# Coincidence of spacetime events & frame independence

• "Don't panic!"
In summary, the conversation discusses the concept of coincident spacetime events and their frame independence, as well as the reason for expressing Lagrangian densities in terms of field values at a single spacetime point. The conversation also touches on the idea of non-locality and the requirement for interactions to be localized in both space and time. It is stated that the Lorentz transformation can be used to show that if two events are coincident in one frame, they are coincident in all inertial frames. The importance of locality and Lorentz invariance in interactions is also mentioned.
"Don't panic!"
I have been reading these notes: http://isites.harvard.edu/fs/docs/icb.topic455971.files/l10.pdf
in which they claim that if two spacetime events are coincident in one frame of reference then they are coincident in all inertial frames of reference, thus spacetime events are absolute i.e. they exist independently of any reference frame.
Is this simply to do with the fact that if two spacetime events are coincident then their spacetime interval is null, i.e. ##ds^{2}=0##, and as this is a frame independent quantity this must hold in any inertial reference frame, hence they are coincident in all inertial frames? Or does it simply follow from the fact that spacetime is modeled as a 4-D smooth manifold, and the points on a manifold exist independently of how you choose to represent them?
Is there anyway to prove the statement that if two spacetime events are coincident in one frame of reference then they are coincident in all inertial frames?

One of the reasons for me asking this is to try and understand why Lagrangian densities ##\mathscr{L}## are always expressed in terms of field values (and their derivatives) at a single spacetime point? Is this simply because the Lagrangian At a given instant in time is given by $$\mathcal{L}(t)=\int d^{4}x\;\mathscr{L}(t,\mathbf{x})$$ (I know that there is no explicit dependence on spacetime points, but this is just to highlight my question) and so fields can only interact with objects at a single spatial point (or infinitesimally close to that point), as otherwise they would be separated by a spacelike interval and their interaction would correspond to an instantaneous action at a distance (prohibited by the finite speed of light)?

"Don't panic!" said:
Or does it simply follow from the fact that spacetime is modeled as a 4-D smooth manifold, and the points on a manifold exist independently of how you choose to represent them?

This.

"Don't panic!" said:
One of the reasons for me asking this is to try and understand why Lagrangian densities ##\mathscr{L}## are always expressed in terms of field values (and their derivatives) at a single spacetime point?

If you had terms which depended on different points in space-time, your theory would be non-local. Also note that the integral to obtain a Lagrangian is over space only, not over space-time. If you integrate over space-time, you obtain the action (which to some extent is more fundamental).

"Don't panic!" said:
Is there anyway to prove the statement that if two spacetime events are coincident in one frame of reference then they are coincident in all inertial frames?
In SR just take the Lorentz transform of two events, and show that if ##a=b## then ##a'=b'##

In GR it follows from the fact that coordinate charts are defined to be 1 to 1.

So when texts (such as the one that I linked to in my original post) state that the coincidence of spacetime events is frame independent (is this the statement that spacetime points are frame independent, i.e. if two spacetime events are coincident then they are labelling the same spacetime point?) then they are simply relying on the fact that spacetime is a 4-D manifold and nothing else? Is there any physical intuition to this as well?

Orodruin said:
If you had terms which depended on different points in space-time, your theory would be non-local.

I understand that interactions between two objects that are spatially separated at a given instant in time are non-local as they correspond to action at a distance (instantaneous influence of one object on another when they are spatially separated by some finite distance), but for instance, why couldn't one consider a Lagrangian density that involves fields evaluated at the same spatial point, but at different times? Or is it simply that the Lagrangian describes the dynamics of the state at a given instant in time and thus the Lagrangian density can only be local if it's value at each spatial point at a given instant only depends on the field values at that point (as otherwise this would correspond to instantaneous propagation of information which is prohibited due to the finiteness of light speed)?!

