Coincidence of spacetime events & frame independence

["Don't panic!"]

Not multiple timelike paths; multiple inertial frames, but only one in which they are located at the same spatial point.

OK, let me have another attempt to try and consolidate the concept correctly in my brain.

The concept of locality in physics requires that direct interactions between two objects can only occur through physical contact, i.e. no action-at-a-distance. Relativity in fact demands that all interactions are local due to the finite speed of light limiting the speed at which interactions can be mediated between objects. Requiring that interactions are described in a Lorentz invariant manner (such that physics is frame independent) and that direct interactions can only occur through physical contact (implying that the interaction takes place at a single point in space) leaves us with 3 possibilities to consider:

1. Time-like separation :

In this case there will be multiple inertial frames in which the two objects will be spatially separated, but only one in which they are located at the same point (in space). Thus, if they are time-like separated there is no consistent way to construct a local Lorentz invariant interaction.

2. Space-like separation :

In this case in all inertial frames the two objects will be spatially separated and can not interact at all (even by mediation of local interactions) and thus any direct interaction will be non-local in all cases. (This will be most explicitly obvious in the one inertial frame in which the two objects will be located at the same instant in time and as such the interaction would have to be instaneous across an arbitrary distance, which is clearly forbidden in SR).

3. Light-like separation :

In this case (as you said) there will be no inertial frames in which the the two objects will be located at the same point in time or space.Consequently, the only possible case in which their can be a local Lorentz invariant interaction between the two objects is if the spacetime points at which they are both located are coincident, i.e. they interact at a single point in spacetime.Would something like this be correct?

Not really, because "infinitesimally" is not a precise term. Also remember that when we start talking about quantum fields, your ordinary intuitions about "objects interacting" don't apply. Quantum fields are not objects, and quantum field interactions are not like ordinary "forces" in your everyday experience.

What would be a precise why to describe the situation for quantum fields? Why must they interact at a single spacetime point? Is the argument essentially the same as the reasons I've listed above?

 

[PeterDonis]

The concept of locality in physics requires that direct interactions between two objects can only occur through physical contact, i.e. no action-at-a-distance.

Right here is where I think you're going wrong. This is one concept of locality, but it only works for theories in which there are "objects" with well-defined physical boundaries, so that "physical contact" between objects has a well-defined meaning. In both GR and quantum field theory, this concept doesn't work, because these theories include fields, which are not "objects" with well-defined physical boundaries.

Since your original question in this thread was about Lagrangian densities, which already implies that you're talking about fields, the appropriate concept of "locality" has to be one that applies to fields. That concept is just what has already been said several times in this thread: the only way to write a Lorentz-invariant Lagrangian density for fields is for the Lagrangian density to be a function of fields and their derivatives at a single spacetime point.

If you want to understand why that is, heuristically, the reason is simply that any two distinct spacetime points will be at "different points in space" in at least one inertial frame; this is true for spacelike, null, and timelike separations. And "different points in space" means "not local". But that's only a heuristic understanding; it doesn't prove the statement I said above, it just makes it plausible. To prove it, you would need to study quantum field theory and understand how QFTs are constructed and what is required, mathematically, to guarantee that the theory is Lorentz invariant.

 

["Don't panic!"]

Right here is where I think you're going wrong. This is one concept of locality, but it only works for theories in which there are "objects" with well-defined physical boundaries, so that "physical contact" between objects has a well-defined meaning. In both GR and quantum field theory, this concept doesn't work, because these theories include fields, which are not "objects" with well-defined physical boundaries.

Ah ok, so this concept of locality applies to classical theories where we are describing point-like particles and their interactions then?

If you want to understand why that is, heuristically, the reason is simply that any two distinct spacetime points will be at "different points in space" in at least one inertial frame; this is true for spacelike, null, and timelike separations. And "different points in space" means "not local".

So when textbooks talk of locality in QFT is this simply that fields located at distinct spatial points cannot interact with one another directly?
Then for a Lorentz invariant notion of locality the only possible case is if the fields interact at single spacetime points, right?

Are the arguments I gave in the numbered list in my previous post about why this is the case correct at all?

I think I've confused myself a bit over the notion of locality as several of the texts that I've read quote Einstein when discussing locality, and he says something along the lines of:

"two objects that are physically separated in space cannot directly influence one another.
This notion of locality is used consistently only in field theory".

This is what I've been trying to parse and base my understanding on, but I fear that I'm confusing the situation?!

 

[PeterDonis]

this concept of locality applies to classical theories where we are describing point-like particles and their interactions then?

Basically, yes. (Often "objects" in classical, i.e., pre-relativistic, theories aren't point particles; they can have finite sizes. We just ignore all their internal structure; in many cases that's a perfectly good approximation.)

So when textbooks talk of locality in QFT is this simply that fields located at distinct spatial points cannot interact with one another directly?

Basically, yes; that's just another way of saying that the Lagrangian density has to be a function of the fields and derivatives at a single spacetime point.

Are the arguments I gave in the numbered list in my previous post about why this is the case correct at all?

They're not incorrect, but they're overly complicated. What you quoted from my post (the part after "heuristically") is a simpler way of saying the same thing that you were trying to say.

several of the texts that I've read quote Einstein when discussing locality, and he says something along the lines of:

"two objects that are physically separated in space cannot directly influence one another.
This notion of locality is used consistently only in field theory".

This is what I've been trying to parse and base my understanding on, but I fear that I'm confusing the situation?!

It seems so, yes. Pop science references, even ones by Einstein, are not good sources if you are trying to actually understand the science. Ordinary language is too imprecise to really pin down scientific concepts. You need math, or at least something that describes a mathematical requirement, like the requirement that the Lagrangian density be a function of fields and their derivatives at a single spacetime point.

 

["Don't panic!"]

It seems so, yes. Pop science references, even ones by Einstein, are not good sources if you are trying to actually understand the science. Ordinary language is too imprecise to really pin down scientific concepts. You need math, or at least something that describes a mathematical requirement, like the requirement that the Lagrangian density be a function of fields and their derivatives at a single spacetime point.

Yes, I don't generally like to source my information from pop-sci books (actually, this one was from Wikipedia, my bad), but I have been struggling to understand exactly what the notion of locality is in the case of field theory?!

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?

Is the principle of locality in general simply that we require that the physics at each spacetime point is only directly affected by its immediate surroundings (i.e. an arbitrarily small neighbourhood of the point), and then Lorentz invariance requires that fields should only interact at single points.
Or is it simply that fields 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)?

Sorry to be a pain, but I feel I'm missing something crucial here.

 

[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).