- #1

ianhoolihan

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Reading through Peskin and Schroeder, I came across the following statement, with regards to propagators:

To really discuss causality, however, we should ask not whether particles can propagate over spacelike intervals, but whether a measurement performed at one point can affect a measurement at another point whose separation from the first is spacelike. The simplest thing we could try to measure is the field [itex]\phi(x)[/itex], so we should compute [itex][\phi(x),\phi(y)][/itex]; if this commutator vanishes, one measurement cannot effect the other.

Could someone explain how the commutator is related to the measurement of the field in this context? Searching online, the only thing that crops up is the usual uncertainty principle and its relation to the commutator.

Secondly, as a somewhat aside point, PS go on to say that for a spacelike separation [itex]x-y[/itex], one can perform a continuous Lorentz transformation such that [itex]x-y \longrightarrow -(x-y)[/itex]. On the other hand, if the separation is timelike, this is not possible. Is there a nice way to prove this in matrix notation? E.g. one can show that [itex]\Lambda^2 = 1[/itex] and so on, but I think things would then get messy with components etc to show it is not possible for [itex](1,0,0,0)[/itex] but is for [itex](0,0,0,1)[/itex]...