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I ask this question as the two seem to be very closely related and are sometimes taken to be one and the same (in the notion of microcausality in QFT), which has left me confused as to what meaning of the two concepts.

  1. Locality:

    My understanding of locality (which, if incorrect, please let me know) is that particles can only directly interact with one another if they are in contact with one another (implying that the interaction occurs at a single point), i.e. no action-at-a-distance (particles at distinct points cannot exert a direct influence on one another).

    Thus, in the case of QFT, the value of a Lagrangian density at a given point should only depend on the values of the fields at that point, along with a finite number of their derivatives (to explain interactions with fields infinitesimally close to that particular point). In summary, the dynamical state of a QFT at a given point in spacetime should be locally determined (i.e. the dynamics of a physical system should only depend on the local behaviour of the fields, and not on their global behaviour)

  2. Causality:

    As far as I understand it, causality is the statement that two physical systems cannot "communicate" if they are separated by a space-like interval.

    In QFT there is the concept of microcausality, in which fields must commute with one another if they are separated by a space-like interval. However, it is often formulated by saying that two fields measured simultaneously must commute unless they are located at the same spatial point. To me this seems almost the same statement as given by locality?!

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  • $\begingroup$ nice question, without the words past, future and time $\endgroup$ – user46925 Jul 12 '15 at 12:49
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I agree with your definition of locality (probably not surprising :)).

Causality I would say is the statement that an event in the future should not affect an event in the past. We can formulate this in classical physics terms. Causality is necessary in order for there to be a well defined initial value problem: I should be able to choose an initial time slice, specify the field values and derivatives on that slice, and evolve the system forward from there unambiguously. Acausality would allow an event from the future to come back and affect what's going on in the past--in principle that would allow the field evolution to change the initial conditions you started with.

If you like, causality is the requirement that there should be no time machines that allow me to send information into my past--I should not be allowed to kill my own grandfather.

If you don't demand Lorentz invariance, locality and causality are distinct concepts. I can certainly imagine non-local theories that are causal--Newton's action at a distance version of gravity is certainly causal, but it is nonlocal. Similarly, I can imagine a universe where I can press a button and reverse the flow of time for me (ie, my clock runs in the opposite direction of the rest of the universe), where I can only interact with things locally but I now have clearly violated causality.

These notions however become related once you demand Lorentz invariance. The reason is that the notion of simultaneity is relative. In particular, the time ordering of spacelike separated events becomes observer dependent. So if two spacelike separated events can affect each other (which is definitely non-local), there is a frame where I am using this spacelike communication to talk to someone in my past. She can then (provided that she can also perform spacelike communication) talk to someone in my past but also inside my past light cone. So you can create a loop of communication that ends up in my past light cone. In this example, no one is moving faster than light (or, maybe more accurately, the non-local communication allows for superluminal transfer of information), but the nonlocal transfer of information has allowed something I said now to end up in my past light cone.

So if we don't want to allow spacelike transfer of information, what can we do? Well at a fixed time the only event that is not spacelike separated from me is the event where I am located. So I can only affect the fields and their derivatives at my location.

As a warning, in gravity when the spacetime metric becomes dynamical, all of this becomes more complicated! In special relativity when the metric is fixed, things are more clear.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – David Z Jul 14 '15 at 7:00
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Causality means that if something happens before in one reference frame of your choice, it happens before in any other existing reference frame in the universe.

Locality means that if two events are space-like separated then it exists at least one reference frame where they happen at the same time; if two events are time-like separated, then it exists at least of reference frame where they happen at the same point.

In order causality to be preserved we have to ask that the physical measurements and observables are of time-like types, which in QFT translates in turn into the statement that observables must commute if space-like separated.

How this translates into all the other formal rules is in general an a posteriori check. In principle the Lagrangian may depend on everything, but after having worked out the equations of motion you can realise that some constraints and dependences must be taken out to ensure the uniquiness of the solutions as well as the Cauchy problem to make sense and so on. That the Lagrangian only depends on the value of the fields in one point, well, this just follows from the fact that it is a function of the fields, which in turns are maps from one point of the space-time onto the bundles. Moreover, the Lagrangian only depends on a finite number of derivatives because we want the equations of motions to be a finite order differential equation (because that is how mechanics is), which can be solved specifying a finite number of initial conditions (because that is how mechanics is); this would not be fulfilled if you allow infinite order of derivatives of the fields as dependency in the action principle.

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  • $\begingroup$ It's just that I've read in various notes (including D. Tong's notes) that if a Lagrangian is expressed as a function of field values at different points (e.g. $\phi(\mathbf{x})\phi(\mathbf{y})$), then the theory is non-local. Similarly, if the Lagrangian contains derivatives to infinite order, then the Lagrangian is also non-local. I understood from this that requiring knowledge of how the fields behave at two distinct points (corresponding to infinite number of derivatives, via a Taylor expansion)... $\endgroup$ – Will Jul 8 '15 at 13:02
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    $\begingroup$ ...then one needs to know how the fields behave globally (over spacetime), which is clearly non-local and implies action-at-a-distance, which is undesirable. $\endgroup$ – Will Jul 8 '15 at 13:02
  • $\begingroup$ that's an unusual definition of locality, isn't it? can you explain that choice or provide a ref? $\endgroup$ – innisfree Jul 8 '15 at 16:26
  • $\begingroup$ @innisfree This is what I understood from reading page 10 (pg 16 of the pdf notes) of Tong's notes: damtp.cam.ac.uk/user/tong/qft/qft.pdf Locality as far as understand it can be introduced without the concept of special relativity. It is the notion that physical objects should only be able to exert a direct influence on one another if they are in direct contact, such that the interaction occurring at a given point is only dependent on the physics at that point. $\endgroup$ – Will Jul 8 '15 at 16:33
  • $\begingroup$ @Will my comment is about the definition of locality in this answer. $\endgroup$ – innisfree Jul 8 '15 at 16:39

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