Matterwave said:
In Peskin and Schroeder, we calculate that the propagator D(x-y)=\langle 0|\phi(x)\phi(y)|0\rangle to be non zero even if x and y are space-like separated (although it's exponentially decaying). This suggests (given our interpretation of this quantity) that a particle CAN propagate between these two points (albeit with vanishingly small probability).
Peskin argues, however, that this is not a violation of causality, because causality should only prevent measurements from affecting each other over space-like separations. He goes on to say that, we really should calculate:
\langle 0|[\phi(x),\phi(y)]|0\rangle
And that since this is 0, causality is preserved. Can someone elaborate this point for me? I'm not sure I quite understand what the commutator has to do with measurements, and why it is this quantity that is important and not the D(x-y) which is important.
The propagator treated in P&S is the "
propagator from source" of the
complex Klein Gordon field. It's a somewhat unlucky example because
this particular propagator is non-zero outside the light cone while the
really important propagators don't have this problem.
It's also a propagator from a source (rather than a self propagator) just
like A^\mu is propagated from j^\mu
The propagator from source of the real valued Klein Gordon field is zero
outside the lightcone. You can read this back in the P&S text if you read
between the lines where they talk about the requirement of having to take
positive and negative charged particles together in order to eliminate the
part outside the lightcone.
The propagator from source of the Real Klein Gordon field.
<br />
\tilde{\mathcal{K}}(x^i,t) ~=~<br />
\tfrac{1}{4\pi}\left[~\delta\left(\tfrac12 s^2\right)-\frac{m}{s}J_1(ms)~\theta\left(\tfrac12 s^2\right)\right]<br />
~~~~~~~~\mbox{with} ~~ s^2=t^2-x^2 \gt 0
The most relevant propagator is the
Dirac self propagator, as used in the
Feynman diagrams,which is strictly zero outside the lightcone. The easiest
way to derive it is with the help of the real Klein Gordon propagator {\cal K}
The self-propagator of the Dirac field
<br />
\tilde{\mathbb{D}}(x^i,t) =<br />
\left[<br />
\left(\begin{array}{ll} 1 ~~ 0 \\ 0 ~~ 1 \end{array}\right)<br />
\frac{\partial }{\partial t} -<br />
c\left(\begin{array}{ll} \sigma^i~~~~0\\0\,-\sigma^i\end{array}\right)<br />
\frac{\partial}{\partial x^i} -<br />
i\left(\begin{array}{ll} 0 ~~ 1 \\ 1 ~~ 0 \end{array}\right)<br />
\frac{mc^2}{\hbar}<br />
\right] \tilde{\mathcal{K}}(x^i,t)<br />
The space and time derivatives work first on {\cal K} and the result is the
appropriate Green's function which is has to be convoluted with the Dirac
field to obtain the time evolution of the field. The time and space derivatives
of {\cal K} are the second and third function in this table.
<br />
\begin{array}{|c|c|}<br />
\hline & \\<br />
& \\<br />
\mbox{Space-Time domain } & \mbox{Momentum-Time domain} \\<br />
& <br />
\\ & \\<br />
& \\<br />
\tfrac{1}{4\pi}\left[~\delta\left(\tfrac12 s^2\right)-\frac{m}{s}J_1(ms)~\theta\left(\tfrac12 s^2\right)\right] &<br />
\frac{\sin\left(t\sqrt{p^2+m^2}~\right)}{\sqrt{p^2+m^2}}<br />
\\ & \\<br />
\tfrac{1}{4\pi}~t\,\left[~\delta'\left(\tfrac12 s^2\right)-<br />
\frac{m^2}{2}\delta\left(\tfrac12 s^2\right)+<br />
\frac{m^2}{s^2}J_2(ms)~\theta\left(\tfrac12 s^2\right)\right] &<br />
\cos\left(t\sqrt{p^2+m^2}~\right)<br />
\\ & \\<br />
\tfrac{1}{4\pi} x^i \left[~\delta'\left(\tfrac12 s^2\right)-<br />
\frac{m^2}{2}\delta\left(\tfrac12 s^2\right)+<br />
\frac{m^2}{s^2}J_2(ms)~\theta\left(\tfrac12 s^2\right)\right] &<br />
-\frac{ip^i \sin\left(t\sqrt{p^2+m^2}~\right)}{\sqrt{p^2+m^2}}<br />
\\ & \\<br />
\hline<br />
\end{array}<br />
With \delta' (z)= \partial\delta(z)/\partial z
The causality requirement
The requirement for the propagator to be causal (inside the light-cone)
is that it does not contain the non-local operator in the space-time domain:
\sqrt{-\frac{\partial^2}{\partial x^2} + m^2}
Or as expressed in the momentum-time domain
\sqrt{p^2 + m^2}
This operator is a non local convolution with a Bessel K function which
causes the part outside the light cone. You can check the momentum-
time propagators in the table. If you expand the sine and cosine functions
then you'll find that all the square roots are eliminated because there are
only even powers of the square root and no odd powers.
This is the reason the propagators are local, which is not the case with
the propagator treated in P&S which contains \exp(-i\sqrt{p^2+m^2}\,t)
Regard, Hans
