This text on QFT defines a free theory as that in which dynamics of the field for each degree of freedom evolves independently from all the other. In principle we have an infinite degrees of freedom, at least one one for every point in space. KG equation is the simplest example.
Locality is defined as when there are no terms in the Lagrangian coupling field at one point in space directly to the field at another point in space.
Causality is defined as when all spacelike separated operators commute so that a measurement at a point in spacetime cannot affect measurement at another point in spacetime when they are not causally connected. Slightly more sophisticated: in any measurement the amplitudes of events in which particles travel between these two points cancel.
Is this causality condition equivalent to Lorentz invariance?
Now the question is which of the conditions above is more strong.
I mean is a free theory necessary and sufficient condition for causality or locality.
How about causality and locality, which one implies the other?
If possible I'd prefer to see a mathematical proof using definitions above.