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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.

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    $\begingroup$ What textbook are you using? $\endgroup$ – Okazaki Jan 22 '16 at 10:47
  • $\begingroup$ David Tong lecture notes here damtp.cam.ac.uk/user/tong/qft.html $\endgroup$ – user56963 Jan 22 '16 at 10:50
  • $\begingroup$ What? Of course "freeness" is not necessary for causality, otherwise, what would be the point of causality? Why would causality be equivalent to Lorentz invariance? Causality is one of the Wightman axioms which every QFT "should" fulfill, but what would it have to do with Lorentz invariance? $\endgroup$ – ACuriousMind Jan 22 '16 at 13:55
  • $\begingroup$ I think this is a interesting question - can one construct a simple example of a causal (operator commutativity on spatial slices) field theory which is not Lorentz invariant? $\endgroup$ – zzz Jan 24 '16 at 1:59
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"Is this causality condition equivalent to Lorentz invariance?" No. Lorentz invariance ensures that points in spacetime that are spacelike (timelike) separated in one frame stay spacelike (timelike) in all inertial frames. Causality, in this context, is the notion that an event can not have effect at any spacelike separated point in spacetime. "Now the question is which of the conditions above is more strong. " Define strength. "I mean is a free theory necessary and sufficient condition for causality or locality." Freeness is unrelated to causality. However, a free theory is local by definition --- there are no interactions that couple two different points in spacetime. It is not necessary for a theory to be free to be local --- interactions that couple fields at the same point in spacetime will make it non-free but keep it local.

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