DaleSpam said:
In SR just take the Lorentz transform of two events, and show that if

So would something like this be correct:

Let event ##A## be labelled by spacetime coordinates ##x^{\mu}## and event ##B## be labelled by spacetime coordinates ##y^{\mu}## in frame ##S##. Furthermore, in this frame, let ##x^{\mu}=y^{\mu}##. Now, in another inertial frame ##S'## we have that the coordinates of ##A## in this frame are related to its coordinates in ##S## in the following manner, $$x'^{\mu}=\Lambda^{\mu}_{\;\;\nu}x^{\nu}$$ Similarly, we have that the coordinates of ##B## in this frame are related to its coordinates in ##S## as, $$y'^{\mu}=\Lambda^{\mu}_{\;\;\nu}y^{\nu}$$ It follows then, that $$x'^{\mu}=\Lambda^{\mu}_{\;\;\nu}x^{\nu}=\Lambda^{\mu}_{\;\;\nu}y^{\nu}=y'^{\mu}$$ (as ##x^{\mu}=y^{\mu}##). Therefore, if spacetime events are coincident in one frame they are coincident in all inertial frames (as the Lorentz transformation between frames was chosen arbitrarily).

Yes, exactly.

"Don't panic!" said:
I understand that interactions between two objects that are spatially separated at a given instant in time are non-local as they correspond to action at a distance (instantaneous influence of one object on another when they are spatially separated by some finite distance), but for instance, why couldn't one consider a Lagrangian density that involves fields evaluated at the same spatial point, but at different times?

Is the reason why we require interactions to be localised in both space and time because intuitively (and philosophically so) it makes sense that objects (fields etc) should be in direct contact. If there is any finite separation between them, be it temporally, spatially, or both, then this would imply action at a distance (i.e. The two objects could immediately affect each other regardless of their temporal and spatial separation from each other without any intermediary medium in which the information can propagate between the two)?

Is it also the case that the only case in which an interaction is Lorentz invariant is when it occurs at a single point in spacetime - in all other cases Lorentz invariance is broken?!

(As an aside, in simplest terms, is locality simply the statement that we should be able to specify the location at which an interaction occurs. If it depends on two distinct points finitely separated, then it is not possible to specify an exact location at which the interaction occurs?)

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Indeed we use the field description of interactions in relativity as the most simple form to accommodate interactions with the relativistic notion of causality. "Action at a distance" means that there would be causally connected events that are space-like separated, but this would violate the fundamental notion of cause and effect. The cause should always be temporally before the effect, and this implies that the effect should be within (or at) the future lightcone of the cause. This is most simply achieved by describing, e.g., forces on point particles as mediated via a field, i.e., you have a local Lagrangian for the motion of the particle within a field. E.g., for a particle in an electromagnetic field you have
$$L=-m c^2 \sqrt{1-\dot{x} \cdot \dot{x}}-\frac{q}{c} F_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu},$$
where $$F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$$ is the Faraday tensor of the electromagnetic fields (its components are the usual electric and magnetic field components wrt. to a given reference frame). The dot indicates the derivative with respect to an arbitary scalar world-line parameter, and ##m## is the invariant mass of the particle.

$$m \ddot{x}^{\mu}=\frac{q}{c} {F^{\mu}}_{\nu} \dot{x}^{\nu}.$$
From here on we use the proper time ##\tau## of the particle as the world-line parameter.

The equations of motion which are automatically compatible with the mass-shell condition
$$p \cdot p=m^2 \dot{x} \cdot \dot{x}=m^2 c^2=\text{const},$$
because of
$$\dot{p} \cdot p=m^2 \ddot{x} \cdot \dot{x}=m \frac{q}{c} F_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}=0.$$
Thus the on-shell condition is fulfilled at all times, if it is fulfilled by the initial values of the momenta, and this one must obey for consistency with the classical particle picture. It implies that the tangent vectors on the the space-time trajectory of the particle are always time-like and thus the causality constraint is fulfilled by such manifestly covariant equations of motion automatically.

This building principle of manifestly covariant Lagrangians can be generalized to arbitrary types of fields.

The realization of Einstein causality in quantum theory is more complicated. So far the only successful realization is again in terms of local, microcausal quantum field theory. For a detailed explanation see my lecture notes on QFT:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

vanhees71 said:
Indeed we use the field description of interactions in relativity as the most simple form to accommodate interactions with the relativistic notion of causality. "Action at a distance" means that there would be causally connected events that are space-like separated, but this would violate the fundamental notion of cause and effect.

