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marcus

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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

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

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

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 L

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 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 L

At this point there is a titanic struggle to get an orthogonal set of basis vectors for L

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

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 L^{2}(**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 L^{2}(**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 L

^{2}(**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 L

^{2}(**A**) appear (and in fact it does)At this point there is a titanic struggle to get an orthogonal set of basis vectors for L

^{2}(**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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