# Exact meaning of locality and its implications on the formulation of a QFT

As far as I understand it, locality in physics is the statement that interactions can only occur between physical objects if the spacetime interval separating them is null or time-like. Thus, if the two events $(t,\mathbf{x})$ and $(t',\mathbf{y})$ occur simultaneously, i.e. $t=t'$, one cannot affect the other (unless $\mathbf{x}=\mathbf{y}$) as they will be separated by a space-like interval.

From reading notes on QFT however I have gathered that a physical theory is local if interactions between the quantum fields (contained in the theory) occur at the same point in spacetime. Why is this so, why can't they be time-like separated?

Is it because theories in QFT are described by Lagrangian densities $\mathscr{L}(\phi (x),(\partial_{\mu}\phi) (x))$ which describe the physics at each spacetime point and as the action $S$ of the theory is the integral of $\mathscr{L}$ over spacetime $$S=\int d^{4}x\mathscr{L}=\int dt\int d^{3}x\mathscr{L}(\phi (t,\mathbf{x}),(\partial_{\mu}\phi) (t,\mathbf{x}))$$ the interactions occur at the same point in time and thus for locality to be obeyed the fields must interact at the same spatial point also (as simultaneous events in which $\mathbf{x}\neq\mathbf{y}$ are always separated by a space-like path)?

Locality is a physical requirement we impose (for good reasons). Locality is implemented in the theory by using fields, with local interactions in a Lagrangian density (ie, the Lagrangian only depends on products of fields and derivatives at a single point). I would definitely not say that locality occurs because Lagrangian densities show up in field theory, I would rather say that we use local Lagrangian densities because we want to implement locality.

We typically want locality because, especially when we also demand Lorentz invariance, locality is deeply related to causality. As you say, we typically don't want to allow for spacelike separated interactions because we could send signals back in time.

We typically don't want timelike separated interactions because they are just acausal--that would allow a field value in the future to interact with the field values now. Furthermore, if your theory is Lorentz invariant, then if you allow for timelike separated interactions you also have to allow for spacelike separated interactions.

There is research into non-local theories, and there is some evidence that quantum gravity is non-local. However, standard QFTs, in particular the Standard Model, are local in the above sense.

• Ah ok, so is the reason why we consider Lagrangians as functions of fields (and their derivatives) at a single point in spacetime because we are considering interactions that are simultaneous (i.e. the interaction between the fields occur at the same point in time and this immediately requires that they occur at the same point in space, as otherwise they will be spacelike separated)? Or is it simple as you said that if we allow timelike interactions then spacelike interactions will necessarily also be allowed, which violates locality?! – Will Jul 7 '15 at 13:10
• ... I find it slightly confusing because in general aren't timelike separations allowed in the principle of locality? What would you say was the definition of locality? Are there any good notes on the subject that you know of. – Will Jul 7 '15 at 13:12
• The physics is that we want interactions to occur at the same spacetime point. It's more or less common sense, if we don't demand that then the theory could describe something here instantaneously appearing over there. To me that's enough to focus on local theories. In particular I wouldn't want two points at different times to interact directly, information could flow back in time. When you add in lorentz invariance as well, the case gets even stronger for focusing on local theories, because you'll probably break causality if the interactions aren't local. – Andrew Jul 7 '15 at 13:15
• Timelike separated events can causally influence each other, but that doesn't mean fields at those points interact directly. Direct interactions between points of any separation, timelike, space like, or null, is inconsistent with locality. The only source I really know on this is weinberg vol 1, he derives the need for a local lagrangian density from the "cluster decomposition principle" (basically experiments at different places should be uncorrelated). – Andrew Jul 7 '15 at 13:20
• For what it's worth maybe I should mention that fields at timelike and null separations don't commute with each other, and this is consistent with locality. But that's different from having points at timelike or null separations directly interacting in the lagrangian. – Andrew Jul 7 '15 at 13:28