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I would like to understand precisely in what sense are relativistic Schrödinger equations (Klein-Gordon,Dirac etc) causal. I'm not referring to the second quantized field or any field theory for that matter but the equations for a finite number of particles. I want to understand if these systems obey a more conventional notion of causality compared to microcausality in field theory where it seems to be well known that while fields can "spill out" outside the light cone, microcausality ensures that quantum mechanical measurements cannot be used to communicate faster than the speed of light. I ask this since in the references I have seen it is only stated that the equations are "causal" but I don't understand in what sense is this supposed to be interpreted. I have heard folklorically that this just means that the propagator for these equations are supported inside the light cone and this is usually contrasted with the propagator for the non-local square root schrödinger equation where the propagator only decays exponentially outside the light cone. My confusion then comes up, since in every introductory QFT textbook one learns that the Feynman propagator for these fields do in fact satisfy the "classical" relativistic equations which means they are also propagators for these equations, but the Feynman propagator does spill outside the light cone. I understand that there are many propagators, and the Feynman propagator isn't necessarily the same as the actual time evolution propagator for the finite particle systems, but I haven't seen this discussed in the references I have read.

So the question is simply in what precise mathematical sense are these systems causal?

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"Causality" may mean everything! However, sticking to the one-particle formalism (not QFT), I think that the textbooks you consider are referring to this fact.

First of all, all these equations (KG, Dirac) satisfy an existence and uniqueness theorem for Cauchy data (in a certain class of functions) which are assigned on Cauchy surfaces of the 4D Minkowski spacetime $M$, typically the 3-rest space $\Sigma$ of a Minkowski reference frame.

Next consider a couple of initial data on $\Sigma$ such that they are identical outside a region $\Omega \subset \Sigma$.

Causality means that the corresponding two solutions $\psi$ and $\psi'$ are identical outside $J^+(\Omega) \cup J^-(\Omega)$.

Above, the causal future $J^+(\Omega)\subset M$ and causal past $J^-(\Omega)\subset M$ of $\Omega$ in the Minkowski spacetime $M$ are defined as follows. $p\in J^\pm(\Omega)$ iff there is causal segment form a point $q\in \Omega$ to $p$, respectively, future-pointing or past-pointing.

All that means that a change in the initial condition cannot propagate faster than light in the corresponding solution.

There is no relation with the Feynman propagator, since it referes to a special quantum state: the vacuum state which is a non-local entity. The mathematical object that enters the play in the discussion above is the advanced/retarded fundamental solution which is, in fact, used to construct the solution out of the initial data.

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  • $\begingroup$ I think I understand now, but just to be absolutely sure this implies that if we then define the evolution kernel for these solutions, as a combination of the advanced and retarded greens functions, this Kernel would indeed vanish outside the light cone? $\endgroup$
    – Fiter
    Commented Jun 14 at 19:06
  • $\begingroup$ Yes, it vanishes for non-causally related arguments. $\endgroup$ Commented Jun 15 at 8:16

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