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What does Weinberg–Witten theorem want to express?

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[Weinberg-Witten theorem](http://en.wikipedia.org/wiki/Weinberg%E2%80%93Witten_theorem) states 
that massless particles (either composite or elementary) with spin $j > 1/2$ cannot carry a Lorentz-covariant current, while massless particles with spin $j > 1$ cannot carry a Lorentz-covariant stress-energy. The theorem is usually interpreted to mean that the graviton ( $j = 2$ ) cannot be a composite particle in a relativistic quantum field theory.

Before I read its proof, I've not been able to understand this result. Because I can directly come up a counterexample, massless spin-2  field have a Lorentz covariant stress-energy tensor. For example the Lagrangian of massless spin-2 is massless Fierz-Pauli action:

$$S=\int d^4 x (-\frac{1}{2}\partial_a h_{bc}\partial^{a}h^{bc}+\partial_a h_{bc}\partial^b h^{ac}-\partial_a h^{ab}\partial_b h+\frac{1}{2}\partial_a h \partial^a h)$$

We can calculate its energy-stress tensor by $T_{ab}=\frac{-2}{\sqrt{-g}} \frac{\delta S}{\delta g^{ab}}$, so we get

$$T_{ab}=-\frac{1}{2}\partial_ah_{cd}\partial_bh^{cd}+\partial_a h_{cd}\partial^ch_b^d-\frac{1}{2}\partial_ah\partial^ch_{bc}-\frac{1}{2}\partial^ch\partial_ah_{bc}+\frac{1}{2}\partial_ah\partial_bh+\eta_{ab}\mathcal{L}$$ 
which is obviously a non-zero Lorentz covariant stress-energy tensor.

And for U(1) massless spin-1 field, we can also have the energy-stress tensor $$T^{ab}=F^{ac} F^{b}_{\ \ \ c}-\frac{1}{4}\eta^{ab}F^{cd}F_{cd}$$
so we can construct a Lorentz covariant current $J^a=\int d^3x T^{a 0}$ which is a Lorentz covariant current.

Therefore above two examples are seeming counterexamples of this theorem. I believe this theorem must be correct and I want to know why my above argument is wrong.

asked Apr 20, 2016 in Theoretical Physics by Alienware (185 points) [ no revision ]

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