Fermionic Fields in Einstein Field Equations | Explained

[ShayanJ]

In the Einstein-Hilbert action wikipedia page, the following paragraph is written:

The Palatini formulation of general relativity assumes the metric and connection to be independent, and varies with respect to both independently, which makes it possible to include fermionic matter fields with non-integral spin.

I thought for treating spin, we need to consider Einstein-Cartan theory! This is really surprising to me. Can anyone suggest a paper or book that explains this in some detail?
Thanks

 

[JorisL]

The statement isn't complete.
In fact, when you include fermionic matter the connection needs to get a non-zero torsion.
This is exactly the main 'assumption' in Einstein-Cartan theory.

The Palatini formulation gives a way to find this connection. The connection is often called the spin connection.

If you like GR, do this stuff if you have time. It's pretty elegant in my opinion.

 

[ShayanJ]

The statement isn't complete.
In fact, when you include fermionic matter the connection needs to get a non-zero torsion.
This is exactly the main 'assumption' in Einstein-Cartan theory.

The Palatini formulation gives a way to find this connection. The connection is often called the spin connection.

If you like GR, do this stuff if you have time. It's pretty elegant in my opinion.

I do like GR and of course have time for such things but I'm at the level of learning that can only understand people's calculations but can't do such calculations myself!

 

[haushofer]

Look e.g. at the Supergravity notes of Samtleben :) They are very pedagogical, and also treat this issue. I'd say to include fermionic fields, you need the spin-connection and vielbein, which means you write everything in terms of inertial coordinates (fermionic rep's are only describable in the tangent space!). This can be done without the Palatini formulation, so I'm not sure I understand the Wikiquote.

 

[JorisL]

I'm using the Supergravity book by Freedman and Van Proeyen.
It's nicely written, exercises throughout the text when they are appropriate.

 

[dextercioby]

As said, the statement is inaccurate. The inclusion of spinor fields in curved spacetime (hence in the presence of gravity) needs gravity treated in the viel/vierbein-spin connection formulation which in turns comes nicely as the fiber bundle formulation of GR. The Palatini formulation is the first-order formulation of the H-E action. The connection is purely classical and no intepretation in terms of fiber bundles is made. It's not reformulated in terms of the viel/vierbein field. I think in order to reach supergravity, one needs Poincare gauge theory first.

 

[ShayanJ]

Can someone suggest a good mathematical book covering tetrads(or more generally Cartan's formalism)?
You know, I'm a proponent of dexterciboy's signature!

Learn mathematics from books written by mathematicians

Also is there any book out of supergravity literature that focuses on this issue?
Because I'm afraid in such books, any other issue somehow gets "supergravitized"!
I think its better to learn something in the place it belongs to, not in the place where its used.(A natural generalization of dexterciboy's signature:D)

Thanks all

 

[haushofer]

Every sugra book should focus on this issue. Van Proeyen's book or his online lecture notes are good. Samtleben's notes are the most basic ones. But both are physicists :P

 

[dextercioby]

Differential Geometry has its own classical books: Lee's books, Spivak's books, 2 vols of Kobayashi and Nomizu and last but not least Husemoller. Books written especially for physicists: T. Frenkel's , either edition of Nakahara or Aldrovandi & Pereira.