# Weinberg-Witten Theorem and Quantum Gravity

1. Nov 2, 2012

### stevendaryl

Staff Emeritus
Can someone who understands it explain the significance of the
Weinberg-Witten Theorem to the possibility of quantum gravity?
(Described here: http://en.wikipedia.org/wiki/Weinberg-Witten_theorem)

From the Wikipedia article, the conclusion is
I don't understand how that can be correct. Deser and Feynmann
(independently?) showed that General Relativity can be understood (at
least in asymptotically flat spacetimes) as the field theory of a spin-2
massless particle whose source is the total stress-energy tensor. I know
that this approach is not renormalizable, at least not in any obvious
way, but it seems to conflict with the Weinberg-Witten Theorem. Is the
WWT in some way a proof that the Deser/Feynmann theory is not
renormalizable?

One thing that is a little confusing about the Deser/Feynmann theory is
exactly what the stress-energy tensor is. The assumption is that the
spin-two particle couples to the total stress-energy, including the
stress-energy of the particle itself. This is apparently very different
from GR, in which the appropriate stress-energy tensor has no
contribution due to gravity. I don't have a good grasp of how this is
reconciled.

2. Nov 2, 2012

### atyy

http://arxiv.org/abs/1007.0435
"The Weinberg–Witten theorem states that a massless particle of spin strictly greater than one cannot possess an energy-momentum tensor Tμ which is both Lorentz covariant and gauge invariant. Of course, this no-go theorem does not preclude gravitational interactions. In the spin-two case, it implies that there cannot exist any gauge-invariant energy-momentum tensor for the graviton."

More discussion here:

http://pubman.mpdl.mpg.de/pubman/item/escidoc:33005:2/component/escidoc:33006/AnnPhys17-803.pdf
"However, in [24, 25] it is pointed out that the obtained pseudotensor of the gravitational ﬁeld is still Lorentz covariant, even though not generally covariant. Thus, at ﬁrst sight, the Weinberg-Witten theorem appears to be applicable to this tensor. However, the crucial point seems to be again that these pseudotensors are constructed in the context of a classical, i.e. a non-quantum theory. ...

"The process of constructing the energy-momentum pseudotensor for the gravitational ﬁeld in a classical theory only breaks general covariance, not Lorentz covariance. Constructing a quantum ﬁeld then requires additional terms under Lorentz transformations. To maintain Lorentz covariance of the theory, these additional terms would have to be eliminated by gauge transformations requiring general covariance. Like for the gluon, this gauge invariance of the current is not given for the graviton."

Last edited: Nov 2, 2012