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I just started reading about Feynman diagrams in Griffith's Introduction to Elementary Particles. Using the convention that time goes to the left, and arrows pointing forward in time represent particles and those pointing backward represent antiparticles, what does it mean if a particle points straight up, i.e. at a constant time?

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It means nothing.

If you want to look at such a diagram as a diagram of worldlines (as you seem to be implying), this isn't what a Feynman diagram is. A Feynman diagram is a graphical object that encodes certain integrals in perturbative quantum field theory. The lines in it are not worldlines of actual particles, and to say that "time goes to the left" is totally meaningless. This is exacerbated by the fact that for computational purposes one usually prefers to draw those diagrams in momentum space where there is no "direction of time". The only meaningful thing that "time goes to the left" tells you is that the lines at the left end represent the input particles and the lines at the right end represent the resulting particles of this scattering process.

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    $\begingroup$ While in some sense this is true, this is also a terrible answer. If you adapt the philosophy "Nothing means anything; just shut up and calculate," you will never develop any actual intuition for physics, you will never truly understand physics, and you will never make any advances in physics research. $\endgroup$ Sep 10, 2016 at 15:04
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    $\begingroup$ I think it's much better for your intuition to think about lines between spacelike separated particles as "virtual particles", which can only exist for short periods because of the Heisenberg Uncertainty Principle. Is this "true"? Ask a philosopher. $\endgroup$ Sep 10, 2016 at 15:05
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    $\begingroup$ @PeterShor That's an abuse of the uncertainty principle, for the correct interpretation of a time-energy uncertainty principle see this question. As to whether it is "much better for intuition": Yes, thinking about "virtual particles" is more intuitive for humans than about the abstract workings of quantum field theory. That doesn't make it any more correct. $\endgroup$
    – ACuriousMind
    Sep 10, 2016 at 15:09
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    $\begingroup$ It may not make it any more correct, but is it more useful for reasoning about what's going on? That's the real question. $\endgroup$ Sep 10, 2016 at 15:18
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    $\begingroup$ @PeterShor could you elaborate a bit on what you mean by 'because of the Heisenberg Uncertainty Principle'? I don't see exactly how the virtual particles follow from the principle. $\endgroup$ Sep 18, 2016 at 1:30

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