Spin Geometry: Introduction & Overview

[Kontilera]

Hello!
I have done some quantum mechanics, quantum field theory and general relativity. Not much, but enough to say that I have the big picture. Aside from this I have read about analysis on manifolds, functional analysis, Lie algebras and topology.
Now there is a red book in my bookshelf that goes by the name of Spin Geometry by Lawson and Michelson and it is still to complex for me to grasp what it is about.

This leaves basically two questions,
i) What is spin geometry, does it have anything to do with quantum gravity?
ii) Let's say it is a really interesting subject, is there any easier introduction to the subject that I should start with instead? Have I missed any important mathematics that I should grasp before?

 

[Naty1]

I recently posted about what appears to be a related subject:
[in the Cosmology forum as my interest was in high density big bang conditions.]

Can torsion avoid the big bang singularity

https://www.physicsforums.com/showthread.php?t=664451&highlight=torsion+cosmology

If your SPIN GEOMETRY involves the coupling between Dirac spinors and geometric torsion, this could offer some insights... and a number of possible avenues for further reading. [I never heard of this before I posted and don't remember how I stumbled across it, so what little I learned is in the post.]

...we show that extending Einstein’s general relativity to include the intrinsic angular momentum (spin) of matter, which leads to the Einstein-Cartan-Kibble-Sciama (ECKS) theory of gravity, naturally explains why the Universe is spatially flat, homogeneous and isotropic, without invoking inflation. We also propose that the torsion of spacetime, which is produced by the spin of quarks and leptons filling the Universe and prevents the formation of singularities (points of spacetime with infinite curvature and matter density), provides a physical mechanism for a scenario in which each collapsing black hole gives birth to a new universe inside it. Gravitational repulsion induced by torsion, which becomes significant at extremely high densities, prevents the cosmological singularity...

 

[Naty1]

I did a quick search [spin geometry] in these forums and found

https://www.physicsforums.com/showthread.php?t=664423&highlight=spin+geometry

which also offer some leads.

Try a search yourself to sift through previous discussions.

good luck.

 

[Phys-excess]

Consider a spin-1/2 particle in a trap. If you rotate the trap through 2π rad of space, the spin vector only gets rotated through π rad. This is explained in most intro QM classes. Spin geometry attempts to build a geometry that contains this kind of thing. The goal of that inquiry is to rigorously probe all the different quirky things spin can do using geometric tools. I think for a physicist spin geometry is not something you would consider a useful tool. If you do want to learn it I would tear through all the differential geometry and differential topology you can find. After that, you may possibly acquire enough skill to actually read the spin book.

 

[Naty1]

I was reminded: You know spin networks...

http://en.wikipedia.org/wiki/Spin_networks#In_the_context_of_loop_quantum_gravity

The wiki explanation is not particularly interesting, but in THREE ROADS TO QUANTUM GRAVITY Lee Smolin does a nice job of explaining how a spin network structure generates discrete spacetime...and links them to Wilson+ Polyakov quantized loops without any field
dependence...each spin network can be associated with a possible quantum state for the geometry of space...
all in Chapter 10...

 

[Kontilera]

As I understand it seems like a "mathematical detour" without any distict goal regarding physics... I remember that I mailed a quite famous physicists some years ago and asked about the subject of Hestenes' geometric algebra (about its relevance and meaning) whereby he answered that it was not something new or original, instead he advised me to look up spin geometry.

unfortunately I think the subject lost some of its appealing mystique. :(

 

[torsten]

As I understand it seems like a "mathematical detour" without any distict goal regarding physics... I remember that I mailed a quite famous physicists some years ago and asked about the subject of Hestenes' geometric algebra (about its relevance and meaning) whereby he answered that it was not something new or original, instead he advised me to look up spin geometry.

unfortunately I think the subject lost some of its appealing mystique. :(

The topic of the book "Spin geometry" by Lawson, Michelson has (nearly) nothing to do with spin networks of LQG. It is a pure math book about the concept of a spin structure on a smooth manifold. It try to answer questions like: "When does a spin structure on a manifold exists?" or "What are the geometrical and topological consequences that a Dirac operator exists?" With these methods, one undrestands which manifolds carry a metric of positive scalar curvature etc.
But an advice: start with a simplier book like Nakahara: Geometry, Topology and physics