As far as I understand it locality is the rejection of action-at-a-distance. By this I mean that in a given frame of reference at a given instant of time (in that reference frame), two physical objects can only interact if they are in physical contact. Lorentz invariance requires that this is the case in all frames of reference and so direct interactions between physical objects can only occur if they are located at coincident spacetime points.

If this is correct, would it then be correct to say that locality is enforced in QFT by requiring that spacelike separated fields commute, i.e. $$[\phi(x),\phi(y)]=0\qquad\text{for}\quad (x-y)^{2}<0$$ and then following the argument in the classical case that I gave above, lorentz invariance requires that interactions between fields are point-like? Do we require locality of interactions to ensure causality?

I've been quite confused with the conceptual notion of locality and thought that I'd finally sorted it in my head, but now I'm not so sure. I'd really appreciate some help in understanding the conceptual notion of locality and why we require interactions to be point-like?

  • $\begingroup$ I'm pretty sure (not my field and its quite a while since I read the book) that Zee, "QFT in a nutshell" deals with exactly your issue quite thoroughly very early in the book. And IIRC your equation is exactly how locality described in QFT. $\endgroup$ Aug 12 '15 at 14:50
  • $\begingroup$ @WetSavannaAnimalakaRodVance Thanks for the tip, I'll take a look. $\endgroup$
    – Will
    Aug 12 '15 at 15:12
  • $\begingroup$ @WetSavannaAnimalakaRodVance I wasn't able to find anything in Zee, I don't suppose you can remember the chapter or section that he discusses it in? $\endgroup$
    – Will
    Aug 12 '15 at 15:36
  • $\begingroup$ How does $(x-y)^2 < 0$ even make sense? $\endgroup$
    – user193319
    Nov 22 '18 at 23:20
  • $\begingroup$ @user193319 it's a shorthand notation for the spacetime interval, e.g. $(x_0 - y_0)^2 - (x_1 - y_1)^2 - (x_2 - y_2)^2 - (x_3 - y_3)^2$. This will be less than 0 (in this sign convention) when the points are spacelike separated. $\endgroup$ Jun 23 at 1:10

Indeed, the QFT notion of locality is that observables at space-like separation commute, i.e. $$ (x - y)^2 < 0 \implies [\mathcal{O}_1(x),\mathcal{O}_2(y)] = 0 $$ for all local observables $\mathcal{O}_1,\mathcal{O}_2$, which are generically polynomials in the fields and their derivatives. This is our notion of locality because, classically, we know that measurements at spacelike separated events should not be able to influence each other in the sense that the expectation value of one measurement is influenced by the other. This is local in the sense that, by this, you cannot determine whether $\mathcal{O}_1(x)$ was measured by measuring $\mathcal{O}_2(y)$.

Note that this does not prohibit correlations, such as the one entanglement produces, between the results of spacelike separated measurement, which are results of the states being "non-local", since they are functionals of the total field configuration, not of spacetime events.

Note also that this contains the usual abuse of notation that the fields and observables are supposed to be functions on spacetime rather than operator-valued distributions acting on test functions. More formally, one should impose that all observables commute when they are applied to two functions whose supports are space-like separated.

Locality of a QFT may be easily established for free fields by using the mode expansion. In general, it is unknown whether interacting theories fulfill locality, but one usually assumes that the Wightman axioms, which include locality, hold even if they are not proven to do so. Only for very few and low-dimensional theories, e.g. scalar fields in 2D, it is proven that a theory with arbitrary polynomial interactions obeys the Wightman axioms.

Therefore, it is not, in general, known that "point-like interactions" (by which I supposed you mean the usual polynomial interactions in the fields) suffice to produce a QFT in which one could rigorously show that locality in the quantum field theoretic sense is preserved.

  • $\begingroup$ Thanks for your detailed explanation. Is what I put at the start of my post about a classical notion of locality correct at all? I've found that a lot of my confusion arises from why locality of a theory is taken to mean that interactions between fields occur at single spacetime points? Is this simple to provide a Lorentz invariant notion of locality, and also to ensure causality, or is there something else to it? $\endgroup$
    – Will
    Aug 12 '15 at 15:10
  • $\begingroup$ @Will: I'm not sure if there is a single accepted definition of classical locality, but I'd say it like this: A classical theory is local if changing the initial conditions for the equations of motion in a certain area does not change the result of the equations of motion in spacelike separated areas, in other words, if "changes" only propagate at the speed of light at most (as is the case e.g. in classical electromagnetism, cf. Lienard-Wichert potentials). $\endgroup$
    – ACuriousMind
    Aug 12 '15 at 15:14
  • $\begingroup$ Ah ok, so locality is purely the statement that spacelike separated objects cannot interact, such that we can localise the source of an interaction to within its immediate neighbourhood? $\endgroup$
    – Will
    Aug 12 '15 at 15:25
  • $\begingroup$ Why in QFT is locality prescribed by requiring that fields interact at single spacetime points? Is this simple to provide a Lorentz invariant notion of locality, and also to ensure causality, or is there something else to it? $\endgroup$
    – Will
    Aug 12 '15 at 15:27
  • $\begingroup$ @Will: I'm not sure what you're asking - how does QFT "prescribe that fields interact at single spacetime points"? $\endgroup$
    – ACuriousMind
    Aug 12 '15 at 15:37

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