### 7.6 Weinberg–Witten theorem

The Weinberg–Witten theorem [673] 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 [673] 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 [673]:
Of course, there are acceptable theories that have massless charged particles with spin
(such as the massless version of the original Yang–Mills theory), and also
theories that have massless particles with spin (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, 366, 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.