circumflex symbol which I cant type.

As you know the Planck length is already very small and we are talking about a difference in length that is 30 orders of magnitude smaller than Planck length.

LQG is not a lattice theory or a theory in which space is broken up into little steps of uniform size. You probably realize this. The underlying space is a smooth manifold, a continuum. So I am left wondering how the astronomical observation you mentioned apply.
There definitely are dispersion relations that should be testable using gamma ray bursts---high energy light that has traveled for long distances. This kind of testing is in progress and more is planned---it is of vital interest. But it is hard to tell whether the particular article you mentioned is relevant or not without reading it.

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