Would you be able to comment on whether my following summary is correct or not?

Pre-relativity, the notion of action-at-a-distance was simply that one object could affect another instantaneously, without any mediating agent, regardless of their physical (i.e. spatial) separation. This in itself is philosophically undesirable as why should one object be able to affect another without being in contact, or without the affect being transmitted from one to the other without a medium. When relativity is taken into account this situation becomes untenable as even if one introducs a medium to transmit the interactions between the two objects, the finiteness of light speed means that no two objects that are physically separated can interact with one another without violating causality. Thus we implement locality and require that objects can only be affected by physics in their immediate neighbourhood, i.e. an opoint located at a specific point can only directly interact with objects at adjoining points.

Enforcing locality requires that at a given instant in time, two objects can only interact if they are located at the same spatial point, right? Why is it though that when we construct a Lagrangian density to describe an interaction between two fields the interaction terms must be defined at the same spacetime point? Why couldn't we consider interaction terms that are evaluated at the same spatial point, but at different instants in time?! Is this simply because the Lagrangian density is constructed in order to describe the Lagrangian at a given instant in time (by integrating over a spatial volume), $$\mathcal{L}(t)=\int\;d^{3}x\mathscr{L}(t,\mathbf{x})$$ and thus all fields in the Lagrangian density are necessarily evaluated at the same instant, and this in turn forces them to be evaluated at the same spatial point to ensure locality (two fields can only interact instantaneously if they are located at the same spatial point)?! Or is there something else to it that I'm missing?

"Don't panic!" said:
(As an aside, in simplest terms, is locality simply the statement that we should be able to specify the location at which an interaction occurs. If it depends on two distinct points finitely separated, then it is not possible to specify an exact location at which the interaction occurs?)

Also, is the above description correct at all?

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"Don't panic!" said:
Is this simply because the Lagrangian density is constructed in order to describe the Lagrangian at a given instant in time (by integrating over a spatial volume),

In a relativistic theory, the Lagrangian density is a function on spacetime, not space, and the action is the integral of the Lagrangian density over a spacetime volume, not a spatial volume:

$$S = \int d^4 x \mathscr{L} \left( x \right)$$

where ##x## labels a point in spacetime, not space. This form of the Lagrangian density automatically ensures locality. It is also required for Lorentz invariance, since in a Lorentz invariant theory there is no invariant notion of "an instant in time" for spatially separated points; or, to put it another way, there is no invariant way to split up spacetime into "space" and "time", so you can't separate out integration over "space" from integration over "time".

PeterDonis said:
This form of the Lagrangian density automatically ensures locality.

Is this simply because the Lagrangian density depends only on a single spacetime point and not on any finitely separated points (infinitesimally close points are ok though, right? They appear implicitly in the dependence of first-order spacetime derivatives of the fields?), and so the automatically ensures that spatially separated fields cannot interact instantaneously with one another?
PeterDonis said:
It is also required for Lorentz invariance, since in a Lorentz invariant theory there is no invariant notion of "an instant in time" for spatially separated points

Is it correct to say then that the only case in which an interaction can be local in all inertial reference frames (i.e. To ensure that the locality of the interaction is Lorentz invariant) is if the interaction (described by the Lagrangian density) occurs at a single spacetime point?

Why wouldn't it be ok to include interactions that are timelike separated in the Lagrangian density (as isn't the timelike nature a Lorentz invariant concept). For example, why would it be incorrect to have something like $$\mathscr{L}\sim\phi (t,\mathbf{x})\psi (t',\mathbf{x})$$ in which ##(t,\mathbf{x})## and ##(t',\mathbf{x})## are timelike separated?

"Don't panic!" said:
Is this simply because the Lagrangian density depends only on a single spacetime point and not on any finitely separated points (infinitesimally close points are ok though, right? They appear implicitly in the dependence of first-order spacetime derivatives of the fields?)

Yes.

