Question

In quantum field theory, the fact that space-like separated observables commute, i.e. [ϕ(x)^,ϕ(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. {ϕ(x),ϕ(y)}=0. Why and how can this Poisson bracket be interpreted in terms of causality of the theory?

Answer 1

In classical field theory, the fact that the Poisson bracket of space-like separated observables is zero implies that there is no causal relationship between them.

Step-by-step explanation:

The Poisson bracket of space-like separated observables in classical field theory being zero can be interpreted in terms of the causality of the theory. Causality refers to the idea that cause and effect relationships between events should occur in a specific order, with cause preceding effect. In classical field theory, if the Poisson bracket of two observables is zero, it means that the two observables commute, or can be measured simultaneously, regardless of the spatial separation between them.

This implies that the values of the observables do not depend on each other's position in space. Therefore, if the Poisson bracket of two observables with space-like separation is zero, it suggests that the values of these observables do not influence each other, and thus, there is no causal relationship between them. This supports the idea that cause and effect relationships in classical field theory are independent of spatial separation.