Dear all,
Suppose, [math]a,b \in R^2[/math].
Then, [math]a^{T} = \int_0^{2\pi} d\phi \delta(\phi-\phi_a)(\sin(\phi),\cos(\phi))[/math].
Hence, [math] a^{T}.b=\int_0^{2\pi} d\phi_1 \delta(\phi_1-\phi_a)\int_0^{2\pi} d\phi_2 \delta(\phi_2-\phi_b)\cos(\phi_1-\phi_2)[/math].
Now, [math]\cos(\phi_1-\phi_2)=sin(\frac{\pi}{2} \phi_1-\phi_2)[/math].
Hence, [math]\cos(\phi_1-\phi_2)=2\sin(\psi)\cos(\psi)[/math] and so, we may define
[math]A(\phi_1,\phi_2)=\alpha(\psi)\sin(\psi)[/math] and [math]B(\phi_1,\phi_2)=\frac{2}{\alpha(\psi)}\cos(\psi)[/math] with
[math]\psi=\frac{\pi}{4} \frac{1}{2}(\phi_1-\phi_2)[/math]
It can be demonstrated that there are [math]{\alpha \in [1,2]}[/math] such that [math]|A|\leq 1[/math] and [math]|B|\leq 1[/math].
Hence, a Bell form,
[math] a^{T}.b=\int_0^{2\pi} d\phi_1 \delta(\phi_1-\phi_a)\int_0^{2\pi} d\phi_2 \delta(\phi_2-\phi_b)A(\phi_1,\phi_2)B(\phi_1,\phi_2)[/math] has been found.
So LHV are not impossible when we consider that the setting parameters are averages of the probability densities.
Cheers
Han Geurdes