"Don't panic!" said:
why would it be incorrect to have something like

$$\mathscr{L}\sim\phi (t,\mathbf{x})\psi (t',\mathbf{x})$$

in which ##(t,\mathbf{x})## and ##(t',\mathbf{x})## are timelike separated?

Because there will be multiple timelike paths between those two events, and the action might depend on which path is taken. (In fact, this is exactly what happens in the path integral formulation of quantum mechanics.) In other words, timelike separated events are still not "local" (unless they are infinitesimally close). Another way to put it might be that the Lagrangian density you wrote down leaves out intermediate steps: what happens at events between the two you included?

PeterDonis said:
Because there will be multiple timelike paths between those two events, and the action might depend on which path is taken. (In fact, this is exactly what happens in the path integral formulation of quantum mechanics.) In other words, timelike separated events are still not "local" (unless they are infinitesimally close). Another way to put it might be that the Lagrangian density you wrote down leaves out intermediate steps: what happens at events between the two you included?

So is the point that we require the Lagrangian density to be local in time as well? Also, is it correct to say then that the only case in which an interaction can be local in all inertial reference frames (i.e. To ensure that the locality of the interaction is Lorentz invariant) is if the interaction (described by the Lagrangian density) occurs at a single spacetime point?

(As an aside, in simplest terms, is locality simply the statement that we should be able to specify the location at which an interaction occurs. If it depends on two distinct points finitely separated, then it is not possible to specify an exact location at which the interaction occurs?)

"Don't panic!" said:
So is the point that we require the Lagrangian density to be local in time as well?

The point is that, in a relativistic (Lorentz invariant) theory, there is no such thing as "local in space" vs. "local in time"; there is only "local in spacetime".

"Don't panic!" said:
is it correct to say then that the only case in which an interaction can be local in all inertial reference frames (i.e. To ensure that the locality of the interaction is Lorentz invariant) is if the interaction (described by the Lagrangian density) occurs at a single spacetime point?

Yes. If you think about it, you will see that this is the same thing I said just above, just in different words.

"Don't panic!" said:
(As an aside, in simplest terms, is locality simply the statement that we should be able to specify the location at which an interaction occurs. If it depends on two distinct points finitely separated, then it is not possible to specify an exact location at which the interaction occurs?)

This is a reasonable way of putting it, yes.

PeterDonis said:
The point is that, in a relativistic (Lorentz invariant) theory, there is no such thing as "local in space" vs. "local in time"; there is only "local in spacetime".

Is this because in a relativistic framework it is simply not possible to separate time and space into separate entities and thus something can only be local if it is local in spacetime (as you said)?

"Don't panic!" said:
Is this because in a relativistic framework it is simply not possible to separate time and space into separate entities and thus something can only be local if it is local in spacetime (as you said)?

Yes.

Does the same kind of idea apply in the classical mechanics case, i.e. Before constructing a Lagrangian I'm I correct in saying that we require it to be local in time, such that the dynamics of a physical system at a given instant in time are only affected by the local behaviour of the environment at that given point in time (and infinitesimally close to it)?

"Don't panic!" said:
Does the same kind of idea apply in the classical mechanics case

Sort of. You can formulate Newtonian mechanics using a Lagrangian that is apparently local, but it still doesn't work quite the same as in the relativistic case.

For example, Newton's law of gravity in its usual formulation is explicitly nonlocal; it describes a force that acts instantaneously at a distance. We can also formulate the law using an apparently "local" Lagrangian that includes a potential energy term due to the "gravitational field". But all that really does is hide the nonlocality in the potential energy term.

PeterDonis said:
Sort of. You can formulate Newtonian mechanics using a Lagrangian that is apparently local, but it still doesn't work quite the same as in the relativistic case.

How does one motivate the Lagrangian depending on only one instant in time and it being dependent on position and velocity at that time then? I thought it was simply a requirement of locality again, i.e. that the dynamics of a physical system should depend only on the physics in its immediate neighbourhood? Or, in the classical mechanics case, is it's functional dependence purely based on empirical evidence (that we can determine the dynamic evolution of a system given its initial position and velocity), and the fact that Newton's laws hold?

