# What is the difference between locality and causality?

Tags:
1. Jul 10, 2015

### "Don't panic!"

This has been causing some confusion to me as the two concepts seem very similar, if not the same (especially when taking special relativity into account).

As far as I understand, even in classical physics (i.e. even before considering QFT and the like), one requires that interactions are local, i.e. the dynamics of a physical system are only affected by their immediate surroundings (an object can only exert a direct influence on another object, at a given instant in time, if they are in direct contact with one another - the interaction occurs at a single spatial point).
If I understand correctly, in classical mechanics this doesn't imply causality between events as propagation of information is not bounded by the speed of light. Thus, Newton's law of gravity $$\mathbf{F}=\frac{GMm}{\vert\mathbf{r}-\mathbf{r}'\vert^{2}}$$ is causal because it describes the influence of a body of mass $M$ on a body of mass $m$ - it unambiguously describes the effect on a body caused by another body. It is, however, non-local as it describes a direct influence on one body by another that are spatially separated by a finite distance $\vert\mathbf{r}-\mathbf{r}'\vert$ (i.e. they are not in direct contact).

The issue I struggle with is that when we introduce special relativity the two concepts no longer seem distinct as a space-like separation between two physical systems immediately implies that they are not in causal contact (causality is in some sense required by demanding locality)?!

Finally, is the reason why in textbooks it is explained that interactions are local if they occur at a single point in spacetime because locality is the requirement that two physical systems must be in direct contact in order to influence one another directly? (If they are at the same spatial location, but at different points in time, then they can not directly influence each other. Similarly, if they are at the same point in time, but they are located at different spatial points, then they can not directly influence one another. It is only when they are located at the same point (or infinitesimally close to one another) in space and time that they can directly influence one another)

2. Jul 10, 2015

### atyy

Why do you keep refusing to acknowledge that there are many meanings to these words?

Why do you keep insisting on confusing the meaning of "manifest locality" that is used to construct Lagrangians with the locality defined by "no superluminal signalling"?

If you cannot even accept that, then asking more and more questions about more and more different definitions of these terms will only result in more and more confusion.

3. Jul 10, 2015

### Staff: Mentor

Atyy makes an important point here. Natural languages such as English are inherently somewhat imprecise, so struggling to find completely satisfactory universal definitions of words like "locality" and "causality" is somewhat futile. Instead, when you see these words, you have to try to understand what the author intends them to mean in that particular context.

4. Jul 10, 2015

### "Don't panic!"

Sorry, it just seems to have got stuck in my brain. I think I understand the difference, but I have a real problem in doubting myself!

Here's a summary of the "musings in my head" on the subject (I'll put this and then "shut up". Sorry to be such a nuisance):

"Manifest locality" is then the notion that physical objects should be in direct contact to have a direct influence on one another, i.e. no action at a distance. (This notion is regardless of causality, or "Einsteinian locality", as it says nothing about the speed at which information can propagate and simply follows the intuitive notion that direct interactions between objects should only occur between objects in direct contact).

Then "Einsteinian locality" is the notion that interactions between objects that are spatially separated occurs through signals propagating between the two at finite (light) speed.

5. Jul 10, 2015

### atyy

I think the most important point is that the "manifest locality" often used for Lagrangians is just a mathematical convenience and not a physically meaningful concept (at the non-rigourous level that QFT is usually formulated). Gauge theories are an example where one might imagine that gauge redundancy and manifest locality are not needed, but we choose to use gauge redundancy and manifest locality just for mathematical convenience. See http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf, section 9.8.

In special relativity, ie. if we assume that spacetime is a fixed Lorentzian manifold, there are at least two physically important notions of locality or causality. Again, don't worry about the terms, each of these is called locality or causality by some people. What I am calling "Signal causality" is called "Einstein causality" by others.

1) Einstein causality or local causality. This is essentially the causality of classical special relativity that the cause of an event is in its past light cone. You can find notions of this type of causality defined in http://arxiv.org/abs/1311.6852 and http://arxiv.org/abs/1208.4119.

2) Signal causality or signal locality. This is an operational definition, and simply means no faster than light signalling. This is implemented mathematically by spacelike operators commuting, eg. http://cds.cern.ch/record/980036/files/197508125.pdf, section 7.

There are different inequalities corresponding to these two types of locality. If Einstein locality is obeyed, then the Bell inequality will not be violated. If Signal locality is obeyed, then a different inequality holds, shown by the "no signalling polytope" in Fig. 4 of http://arxiv.org/abs/1303.2849.

When relativity is not obeyed, eg. if we assume Galilean symmetry, the locality needs to be defined not in an all or none manner. Let's again go to the quantum case, and an operational definition, defined using correlations of spatially separated observables. In principle, faster than light communication is possible, but one can still define locality by saying how small the signal becomes. This is the idea behind the locality in http://www.scholarpedia.org/article/Lieb-Robinson_bounds.

For many more definitions and possible relationships between them, you can try:
http://arxiv.org/abs/1311.6852
http://arxiv.org/abs/1403.4621
http://arxiv.org/abs/1507.01588

Last edited: Jul 10, 2015
6. Jul 10, 2015

### "Don't panic!"

Thanks for your comprehensive answer. I think, as you have pointed out earlier, that I am trying to be too general in my understanding of locality and this is holding me back in my understanding. I appreciate your help and patience.

7. Jul 11, 2015

### atyy

I guess I should add that if one only does coarse measurements, then it may seem that special relativity is true although it isn't.
http://www.physics.upenn.edu/~kane/pedagogical/295lec3.pdf
http://arxiv.org/abs/1407.6532

It is also true that if special relativity is true, if one only does measurements on slowly moving objects, then it can seem like Galilean relativity is true. The example here is that the standard model of particle physics is relativistic, but condensed matter physics is mostly non-relativistic.