I took some trouble with my own answer to the following question on Physics StackExchange. So I have copied the details here so it doesn't just get buried. However, all the answers at the question are worth a look.
"I have never seen a “causality operator” in physics. When people invoke the informal concept of causality aren’t they really talking about consistency (perhaps in a temporal context)?
For example, if you allow material object velocities > c in Special Relativity (SR) you will be able to prove that at a definite space-time location the physical state of an object is undefined (for example, a light might be shown to be both on and off). This merely shows that SR is formally inconsistent if the v <= c boundary condition is violated, doesn’t it; despite there being a narrative saying FTL travel violates causality?
Note: this is a spinoff from the question: The transactional interpretation of quantum mechanics."
As the originator of this question I have reviewed and learned something from all the answers posted so far. I would like to summarise my own views here.
Where did this query come from? From a question about the Transactional Interpretation of Quantum mechanics (TIQM), where said theory’s reliance upon “retrocausality” (‘causality’ backwards in time) was held to be a fatal defect. This kind of causality-argument is common in physics: we say that faster-than-light travel in SR is ruled out because it would violate causality.
Such causality arguments are conducted in what you might call the “metalanguage of physics”: technical English which supports and explains formal results. However, the arbiter in the end is the maths, so how do we interpret the notion of causality within the formalism?
Physical theories are defined by mathematical relationships between entities (observed and unobserved) usually expressed by equations (think Schrodinger, Dirac, the Lorentz transformation). If we say that event E1 “causes” event E2, several answers here suggest that the interpretation of causality in the formal theory is that: (i) if E1 is postulated to occur then the theory logically implies that E2 must occur as well; (ii) E2 is within or on the future light-cone of E1 (we say “cause precedes effect”).
However, it’s possible that condition (ii) is too stringent. While logical entailment is obviously an essential part of any formalised theory, our smuggling in of the word “future” is already an extra assumption. Our fundamental theories do not impose a specific past-future direction on the time dimension. This means that if you reverse the film, the events you see are still consistent with our fundamental theories.
Sometimes people use causality-like words in the physics metalanguage without conventional time-ordering condition (ii). For example, a possible Feynman diagram for electron-positron scattering has a narrative that an electron travelling backwards in time from the future encounters a (normal) electron, they exchange a virtual photon and continue on their way scattered. The 'cause' of the scattering event was the arrival of the future electron. Many textbooks mention this way of thinking but we don’t mind because the underlying theory gives consistent results which accord with observation. Perhaps TIQM is like this despite its narrative of retrocausation.
So my conclusion is that we have to be careful about arguments concerning a theory’s validity relying upon causation arguments couched in physics’ metalanguage. It’s not a slam-dunk. Sometimes if a theory violates conventional “cause precedes effect” causation it indicates a breakdown in the underlying mathematics, normally inconsistency. At other times a 'causation' argument is just a way of talking about the entailment of the theory in an innovative or whimsical way, and the theory is actually OK. Go look at the maths.
NOTE: there is a whole separate discussion about why, in natural language, we think so naturally in terms of cause and effect. It links to discussions about the arrow of time and why we do seem to be unhappy about running the film backwards as a valid picture of reality. That is a whole separate issue but still, I suspect, part of physics judging by the number of recent books on the subject.