Proposed Resource Sticky for Quantum Differential Geometry

[marcus]

Quantum Geometry is a fairly new field using a wide range of math tools. It would help to have a sticky where links to useful online papers can be kept for ready access.

A lot of the interest in quantum differential geometry stems from efforts to quantize the 3+1 spacetime manifold. Not to divide it up into little bits! But to introduce indeterminacy (a hilbertspace of possibilities) as to its SHAPE. Quantizing geometry means to have a wavefunction over all possible shapes the manifold can have.

Quantizing the geometry, in other words the shape, of a 3+1 dimensional (technically pseudoriemannian) manifold is tantamount to quantizing general relativity----if gravity is geometry, with its effects modeled by spacetime shape, then to quantize shape is to quantize gravity. Not in the sense of string theory but in the sense of general relativity. So this is part of the motivation of quantum differential geometry.

So let's consider Differential Geometry (the basic math underlying General Relativity but also a lot more besides) and see what kind of online resources, tutorials, landmark papers, surveys and review articles, we would need to help find our way into the Quantum version of it.

 

[marcus]

Svetlichny's "Preparation" and Carroll's "Lecture Notes"

Here's a couple suggested IIRC by Ambitwistor

http://arxiv.org/math-ph/9902027

"Preparation for Gauge Theory" by George Svetlichny, of the Catholic University of Rio, Brazil.

http://arxiv.org/gr-qc/9712019

"Lecture Notes on General Relativity" by Sean Carroll

Also, since I have it handy, I will include a Quantum Algebra reference:

http://arxiv.org/q-alg/9710022

"Harmonic analysis on the quantum Lorentz group" by E Buffenoir and Philippe Roche

The quantum Lorentz group---the q-deformed SL(2,C) has suddenly gained enormous interest because of observed positive Lambda (accelerated expansion). But that happened with the supernova observations in 1998. How did Buffenoir and Roche know to do the essential group representation work on it in 1997? Another puzzle.
They do Plancherel with the quantum Lorentz group

 

[marcus]

Background in cosmology

Cosmology is driving an historical change in Differential Geometry IMO because of the observation of dark energy---positive cosmological constant. It means that the groups become q-deformed.

The cosmological constant Lambda is essential the reciprocal of the area of the cosmological event horizon. One over the square root of Lambda is the radius of the event horizon (estim. 62 billion LY)

This is basic cosmology stuff so we need backgrounders in cosmology.
There are online calculators and animations that let you see how things change when you vary the parameters like the Hubble, the dark matter and dark energy fractions, spatial flatness/nonflatness and so on. There are tutorials like Ned Wright, Eric Linder. There are lecturenotes like Siobahn Morgan, George Smoot.

I will mention one link I have found very useful:

http://arxiv.org/astro-ph/0305179

"Inflation and the Cosmic Microwave Background" by Charles Lineweaver

He's got a lot of information condensed into a few pages in very accessible form and some excellent space-time diagrams. If you want to know why the farthest objects we could every expect to see no matter how long we wait are currently 62 billion LY away, and how this relates to the density of dark energy which is 0.6 joules per cubic kilometer (1.3E-123 in Planck units), then you can dig that out from Lineweaver and a whole lot more.

Cosmology has become an observational branch of astronomy and it is influencing what problems in Differential Geometry are attracting interest.

Karim Noui and Philippe Roche used the operative phrase "Cosmological Deformation" in the title of their paper
http://arxiv.org/gr-qc/0211109
the cosmological constant deforms the structures used in Differential Geometry in a certain barely perceptible but very interesting way.
The numbers in certain matrices become very very slightly non-commutative. Heady stuff. Incredible and exciting. Apparently leads to the quantization of speed.