# Martin Kober: The relation of a unified QFT of spinors to the structure of GR

1. Dec 28, 2008

### Hyperreality

As suggested by Marcus, I have read the whole paper and decided to start a discussion of Kober's paper by asking a few questions.

The paper certaintly presents a very interesting view. Part of it is an extension of Heisenberg's work on unified QFT, that the mass originates from self-interaction of spinor fields. However, this theory has chosen the Minkowski metric as a priori, so that it takes the view that the structure of spacetime is more fundamental than the matter fields. Therefore, the thoeory is background dependent.

While Kober in the paper takes the view that mass are generated through self-interaction, he differes from Heisenberg in that the spinor fields are the most fundamental in nature. He does this by constructing the spacetime metric tensor $$g_{\mu\nu}$$ from tetrads. And the tetrad are constructed from two two-component spinors $$\varphi$$ and $$\chi$$ so that, $$g_{\mu\nu}=g_{\mu\nu}(\varphi,\chi)$$.

Using the constructed metric tensor, he then writes down the action for both matter and gravity. Since my knowledge on gravity is limited, I do not have much to say about it.

One interesting thing arises here is that, because the spinor fields $$\varphi$$ and $$\chi$$ are more fundamental than the metric tensor $$g_{\mu\nu}$$. So to quantise $$g_{\mu\nu}$$ one only needs to impose the canonical anticommutation relations on $$\varphi$$ and $$\chi$$.

In my opinion, the essence of this paper is the proposed construction of the metric tensor $$g_{\mu\nu}$$ from spinors, and the view that the spacetime is a consequence of the matter field. This is essentially a "relationalistic" view of spacetime, and I think it is also the view adocated by Julian Barbour in his book the "End of Time".

Another thing that interested me is that the following statements

1. Tetrads can be constructed from spinors

2. The metric tensor can be written in terms of the tetrads

seems to be text-book stuff for knowlegeable people in the field (not me). If so, why hasn't anyone thought of this, or is there some previous objections to this formulation?

2. Dec 28, 2008

### marcus

Thanks for starting a thread on this. It's hard to know in advance how much interest there will be. AFAIK it's a new idea. Kober is a young guy, postdoc I believe. It was John86 who spotted this and called our attention to it:
Hopefully he will check in and notice this thread.

I'll copy the abstract here for easy reference:

http://arxiv.org/abs/0812.0713
The relation of a Unified Quantum Field Theory of Spinors to the structure of General Relativity
Martin Kober
(Submitted on 3 Dec 2008)

"Based on a unified quantum field theory of spinors assumed to describe all matter fields and its interactions we construct the space time structure of general relativity according to a general connection within the corresponding spinor space. The tetrad field and the corresponding metric field are composed from a space time dependent basis of spinors within the internal space of the fundamental matter field. Similar to twistor theory the Minkowski signature of the space time metric is related to this spinor nature of elementary matter, if we assume the spinor space to be endowed with a symplectic structure. The equivalence principle and the property of background independence arise from the fact that all elementary fields are composed from the fundamental spinor field. This means that the structure of space time according to general relativity seems to be a consequence of a fundamental theory of matter fields and not a presupposition as in the usual setting of relativistic quantum field theories."

3. Jan 2, 2009

### Blackforest

Well happy new year to everyone here.

As not suggested by Marcus, that is only because of some intellectual curiosity, I decided to read this article too. I must say: it was an appreciated help for my self. The stuff gave me the energy to reorganize my own construction.

The problem I have is the following: I also get a metric induced by the spinor connection but in a quite different approach. Furthermore I don't need a differential structure at the beginning. The former is just needed to get a GTR conform metric. So much on a personal approach which is never wellcome here. It was just to bring some new stuff into this approach of Kober. Very interesting I must say.

Blackforest

4. Jan 3, 2009

### schieghoven

http://arxiv.org/abs/0812.0713
The relation of a Unified Quantum Field Theory of Spinors to the structure of General Relativity
Martin Kober
(Submitted on 3 Dec 2008)

The tetrad formulation of GR seems to have some significant traction, because it's the only way to define spinor fields on a curved manifold (eg. [1] for a review). Tetrads also make GR look like a gauge theory, which (I think) is quite exciting. [2] has a section about tetrads and includes some good references.

On the other hand, writing the tetrad in terms of spinors is intriguing, I haven't heard of it before. In the sense that the tetrad is a 'square root' of the metric, these spinors provide a sort of 'square root' of the tetrad, or a fourth root of the metric. Interesting.

Can anyone suggest why the authors chose the action (32)? It looks a bit like the Dirac Lagrangian. The immediate question is what the equations of motion for the spinors look like, but it seems the authors weren't able to get them out (page 7). Why not the Einstein-Hilbert action (30)? Surely it's the EH action that will give dynamics equivalent to Einstein general relativity.

Excellent find, by the way. Cheers for the link.

Dave

[1] Brill & Wheeler, Rev Mod Phys 29 465 (1957)
[2] Chamseddine hep-th/0511074 (2005)

5. Jan 9, 2009

### Blackforest

Thanks for the recommanded lectures.

[2] is a very good suggestion. It explains very clearly the "birth" of Ashtekar's work, that means why one can describe the GTR with the tetrads (soldering forms). It also sketches in a very comprehensible language the importance of the relationship between the tetrads and the spinor connections.
By side I get a new manner to relate them together. The small bonus that can be won from that original approach is a possibility to also write some EM fields with the components of the spinor connection only. This is thus completing the work of Kober.
I appreciate the exposé of Einstein's idea which is allowing the introduction of complex numbers into the theory.

So, I have still a lot to learn before continuing the discussion with you.

6. Jan 12, 2009

### Blackforest

First of all I want to apologize if I ask now some stupid question. The fact is that I get in trouble with [02; page 6] because you can read on the same page that ..."Setting the torsion to zero allows to solve for the spinoral connection uniquely provided that the soldering forms are invertible"... and ..." The number of independant components in the torsion matches the number of independant components in the spinoral connection"...

Please stop me if I misunderstand something. Within the GTR: the torsion must vanish. Thus there is only one independant component for the torsion and it owns exactly the value 0. Does it mean that the theory which is exposed in [2; § 2] mainly applies for matter and is not really relevant for vacuum ?

Thanks for explainations.