PeterDonis said:
For example, Newton's law of gravity in its usual formulation is explicitly nonlocal; it describes a force that acts instantaneously at a distance. We can also formulate the law using an apparently "local" Lagrangian that includes a potential energy term due to the "gravitational field". But all that really does is hide the nonlocality in the potential energy term.

In this case am I correct in saying that the force is nonlocal because it acts instantaneously (as you said) without any discernible contact between to massive bodies and without anything to mediate the interaction across the separation?!

How does the gravitational potential "hide" the nonlocality? Is it simply because it's value a a given point depends on masses that are finitely separated from it, i.e. it's of the form $$V\sim \frac{GM}{\vert\mathbf{r}-\mathbf{r}'\vert}$$

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"Don't panic!" said:
How does one motivate the Lagrangian depending on only one instant in time and it being dependent on position and velocity at that time then? I thought it was simply a requirement of locality again, i.e. that the dynamics of a physical system should depend only on the physics in its immediate neighbourhood?

Some history might help here. At the time Newton formulated his law of gravity, the behavior of gravity appeared (given the accuracy of measurement at the time) to be instantaneous action at a distance, so that's how Newton described it. He was uncomfortable with this, and so were others, because instantaneous action at a distance didn't seem physically reasonable; but at the time nobody knew any other way of formulating the theory.

In the 19th century, the Lagrangian method for formulating theories was discovered, and Newtonian gravity was reformulated that way, and people were happy because it appeared that now gravity could be accounted for as a local response to the "gravitational field". They didn't realize (at least, it appears that they didn't until relativity was discovered) that the theory was still nonlocal; the nonlocality had just been hidden inside the dynamics of the "gravitational field" (see further comments below).

Once Einstein discovered special relativity, and began work on trying to come up with a theory of gravity that was consistent with it, it became clear that the nonlocality in the classical theory, even in its Lagrangian formulation, was still there. The only way to fix it was to come up with a fully relativistic theory, based on a fully relativistic Lagrangian (in other words, one that doesn't have even an implicit, "hidden" nonlocality). Once he had this theory (General Relativity), it was easy to show that Newtonian gravity is a very good approximation to it for weak fields and slow motion (all speeds much slower than the speed of light), which is the regime in which Newtonian gravity was discovered. So gravity can "look like" instantaneous action at a distance (to a good approximation) under certain conditions, even though it really isn't.

So I would say it was the discovery of relativity that really made physicists adopt locality as a fundamental requirement.

"Don't panic!" said:
In this case am I correct in saying that the force is nonlocal because it acts instantaneously (as you said) without any discernible contact between to massive bodies and without anything to mediate the interaction across the separation?!

Yes.

"Don't panic!" said:
How does the gravitational potential "hide" the nonlocality?

Because the potential energy itself depends on the instantaneous distance to other gravitating bodies; the "action at a distance" is just reformulated to determine the potential energy directly instead of the force. The force (or "gravitational field") is the gradient of the potential energy, so it is still being indirectly determined by action at a distance.

PeterDonis said:
So I would say it was the discovery of relativity that really made physicists adopt locality as a fundamental requirement.

First of all, thanks very much for the historical context, it's really helped with my understanding of the situation.
How, though, was the functional form of Lagrangians formulated pre-relativity, i.e. how is it theoretically motivated to be dependent on position and velocity at a given instant in time?

Was it the fact that the propagation of information was limited to the finite speed of light that really enforced locality, as information between physical systems that are spatially separated takes a finite amount of time to be transmitted? Interactions can only occur instantaneously if they happen at a single spatial point, and due to relativity rendering the need for spacetime, and the frame independence of physics requiring interactions to be Lorentz invariant, an interaction in a relativistic framework is local and Lorentz invariant only in the case where it occurs at a single point in spacetime?!

PeterDonis said:
In the 19th century, the Lagrangian method for formulating theories was discovered, and Newtonian gravity was reformulated that way, and people were happy because it appeared that now gravity could be accounted for as a local response to the "gravitational field". They didn't realize (at least, it appears that they didn't until relativity was discovered) that the theory was still nonlocal; the nonlocality had just been hidden inside the dynamics of the "gravitational field" (see further comments below).

