Basic tools for quantized differential geometry

[marcus]

One of selfAdjoint's posts in a loop gravity thread pointed out that many of the tools needed belonged in differential geometry.
This is his 22 October 8:33 pm post:

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"Since Greg the magician got us a reprieve on the bandwidth thing, I'll add a little. I just read your very good intro to connection on the other thread, and it set me thinking. Should we try to motivate, say a principle bundle, the lie group and lie algebra acting on the manifold - actually on the tangent bundle, and all that? This is basic stuff, and really belongs on the diff manifolds board that is sort of dormant right now. Just a thought, let me know what you think.

Working on the Thiemann intro, I am now trying to conceptualize the term "anti-self-dual". A few more times around the block and I'll have it.

BTW we should retrieve your explanation of covariant and contravariant, and our discussion of pullbacks, that all goes in here too. Build up a chain of posts like a FAQ that people could use in trying to make sense of these papers.


It's late at night and maybe this is just mindfog speaking, but do let me know what you think."
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What I think is eureka. Yes GR is a subspecialty within the broad field of differential geometry and quantizing GR primarily means facing up to the question of how do you quantize differential geometry.

And the specifically "loop" approach to quantizing GR simply means that you focus on one particular gadget, the connection---so that the quantum states are complex-valued functions defined on the space of all possible connections on the manifold you are studying.

The connection is a very differential-geometry-type idea and all the other stuff you mentioned, that you use in quantizing geometry, are likewise at home here.

so what I think is, why didnt we think of this before? this is obviously the right venue to assemble short explanations of the tools needed both in normal ordinary differential geometry and also in any quantization of it

also differential geometry is the home of the right honorable categorical morphism, the Diffeomorphism, and in the words of Thomas approximately Jefferson:

"We hold these structures to be Invariant..."

(this is from the Declaration of Background Independence, as you will have observed)

 

[marcus]

Somebody named Ambitwistor appeared at one of the loop gravity threads and gave us a link to a really excellent 100-page textbook called "Preparation for Gauge Theory"

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

It's by George Svetlichny, of the Catholic University of Rio, Brazil.

the differential geometry you need for gauge (field) theories turns out to be (not to be surprised) what you need to do loop gravity. that is, if you want to do it like a proper self-respecting mathematician and not by mere animal cunning.

though a lot can be said in favor of brute cunning

we need to make an annotated bibliography of resources useful for understanding the quantization of differential geometry---loop gravity in particular.

 

[M98Ranger]

I completely agree with selfAdjoint's post that the basic tools for quantized differential geometry are rooted in differential geometry itself. In fact, quantizing GR essentially means quantizing differential geometry, and the loop approach focuses on the connection as the key element in this process.

The concept of a principle bundle, the use of lie groups and lie algebras to act on the manifold, as well as the understanding of covariant and contravariant objects and pullbacks are all essential tools in differential geometry that play a crucial role in the quantization process.

I also appreciate the idea of creating a chain of posts that can serve as a FAQ for those trying to make sense of the papers in this field. This would be a valuable resource for anyone interested in understanding the intricacies of quantized differential geometry.

Furthermore, I agree with the notion that differential geometry is the home of categorical morphisms, particularly the Diffeomorphism, which is a fundamental concept in background independence. This highlights the importance of differential geometry in the study of quantum gravity and the need to fully understand its tools in order to successfully quantize it.