Special relativity has changed the way we think about time and the order of events. Einstein famously showed that two events can look simultaneous to one observer but not to another. In fact, it’s possible to make two spatially separated events appear in any order by choosing to view them from different frames of reference . 

But what of three or more events? “Do all possible time orderings occur in some frame or other? If not, what are the restrictions?” ask Alfred Shapere at the University of Kentucky in Lexington and Frank Wilczek of the Massachusetts Institute of Technology in Cambridge. Today, they give us the answer.

Their reasoning  turns out to be simple. Given three events–A, B and C–six possible orderings are possible. These events occur in a universe with three dimensions of space and one of time. In this four dimensional world, it’s always possible to connect theses events by a triangle lying on a two-dimensional plane.

Shapere and Wilczek show that if the plane of this triangle is entirely spacelike–in other words, if it is perpendicular to the dimension of time–then there is a frame of reference in which the events are simultaneous and small transformations can produce all  six orderings.

But if the plane is timelike in any way, then certain orderings become impossible. 

They go on to generalise this result for any number of events in any number of dimensions for both flat and curved spacetimes.

And with a final flourish, they show how to do the same trick backwards: they work out the properties of a spacetime given the nature of the chronologies that it allows. 

That may have interesting implications. The chronology of events plays an important role in quantum mechanics.  For example, imagine three spatially separated events A, B and C. One observer might describe these events using a wave function in which A precedes B. In that case, a measurement at A collapses the wave function accessible to B.

However, another observer can describe the events using a wave function in which B precedes A. In this case, a measurement at B collapses the wave function accessible to A. 

No suppose that the chronology of the events is constrained so that A can never precede both B and C. 

“Then we have the peculiar situation that a measurement at A can cause collapse at either B or C, in different frames, but never both,” say Shapere and Wilczek. 

That’s not a paradox but it is strangely artificial. Clearly there is some deficiency in the logic physicists use to think about quantum mechanics in this way.

Shapere and Wilczek say they’ve developed this new approach to chronology specifically to clarify the logic of quantum mechanics. “We plan to elaborate further on the implications of chronology conditions for the logic of quantum theory in future work,” they say.

Using the chronology of special relativity as a testing ground for the logic of quantum mechanics may lead to interesting results. Most exciting would be some kind of conflict between the two theories which make different predictions. That would allow an experimental test. 

Of course, there is already a conflict of sorts between special relativity and quantum mechanics in the form of entanglement, the strange quantum phenomenon in which two bodies share the same wavefunction even though they are spatially separated.    

The fundamental nature of entanglement is one of science’s great mysteries. Perhaps Shapere and Wilczek’s new chronological approach will help.

Ref: arxiv.org/abs/1208.3841: Constraints on Chronologies