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Let’s consider a pair of events $A(t_1,x_1)$ and $B(t_2,x_2)$,having a spacelike separation wrt an inertial frame denoted by K.In the frame K’ moving along the positive x-x’ direction with a constant speed v, the events are denoted by $A’(t’_1,x’_1)$ and $B’(t’_2,x’_2)$.For the unprimed frame we assume $t_2>t_1$ and $x_2>x_1$. Now $$\frac{x_2-x_1}{t_2-t_1}=k>c$$ [considering the space-like separation between the events].

In the transformed frame(K’) we have:

$$ x’_1=\gamma(x_1 – vt_1)$$ $$ t’_1=\gamma(t_1-\frac{vx_1}{c^2})$$ And $$ x’_2=\gamma(x_2 – vt_2)$$ $$ t’_2=\gamma(t_2-\frac{vx_2}{c^2})$$ Now, $$t’_2-t’_1=\gamma[(t_2-t_1)-\frac{v}{c^2}(x_2-x_1)]$$ $$ t’_2-t’_1=\gamma(t_2-t_1)[1-\frac{v}{c^2}\frac{x_2-x_1}{t_2-t_1}]$$ $$t’_2-t’_1=\gamma(t_2-t_1)[1-\frac{v}{c^2}k]$$ Now by our initial assumption $t_2-t_1$ is positive. Therefore $t’_2-t’_1$ will be positive only if $$k<\frac{c^2}{v}$$ To be precise we have for temporal non-reversal: $$c<k<\frac{c^2}{v}$$ For $ k>\frac{c^2}{v}$ the temporal order is reversed. [The ultimate speed(=c) permitted in nature as decided by the principle of causality remains undisturbed. For a pair of events having a spacelike separation in the unprimed frame we can always find a primed frame where the temporal order is reversed.]

Problem Proper:

Let’s examine the above mentioned issues in relation to the particle-antiparticle problem. I would refer to (1) Steven Weinberg,Gravitation and Cosmology, Chapter 2[Special Relativity],Section13—“Temporal Order of Antiparticles”(2)Michael Peskin and Daniel Schroeder,An introduction to Quantum Field Theory.,Chapter (2) The Klein Gordon Field Section 2.1.

The basic idea portrayed in these sections the the probability amplitude of a particle traveling across a spacelike separation[due to quantum mechanical reasons] is cancelled by the amplitude of the antiparticle moving in the reversed direction due to the temporal reversal of events.This protects the principle of causality. But for the events A and B having a spacelike separation in the umprimed frame, temporal reversal does not take place in all boosted frames as indicated in the prelude.

Query: If the particle and the antiparticle move in the same direction in the original[unprimed] and the boosted frame, how do we explain the situation?

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I gave a rewrite, if you don't like it, revert. –  Ron Maimon May 21 '12 at 18:34
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1 Answer

The reason you are confused is because there is no cancellation. The propagator is just nonzero outside the lightcone, there is no particle/antiparticle cancellation in any way.

The thing that you are misreading is only saying that what looks like a particle propagation in some frames looks like an antiparticle propagation in other frames, because of the reversal of the direction of time. The particle can travel a spacelike distance.

This doesn't violate causality, because the particle picture is just not causal. In quantum field theory, there are two notions of causality. One is that if you make a measurement in one region, this does not affect measurements in another region separated by a macroscopic spacelike separation. This is formalized in the requirement of microcausality explained here: Is microcausality *necessary* for no-signaling? . The idea is that local observables are functions of fields, that the fields are attached to space-time points, and the only experiments you can do in region R are those which use the fields in region R.

Then the condition that spacelike separated regions do not affect each other is that

$$ [A(x),B(y)] = 0 $$

For any two points x and y which are spacelike separated, and any measurable fields A and B. This does not include Fermionic fields, these fields are not local observables. You need a fermionic bilinear, like the energy, to make an observable.

The condition of microcausality was suspected to be false right from the beginning, and in the 1960s, there was a lot of effort expended to redefine causality in terms of analytic properties of scattering matrices, so that the property would be formulated in terms of on-shell scattering, rather than off-shell field properties. This is what is asked in this question in a not so great way (http://physics.stackexchange.com/questions/30711/causality-in-string-theory ). The answer is that it is appropriate analyticity of the S-matrix, plus causality on the boundary field theory.

Since modern physics accepts that string theory is a consistent theory of gravity (if maybe not the right one for this universe, although I find that impossible to imagine), it also accepts that there is a failure of microcausality in gravity. This failure is required in gravity just from semiclassical considerations, because if you associate independent quantum fields to every point on the exterior of the black hole solution, the entropy of the black hole is divergent, as noted by t'Hooft. Attempting to make a finite entropy black hole led directly to the holographic principle, and linked back with the 1960s idea for microcausality failure built into string theory.

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So one could add something about commutators and observables here, right? –  NiftyKitty95 Jul 20 '12 at 20:53
@NickKidman: You mean to add that the observables commute in spacelike separated regions? This is true in field theory, it's microcausality, but it's not a fundamental property--- it fails in string theory. I prefer to use analyticity to define causality, but the answer to that is somebody's question about "what is causality in string theory". –  Ron Maimon Jul 21 '12 at 2:49
@RonMaimon, this might get things interesting: arxiv.org/abs/1207.3123 –  lurscher Jul 21 '12 at 4:59
@lurscher: Thanks for the link--- I'll read it, but it looks totally wrong on the surface, but I didn't read the argument. The error might be to do with not including the past-region. –  Ron Maimon Jul 21 '12 at 21:58
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