# Covariant quantum field theories

1. Jun 28, 2007

### Jim Kata

I am not as well read as most of the people in here so I thought I would ask you guys first. What work has been done in the way of developing a covariant field theory? I'm going to ramble for just a little bit so try to follow. It seems to me that QFT is built on two principles Poincare invariance and the Structure Decomposition Principle. A covariant field theory, in my opinion, would be built on basically the same two things except this time it would be General covariance and the Structure Decomposition Principle. The structure decomposition principle implies, in my mind, the need for creation and annihilation operators, and the covariant part would require the generalization of the concept of a field. I am referring to this generalization as a covariant field. The homogeneous Lorentz group is O(3,1) and the corresponding generally covariant homogeneous group, at least locally, is GL(R,4). I guess my question is: is GL(R,4) unitarizable? Have people come up with the concept of a covariant field? If they have, why did it not work? Is it non-renormalizable, power counting sense? The reason I ask is because I think I have a covariant field theory. It's not completely done yet.

2. Jun 28, 2007

### olgranpappy

Perhaps you might find the textbook "Quantum Field Theory in Curved Spacetime and Blackhole Thermodynamics" by Wald useful.

3. Jun 29, 2007

### Demystifier

I think that Jim Kata has something different in mind.

By the way, I was also involved in a problem of constructing a covariant field theory:
http://xxx.lanl.gov/abs/hep-th/0407228
http://xxx.lanl.gov/abs/hep-th/0601027
but it does not seem to be what he has in mind either.

4. Jul 1, 2007

5. Jul 1, 2007

### Haelfix

Im a little confused by what you mean Jim. Write down the difference between a lorentz invariant field theory and a covariant field theory explicitly. I think we have different terminology in mind.

My main objection to not using SL(2,C) as the main isometry group of spacetime, is the fact that the irreducible unitary representations of whatever else you use (if they even exist) would either not match the known particle spectrum and properties, or imply extra unobserved transformation properties.

Last edited: Jul 1, 2007
6. Jul 2, 2007

### Jim Kata

Thanks for the help

I posted this question on a different forum, and didn't really get a good reply, but I typed up my argument in Latex on there. So instead of re-typing the whole thing, I'll just give you the url to the question:

I have way more than what I've written in this link, and I am working on typing it in post script right now, but I suck at Latex so it's going to take me a while to finish it. But when I do, I'll let you take a look, and tell me a.) what I did wrong or b.) who did it before me.

Another question, what tags to do you use to write LaTeX in these forums?

7. Jul 2, 2007

### Hans de Vries

It seems you are reading Weinberg volume I, (which is not really advisable
as an introduction...) Weinberg uses the term "Structure Decomposition
Principle" to state that experiments done in CERN are not correlated to
experiments done in Stanford at the same time.

All Quantum Field Theories involving interaction are covariant theories, and
only free particles are Lorentz (Poincaré) invariant without covariant terms.
I do not see why the General Linear groups should be related to covariant
theories.

Regards, Hans

Last edited: Jul 2, 2007
8. Jul 2, 2007

### Hans de Vries

Covariance results in the most elementary sense from locality. To explain
this a bit look at the following: Let a photon with a linear polarization in
the x direction $\epsilon = (0, A_x, 0, 0)$ arrive at the wave function of an electron
at rest. How will this modify the phase of the electron's wave function in
the x-direction?

Answer: It doesn't, because of locality. The increase in spatial frequency
is nullified by the covariant interaction term:

$$\partial_x \psi - ieA_x \psi \ =\ 0$$

One expects the charge density to be accelerated in the x-direction
as a result of the changing vector potential $E = -\partial A_x /\partial t$, it does indeed
accelerate but the corresponding spatial deBroglie frequency is nullified.

If this wasn't the case then the phase would be dependent on an
instantaneous integration over all of x, meaning that arbitrary distant
events would be of direct influence on the phase of the wave function.

This is similar to the well known fact that a charge in a central potential
has an energy which is independent of the local potential. A particle
descending in the potential well acquires a kinetic energy equal but
opposite to the lowered potential energy.

$$\partial_t \psi - ieV \psi \ =\ 0$$

In the latter case there is a phase change in space for instance
corresponding with the angular momentum. In the first case the
energy of the electron does change as well as the phase in the
z-direction.

This all means that the wave-function of the electron is not longer
Lorentz invariant, but it does become so again when combined
together with the interaction term. Covariant stands for: together
being Lorentz invariant.

Technically the derivatives have to be replaced by the covariant
derivatives to make the interacting theory Lorentz invariant again:

$$\partial_\mu \psi \ \rightarrow\ \partial_\mu \psi - ieA_\mu \psi$$

Regards, Hans

Last edited: Jul 2, 2007
9. Jul 4, 2007

### Jim Kata

response to Hans

This is in response to Hans. What me and you are talking about are two different things, but in the same jest. What you wrote down is called the covariant derivative, but that's not the meaning of covariant to which I am referring. Maybe, I used poor terminology, but I referring to covariant in the sense that a vector transforms like $$\bar p^\alpha = \frac{{\partial \bar x^\alpha }}{{\partial x^\beta }}p^\beta$$. The reason I'm working with $$Gl(4,\mathbb{R})$$ and not $$\frac{{SL(2,\mathbb{C})}}{{Z_2 }}$$ is because I'm considering coordinate changes of the the form
$$\frac{{\partial \bar x^\alpha }}{{\partial x^\beta }}$$ not Lorentz transformations. And to my knowledge the group that represents $$\frac{{\partial \bar x^\alpha }}{{\partial x^\beta }}$$ is $$Gl(4,\mathbb{R})$$. To illustrate what I mean by a covariant field theory, consider the following. All massive fields obey the the Klein Gordon equation $$[\partial ^\mu \partial _\mu - m^2 ]\psi _l$$, well a covariant field theory would obey the same equation except you'd have $$[g^{\mu \tau } \partial _\mu \partial _\tau - m^2 ]\psi _l$$ not
$$[\eta ^{\mu \tau } \partial _\mu \partial _\tau - m^2 ]\psi _l$$. I think my real problem is that I am not very well read on the history of the approaches to quantum gravity. I am sure this has been tried before.

10. Jul 5, 2007

### Haelfix

Hi Jim, theres something a little fishy about this business, but im not going to dwell on it b/c I think I might be off. It seems to me Gl(4,R) wants a linear non symmetric connection to transform properly, but again I think I have something different in mind.

Anyway, I think what you are trying to do is to extend the poincare group to the general affine group GA(4,R) ~ GL(4,R) + translations.

I looked up a reference and it seems Y Ne'eman has done a lot of work on this subject. You can try this:

Ne'eman et al, GL(4,R) (bar) group topology, covariance and curved space spinors Int J Mod physics. A2 (1987) 1655

I gather the difficulty with the whole program is getting the double cover to work out for fermions, but theres probably a reference there to the simpler boson case. Its likely old stuff tho, so you'd have to do some journal hunting. Good luck

Last edited: Jul 5, 2007
11. Jul 9, 2007