General Covariance in Quantum Field Theory

[Newton-reborn]

TL;DR Summary

If Lorentz invariance is a special case of general covariance, why are all laws of physics required to have Lorentz invariance instead of general covariance which is more general?

All physical laws have to be Lorentz invariant according to a lecture I just watched. Why is general covariance (which is more general than Lorentz invariance) not a requirement for all laws of physics? Are there any quantum gravity theories that take the approach of adding general covariance to quantum field theory? Please recommend sources if there are any.

 

[vanhees71]

This you'd only need if you'd do QFT in a general curved spacetime, which I think doesn't exist because it's awfully complicated. Of course such a thing exists for some not too complicated spacetimes. AFAIK, a standard reference for this very challenging topic is

https://doi.org/10.1016/0370-1573(75)90051-4

 

[Tendex]

There's a classical notion of general covariance in general relativity (non-quantized yet for the reasons mentioned in the previous post).

 

[PeterDonis]

AFAIK, a standard reference for this very challenging topic

Wald's monograph, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, is more recent (1993) and AFAIK gives more detailed coverage. I suspect there are good references that are even more recent as well; there has been quite a bit of important theoretical development on this topic in the last couple of decades.

 

[PeterDonis]

All physical laws have to be Lorentz invariant according to a lecture I just watched.

More precisely, all physical laws have to be locally Lorentz invariant. In a curved spacetime, there is no such thing as global Lorentz invariance since there are no such things as global Lorentz transformations.

Why is general covariance (which is more general than Lorentz invariance) not a requirement for all laws of physics?

As far as global coordinate charts and transformations are concerned, it is.

 

[PeterDonis]

a lecture I just watched.

Please give a reference.

 

[vanhees71]

Wald's monograph, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, is more recent (1993) and AFAIK gives more detailed coverage. I suspect there are good references that are even more recent as well; there has been quite a bit of important theoretical development on this topic in the last couple of decades.

That's why I wrote "AFAIK". I'm not an expert in this field. I looked at it some years ago, and found it too complicated given the little direct application, though it's of course utmost interesting from a fundamental point of view.

 

[vanhees71]

There's a classical notion of general covariance in general relativity (non-quantized yet for the reasons mentioned in the previous post).

Sure, but the question was about QFT.

I think to really understand the meaning of "general covariance" in classical GR, one should think about it as a gauge theory, where "gauging" is meant in the sense HEP theorists use this notion, making the Lorentz invariance local. A very good treatment from this point of view is in

P. Ramond, Quantum Field Theory, 2nd Ed.

 

[Tendex]

Sure, but the question was about QFT.

I think to really understand the meaning of "general covariance" in classical GR, one should think about it as a gauge theory, where "gauging" is meant in the sense HEP theorists use this notion, making the Lorentz invariance local. A very good treatment from this point of view is in

P. Ramond, Quantum Field Theory, 2nd Ed.

Yes, that's how I understand general covariance too, I was thinking about the local LI of QFT in analogy with the local LI of GR as a gauge theory. Unfortunately all efforts to quantize GR are fruitless so far.

 

[haushofer]

There is a general argument that general covariance restricts Green's functions depending on observables to be constant, making it hard to define local operators in general covariant QFT's. See e.g. Zee's GR book, appendix 6 of chapter X.8.

 

[A. Neumaier]

More precisely, all physical laws have to be locally Lorentz invariant. In a curved spacetime, there is no such thing as global Lorentz invariance since there are no such things as global Lorentz transformations.

Indeed, general relativity in the tetrad formalism (essential for spinor fields) requires both local Lorentz covariance and covariance under general coordinate transformations.

Wald's monograph, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, is more recent (1993) and AFAIK gives more detailed coverage. I suspect there are good references that are even more recent as well; there has been quite a bit of important theoretical development on this topic in the last couple of decades.

A survey article by Hollands and Wald from 2015 brings this up to date. But here the spacetime manifold is fixed, not dynamical. The unsolved problems in quantum gravity are about a dynamical formulation.

 

[A. Neumaier]

The unsolved problems in quantum gravity are about a dynamical formulation.

A two volume treatise

• Bryce DeWitt, The global approach to quantum field theory, Oxford Univ. Press 2003.

discusses the canonical approach to dynamical quantum gravity in some depth, based on Schwinger's quantum variational principle. Very worthwhile reading. See also this new thread.