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In quantum field theory, the fact that space-like separated observables commute, i.e. $[\hat {\phi (x)}, \hat{\phi(y)}]=0$, is taken as the test for causality. The equivalent statement for classical fields would be that the Poisson bracket of space-like separated observables is zero, i.e. $\{ \phi (x), \phi (y)\}=0$. Why and how can this Poisson bracket be interpreted in terms of causality of the theory?

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Poisson brackets measure the linear independence of two function over classical phase space on the basis of the dependence $m \dot x = p$ in the free case. It boils down to $[x_i,p_k] = \delta_ik$ as always in the differential algebra.

This fact. applied to characteristic functions of volumes in space time at equal time, $$[\chi_{(a,b)(x,p),\chi_{(c,d)(x,p) ]$$ with overlap $\emptyset$ says, these two functions don't interfere and a single particle or a group can be prepared in an ensemble with different state values and probabilties without entangling.

The two phase space volumes then evolve until the they overlap in the future and interact for the following future.

This is - as Pauli says - the reason, why the electron behind the moon can, but is not obliged to take part in the anitisymmetrization concept of fermions. Spaclike separation generates tensor products and statistical independence of the factors.

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