Ah, I think I understand this better now. So, when Newton first wrote down his law of gravitation there was no concept of a gravitational field and hence no way for the force to be mediated across the distance separating two massive bodies, thus it is non-local as the force appears to act without any discernible exchange of information between the two bodies? (Am I correct in thinking that the fact that it was instaneous alone wasn't as huge a problem, as in Newtonian physics information is allowed to propagate infinitely fast, and so if there were something to mediate it the theory wouldn't necessarily be non-local? Or is it considered to be non-local if the two bodies aren't physically touching, regardless of this fact?!)
Then, when the Lagrangian formulation became available and physical interactions could now be described in terms of potentials that Newtonian gravity appeared to become local, as there was now a field to mediate its action, however, this field is itself non-local and so the problem still remained (until relativity came along)?

"Don't panic!" said:
How, though, was the functional form of Lagrangians formulated pre-relativity, i.e. how is it theoretically motivated to be dependent on position and velocity at a given instant in time?

It wasn't. The original discovery of the Lagrangian formulation was based on the principle of least action, and it was thought of as applying to an entire process, not a single event. The particular form in which the Lagrangian is a function of position and velocity at a given instant of time is not general; it was originally discovered as a solution to a particular problem, the problem of the motion of a single particle. Other problems gave rise to other Lagrangians with different forms. As I said, it wasn't until relativity was discovered that requiring the Lagrangian to be local in spacetime became a fundamental principle, and, as we've already said, that was based on the requirement of the interaction being Lorentz invariant.

"Don't panic!" said:
Am I correct in thinking that the fact that it was instaneous alone wasn't as huge a problem, as in Newtonian physics information is allowed to propagate infinitely fast

"Information propagation" wasn't really a concept back then, and people didn't think of instantaneous action at a distance as "information propagating infinitely fast". I'm not sure what they would have thought if they had. But see below.

"Don't panic!" said:
and so if there were something to mediate it the theory wouldn't necessarily be non-local?

The idea of nothing being there to mediate the force between spatially separated objects, like the Earth and the Moon, was a problem, yes, and that may have been a key reason why the field concept was eagerly adopted when it was invented. However, even before relativity there were problems with the field concept; see below.

"Don't panic!" said:
this field is itself non-local and so the problem still remained (until relativity came along)

The key problem, at least in the minds of physicists in the 19th century, wasn't that the field was nonlocal per se. It was that, when they tried to figure out what properties the "medium" that propagated the field would have to have, they turned out to be highly counterintuitive. This problem arose in the case of light as well: many physicists thought of light as being propagated by a medium, the "luminiferous ether", but this created problems even though light, unlike gravity, was known in the 19th century to propagate at a finite speed.

The basic problem is that, in an ordinary material medium, the speed of wave propagation is related to the rigidity of the material: to propagate waves at the speed of light, a material would have to be many orders of magnitude more rigid than steel. Yet this medium, the "ether", had to be indistinguishable from a vacuum. (This problem would be even worse for a medium that would propagate gravity, since its speed of propagation was thought to be effectively infinite in the 19th century.) So the field concept, in a way, turned out to be a "cure" that might have been worse than the "disease".

In our modern conception, the field concept is still there, but the fields are now quantum fields, and they don't work the way an ordinary material medium does; fields can perfectly well propagate in a vacuum at a finite speed.

Mentz114
PeterDonis said:
The idea of nothing being there to mediate the force between spatially separated objects, like the Earth and the Moon, was a problem, yes, and that may have been a key reason why the field concept was eagerly adopted when it was invented. However, even before relativity there were problems with the field concept; see below.

So when one talks of nonlocality of interaction between objects then, it is not in reference to the lack of an agent (field, or other medium) to mediate the interaction between them, but more a statement that two objects that are spatially separated (i.e. not in physical contact) are able to directly affect one another?

"Don't panic!" said:
So when one talks of nonlocality of interaction between objects then, it is not in reference to the lack of an agent (field, or other medium) to mediate the interaction between them, but more a statement that two objects that are spatially separated (i.e. not in physical contact) are able to directly affect one another?

