I am reading Weinberg's first QFT book.

We looked for (and I suppose found) unitary representations of the Lorentz group:

$$U(\Lambda) = 1 - i (\vec{\theta}\cdot\vec{J}-\vec{\eta}\cdot \vec{K})$$

Later, we argued that massless particles must always be in a state such that

$$(J_2 + K_1)|\psi\rangle=0|\psi\rangle$$ $$(-J_1 + K_2)|\psi\rangle=0|\psi\rangle$$

because, as we saw, massless states would have two continuous degrees of freedom if this were not true (pg 71). But such d.o.f have not been seen in experiment. So we concluded, in order for our theory to be consistent, that all existing photons must be in a simultaneous eigenstate of $A=J_2+K_1$ and $B=-J_1+K_2$ with eigenvalue $0$.

I am wondering if it is experimentally verifiable that in fact photons are always in such a state. These definitions of A and B are hermitian, so their sums are as well - this means to me that it may be possible to measure them, no? Has such an attempt been made, in order to confirm that photons do not have continuous internal degrees of freedom?


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