Coincidence of spacetime events & frame independence

[PeterDonis]

I get that mathematically that this means that we should construct Lagrangian densities from fields and their (first-order) derivatives at a single spacetime point, but I'm struggling to understand what the idea is physically?

The Lagrangian density, physically, describes the dynamics of the fields and their interactions. So if the Lagrangian density is a function of fields and derivatives at a single spacetime point, then then dynamics of the fields and their interactions at a given spacetime point is determined by the fields and derivatives at the same point.

 

["Don't panic!"]

The Lagrangian density, physically, describes the dynamics of the fields and their interactions. So if the Lagrangian density is a function of fields and derivatives at a single spacetime point, then then dynamics of the fields and their interactions at a given spacetime point is determined by the fields and derivatives at the same point.

Sorry, yes I understand this part. My confusion is over the notion of locality in field theory (and in general)?
Is it simply that fields (or in general, objects) that are physically separated in space should not be able to directly interact with one another, i.e. any two fields (or in general, objects) that are physically separated in space should not be able to directly interact instantaneously (and Lorentz invariance requires in turn that local interactions should occur at single spacetime points)?

 

[PeterDonis]

My confusion is over the notion of locality in field theory (and in general)?

What I described is the notion of locality in field theory. That's all there is to it. All your other statements, to me, just look like different ways of saying the same thing.

 

["Don't panic!"]

What I described is the notion of locality in field theory. That's all there is to it. All your other statements, to me, just look like different ways of saying the same thing.

Ok, so what I've put in the above post (and perhaps previously) is a correct (but redundant given your description) way of describing what locality "means" then?
I think I'm trying to complicate things because I thought I'd misunderstood earlier. Apologies for that.

As a side note, would it be correct to say that we localise the fields, as we describe them in terms of their values at each spacetime point. Then, interactions between them should clearly be at single spacetime points as any interaction between two fields at distinct spacetime points will certainly be non-local (it will not be possible to localise the exact spacetime point at which the interaction occurred). The principle of locality is then simply the statement that we should be able to localise interactions to the points at which they occur, such that interactions propagate from point to neighbouring point.

Sorry, I realize that the above is a restatement of what we've already discussed, but put in this way it makes particular sense to me and so I'd really appreciate it if you could verify its validity either way.

 

[PeterDonis]

so what I've put in the above post (and perhaps previously) is a correct (but redundant given your description) way of describing what locality "means" then?

I think so. But again, you are using ordinary language, not math, and ordinary language is vague. The same goes for the rest of your post. If you try to express it in math, you will find that you just keep coming back to the plain statement that the Lagrangian density is a function of fields and their derivatives at a single spacetime point. I really think it's better just to keep that, and that alone, as your definition of what "locality" means, and discard the ordinary language descriptions. You can't use them to calculate answers anyway; you need the math. And if you can't calculate answers, you can't be sure that what you're saying is right anyway, because you can't compare your answers with experiment.

 

["Don't panic!"]

I think so. But again, you are using ordinary language, not math, and ordinary language is vague. The same goes for the rest of your post. If you try to express it in math, you will find that you just keep coming back to the plain statement that the Lagrangian density is a function of fields and their derivatives at a single spacetime point. I really think it's better just to keep that, and that alone, as your definition of what "locality" means, and discard the ordinary language descriptions. You can't use them to calculate answers anyway; you need the math. And if you can't calculate answers, you can't be sure that what you're saying is right anyway, because you can't compare your answers with experiment.

You're right, it's much more precise to construct the statement mathematically. I was trying to get a heuristic understanding of it to try and "fully understand" the concept such that I could restate it words to someone else.
I really appreciate your help and I think I understand the idea now, just need to stop doubting myself and overthink things!

 

[vanhees71]

The Lagrangian density, physically, describes the dynamics of the fields and their interactions. So if the Lagrangian density is a function of fields and derivatives at a single spacetime point, then then dynamics of the fields and their interactions at a given spacetime point is determined by the fields and derivatives at the same point.

One can add that not only the Lagrange density is local but then also the observables, i.e., energy-momentum density (through the Belinfante stress-energy tensor, and it must be the Belinfante tensor not the canonical one, because only the former is gauge invariant for e.g., electromagnetism) and the angular-momentum-center-momentum density of the fields.

If you look at a closed system, the total energy, momentum, and angular momentum are conserved and thus the integral of the densities over the entire volume leads to the correct transformation properties of the corresponding total quantities (energy-momentum four-vector, angular-momentum-center-momentum tensor).