It depends; the term "nonlocality" does not have a single meaning that everyone agrees on, nor does its opposite, "locality". The historical stuff is interesting, but not directly relevant to physics if we're talking about theories that are no longer considered correct. In the context of relativistic theories, i.e., our current best theories, "locality" means what we've already discussed: that all the equations have to relate quantities at the same spacetime point.

Ah, OK. Thanks for all your help on the matter.

"Don't panic!" said:
Thanks for all your help on the matter.

You're welcome!

PeterDonis said:
You're welcome!

Sorry, just realized that there's one case that I'm still unsure about that I would hopefully like to clear up.

What about spacetime points that are light-like separated? How does one argue in this case that the Lagrangian density should still be expressed in terms of field values at single spacetime points?

Is it essentially the same issue as time-like intervals, in that there will exist multiple light-like paths between the two spacetime points and hence the ambiguity will make it impossible to localise the interaction within an arbitrarily small neighbourhood of a spacetime point?

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"Don't panic!" said:
What about spacetime points that are light-like separated?

Lightlike separated points are still distinct spacetime points, so the same issue applies. Mathematically, there is a bit of a complication since you can't use arc length along a lightlike path to distinguish points (since it's zero); but that just means you need to find an appropriate affine parameterization, which is easy to do.

"Don't panic!" said:
Is it essentially the same issue as time-like intervals, in that there will exist multiple light-like paths between the two spacetime points

That's more an issue of computing the integral of the Lagrangian along a path to get the action; it's not an issue with localizing the Lagrangian density. Sorry if I wasn't clear about that before.

The issue with localizing the Lagrangian density is that, even if two events are timelike separated, they will still be separated in space in every inertial frame except one. So any interaction that has to be localized in space and also Lorentz invariant has to be localized in time as well, even for timelike separated events.

PeterDonis said:
The issue with localizing the Lagrangian density is that, even if two events are timelike separated, they will still be separated in space in every inertial frame except one. So any interaction that has to be localized in space and also Lorentz invariant has to be localized in time as well, even for timelike separated events.

Ah ok, so when we demand the interaction to be localised in space this is simply the requirement that the two objects must be in physical contact (in order to directly interact with one another), such that the interaction occurs at a single spatial point. Then, for this localisation to be Lorentz invariant we require that it also occur at a single point in time, as otherwise, if they are time-like separated then there will be multiple time-like paths, but only one in which they are located at the same spatial point. Similarly, if they are light-like separated then there will be multiple light-like paths connecting them, but again only one where they are located at the same point in space (and in this case, the same point in time).
The space-like separation case is obvious as there will be no frame in which they will be located at the same spatial point, and as such the interaction will always be non-local in this case (requiring superluminal propagation of the interaction).

Would it also be correct to say that in the continuum case where we consider interacting fields, a field located at a particular spacetime point can also interact directly with its fields in its immediate neighbourhood (i.e. fields located at points infinitesimally close to the point the original field is located at), but only via coupling to the derivatives of the field evaluated at the point it's located at?

Would the above be a correct summary of the situation?

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"Don't panic!" said:
if they are time-like separated then there will be multiple time-like paths, but only one in which they are located at the same spatial point

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

"Don't panic!" said:
if they are light-like separated then there will be multiple light-like paths connecting them, but again only one where they are located at the same point in space

No. Two distinct events that are null separated can never be at the same point in space or the same point in time in any inertial frame.

"Don't panic!" said:
Would it also be correct to say that in the continuum case where we consider interacting fields, a field located at a particular spacetime point can also interact directly with its fields in its immediate neighbourhood (i.e. fields located at points infinitesimally close to the point the original field is located at), but only via coupling to the derivatives of the field evaluated at the point it's located at?

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.

PeterDonis said:
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?
PeterDonis said:
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?

"Don't panic!" said:
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.

PeterDonis said:
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?
PeterDonis said:
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?!

"Don't panic!" said:
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.)

"Don't panic!" said:
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.

"Don't panic!" said:
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.

"Don't panic!" said:
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.

PeterDonis said:
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.

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