Interactions between field operators & locality in QFT

["Don't panic!"]

Why is it required that interactions between fields must occur at single spacetime points in order for them to be local? For example, why must an interaction Lagrangian be of the form $$\mathcal{L}_{int}\sim (\phi(x))^{2}$$ why can't one have a case where $$\mathcal{L}_{int}\sim\phi(x)\phi(y)$$ where $x^{\mu}$ and $y^{\mu}$ are time-like separated?

Is it simply because the fields themselves are localised, i.e. they are described in terms of their values at each spacetime point, and as such to avoid action-at-a-distance they can only interact with one another when they are located at the same spacetime point (as otherwise two fields $\phi(x)$ and $\phi(y)$ at distinct spacetime points $x^{\mu}$ and $y^{\mu}$ could spontaneously interact with one another which would constitute action-at-a-distance)? Hence if they were located at two distinct spacetime points that are time-like separated, then any direct interaction between them would still constitute action-at-a-distance, as they would be able to spontaneously influence one another despite not being at the same spacetime point?

 

[mfb]

If things from the future interact with things in the present, you violate causality.
Not completely impossible, but also nothing that experiments would suggest.

 

["Don't panic!"]

If things from the future interact with things in the present, you violate causality.
Not completely impossible, but also nothing that experiments would suggest.

So is the requirement that interactions occur at single spacetime points as much a statement about causality as it is about locality?
Would it be correct to say that the requirement that causality is obeyed, along with Lorentz invariance, requires that interactions are local and that any direct interaction between two fields can only occur when the fields are located at the same spacetime point?

 

[mfb]

For timelike separations it is causality, for spacelike separations it is locality. I don't think there is a special word for lightlike separation.

Would it be correct to say that the requirement that causality is obeyed, along with Lorentz invariance, requires that interactions are local and that any direct interaction between two fields can only occur when the fields are located at the same spacetime point?

I think so.

 

["Don't panic!"]

For timelike separations it is causality, for spacelike separations it is locality. I don't think there is a special word for lightlike separation.

So is the idea that interactions should be both local in time and space in order to ensure causality and locality are both satisfied?I've had a think about and have come up with the following argument as to why interactions must occur at single spacetime points. It seems to make sense to me, but I'd really appreciate you taking a look at it and letting me know what you think?

First, consider two points $x^{\mu}$ and $y^{\mu}$ that are timelike separated. In such a case it is possible to find a frame in which the two events are located at the same spatial point (but impossible to find one in which there occur at the same time). We see explicitly then that if two fields, located at $x^{\mu}$ and $y^{\mu}$ respectively, are allowed to interact directly, then one would be able to influence the other instantaneously and without any mediation despite their temporal separation. Thus timelike separated fields cannot interact directly.

Secondly, consider two points $x^{\mu}$ and $y^{\mu}$ that are spacelike separated. In such a case it is possible to find a frame in which the two events occur at the same time (but impossible to find one in which there located at the same spatial point). We see explicitly then that if two fields, located at $x^{\mu}$ and $y^{\mu}$ respectively, are allowed to interact directly, then one would be able to influence the other instantaneously and without any mediation despite their spatial separation. Thus spacelike separated fields cannot interact directly.

Finally, consider two points $x^{\mu}$ and $y^{\mu}$ that are lightlike separated. In such a case it is impossible to find a frame in which the two events occur at the same point or to find one in which there occur at the same time). We see explicitly then that if two fields, located at $x^{\mu}$ and $y^{\mu}$ respectively, are allowed to interact directly, then one would be able to influence the other instantaneously and without any mediation despite their temporal and spatial separation. Thus lightlike separated fields cannot interact directly either.

Thus in all three cases we see that any direct interaction between two fields would constitute action-at-a-distance (either through temporal or spatial separation, or both).
Therefore we conclude that in order for an unmediated, direct interaction between two fields to be local it must occur at a single spacetime point.