Causality: Two definitions The general definition of causality is that the principle of the 'effect never occuring before the cause', as in Wikipedia.
The book 'Picturing Quantum Processes' (pg: 262) defines causality as states or processes having the following property:
Discarding the state or process is equivalent to them never having occured.
Are these two definitions equivalent?
 A: Any question about causality itself is addressing an issue at the foundations of science, and touches on the philosophy of science. This should warn you that there will be no quick or simple answer to such a question, if one is seeking a rigorous definition. But we can get a pretty good working definition of causality, good enough for most purposes in science, if we focus on notions such as "If A had not happened then B would not have happened". Or "if one were to change A, then B would subsequently change". In physics we find that such causation never happens faster than the maximum speed for signals (called the speed of light) but here too there are some subtleties.
For example, in some situations one can define useful quantities such as the scalar and vector potentials in electromagnetism, and sometimes influences propagate in these potentials instantaneously from one place to another, but only in such a way that the change in the electromagnetic field takes place in the ordinary speed-limited way. In this example one is using a faster-than-light mathematical tool in order to calculate a light-speed-limited physical process. Owing to cases like this it is not always easy to say what kind of thing we mean by "B" when we say "B was caused by A". "B" needs to be detectable or observable in some way, which amounts to saying that it in turn should be a cause of further phenomena, if we are to arrive at a clear notion of "A caused B."
Such issues comes up in quantum theory, which is no doubt why the author of the book you mentioned chose to say something about causation, with a view to defining it. But the definition you mentioned in your question is much too brief, one cannot make any sense of it.
A: I'm not sure what to think of either of those definitions. In the theory of relativity, causality refers to the fact that what happens at one event (a point in space and time) cannot influence the outcome of another event occurring outside of the light cone of the first event. Basically, this is just a result of the fact that nothing may travel faster than light in the theory of relativity. Therefore, if two events are so far apart in space and time that even light cannot travel between them, one event cannot affect the other.
I suppose one could phrase this as "the effect never occurring before the cause," but this is a bit of an oversimplification. In relativity, there is no absolute notion of events being simultaneous. That's why we need to make sure the "effect" always occurs within the light cone of the "cause." Otherwise, there will exist a reference frame in which the effect comes before the cause.
When we add quantum mechanics into the picture, causality becomes a little more complicated. The wave function of a particle can leak outside of the light cone. However, there is still a sense in which causality is preserved. No measurement that you perform on a particle can affect the outcome of a measurement made outside of the light cone. This is proved within the framework of quantum field theory (I don't think it's true in "classical" quantum mechanics).
I'm not sure what you mean by: "Discarding the state or process is equivalent to them never having occured." That sounds like the definition of "discarding" to me. Maybe if you could clarify, I would be able to comment.
