7.6 Weinberg–Witten theorem

The Weinberg–Witten theorem [673Jump To The Next Citation Point] has often been interpreted as an insurmountable obstacle for obtaining massless spin-two excitations as effective degrees of freedom emerging from any reasonable underlying quantum field theory. However, the status of the Weinberg–Witten theorem [673Jump To The Next Citation Point] insofar as it applies to analogue models is rather subtle. First, note that whenever one’s main concern is in developing an analogue spacetime at the purely kinematic level of an effective metric, then the Weinberg–Witten theorem has nothing to say. (This includes, for instance, all analogue experiments probing the Hawking effect or cosmological particle production; these are purely kinematic experiments that do not probe the dynamics of the effective spacetime.) When one turns to the dynamics of the effective spacetime, desiring, for instance, to investigate quantum fluctuations of the effective geometry (gravitons), then one should bear in mind that the Weinberg–Witten theorem is derived under specific technical assumptions (strict Lorentz invariance in flat spacetime) that are not applicable in the current context. Furthermore, even if the specific technical assumptions are satisfied, then those authors state [673Jump To The Next Citation Point]:

Of course, there are acceptable theories that have massless charged particles with spin j > 1∕2 (such as the massless version of the original Yang–Mills theory), and also theories that have massless particles with spin j > 1 (such as supersymmetry theories or general relativity). Our theorem does not apply to these theories because they do not have Lorentz-covariant conserved currents or energy-momentum tensors, respectively.

Furthermore, when it comes to Sakharov-style induced gravity those authors explicitly state [673]:

However, the theorem dearly does not apply to theories in which the gravitational field is a basic degree of freedom but the Einstein action is induced by quantum effects.

That is: The Weinberg–Witten theorem has no direct application to analogue spacetimes – at the kinematic level it has nothing to say, at the dynamic level its applicability is rather limited by the stringent technical assumptions invoked – specifically exact Lorentz invariance at all scales – and the fact that these technical assumptions are not applicable in the current context. For careful discussions of the technical assumptions see [596, 366Jump To The Next Citation Point, 212, 404]. Note particularly the comment by Kubo [366]

… the powerful second part of the theorem becomes empty in the presence of gravity …

Finally we mention that, though motivated by quite different concerns, the review article [61] gives a good overview of the Weinberg–Witten theorem, and the ways in which it may be evaded.

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