Quantum Causality: Pauli's Definition Explained

[Maximise24]

In a 1940 article in the Physical Review Wolfgang Pauli provides a definition for quantum causality: it is 'implemented microscopically by the requirement that observables commute at spacelike separations'.

I find this confusing. Doesn't spacelike separation by definition exclude causal relationships between events? And how exactly does operator commutativity relate to causality? Might someone be able to offer a qualitative interpretation of this statement?

 

[Bill_K]

Simple example: angular momentum. Suppose a system is initially spin up, J_z = +1. Now measure J_x. But J_z and J_x do not commute, so they have no simultaneous eigenstate. That means that J_z is no longer guaranteed to be +1. Measuring one has influenced the other, and that happened, basically, because they do not commute.

Now suppose instead of angular momentum we consider the Hamiltonian at two points A and B, with a spacelike separation between them. If they don't commute, measuring H at point A can affect H at point B. An influence has just traveled from A to B faster than light. This is obviously unphysical, so we must assume that H at A and H at B do commute, or else we will violate causality.

 

[DrChinese]

Doesn't spacelike separation by definition exclude causal relationships between events? And how exactly does operator commutativity relate to causality? Might someone be able to offer a qualitative interpretation of this statement?

When using the phrase "quantum causality", there is no limitation on distance. So spacelike is not a factor. For that matter, timelike is not a factor either. An example of this would be the measurement of a member of an entangled pair here, which "quantum causes" the other member there (spacelike separated) to have a suitable matching value for the same observable. Such quantum causality may even be observed when particles have never interacted. Or even existed at the same time. In all cases, the uncertainty relations are observed (in consonance with Pauli's comments I suppose - not sure if my comments really address your question).

Please keep in mind that this avoids any discussion of what the word "causality" means itself. Since of course, the outcome of any quantum interaction appears to be random and without any underlying cause.

 

[PeterDonis]

An example of this would be the measurement of a member of an entangled pair here, which "quantum causes" the other member there (spacelike separated) to have a suitable matching value for the same observable.

But this "quantum causality" works the same regardless of which measurement occurs first; in other words, the results are insensitive to the order in which the measurements are made, which is equivalent to saying that the measurements must commute. I think that's what Pauli was referring to. (This is also equivalent to saying that you can't use this "quantum causality" to send information faster than light, as Bill_K pointed out.)

 

[DrChinese]

But this "quantum causality" works the same regardless of which measurement occurs first; in other words, the results are insensitive to the order in which the measurements are made, which is equivalent to saying that the measurements must commute. I think that's what Pauli was referring to. (This is also equivalent to saying that you can't use this "quantum causality" to send information faster than light, as Bill_K pointed out.)

Sure, the ordering is not material. Quantum causality is causality of quite a different kind!

 

[Maximise24]

Thank you for your answers!