Preserving Causality for Field Operators
It is
important to examine the quantity:
\begin{eqnarray}
\langle 0 \vert \phi(x)\phi(y)\vert 0 \rangle
\end{eqnarray}
Or two field operators at different points in space time, acting on the
vacuum. In particular, for a causal theory, we would like all space-like
separated operatores to commute, i.e.:
\begin{eqnarray}
[\mathcal{O}(x),\mathcal{O}(y)] &=& 0 \ \mathrm{if} \ \eta_{\mu
\nu}x^\mu y^\nu <0
\end{eqnarray}
Or two space-like separated observables cannot effect each other. Let us
examine the commutator for our mode-expanded fields:
\begin{eqnarray*}
\phi(x) &=& \sum_k \frac{1}{\sqrt{2L^3
\omega_k}}\left(a_ke^{-ikx}+a^*_k e^{ikx}\right)\\
\left[\phi (x), \phi (y)\right] &=& \sum_{k,p} \frac{1}{2L^3}
\frac{1}{\sqrt{\omega_k \omega_p}}\left[\left(a_ke^{-ikx}+a^*
e^{ikx}\right)\left(a_pe^{-ipy}+a^*_p e^{ipy}\right)\right]\\
&=& \sum_{k,p} \frac{1}{2L^3} \frac{1}{\sqrt{\omega_k
\omega_p}}\left[a_k,a^*_p \right]e^{-ikx+ipy}+\left[a^*_k,a_p
\right]e^{+ikx-ipy}\\
&=& \sum_{k,p} \frac{1}{2L^3} \frac{1}{\sqrt{\omega_k
\omega_p}}\delta_{k,p}\left(e^{-ikx+ipy}-e^{+ikx-ipy}\right)\\
&=& \sum_{k,p} \frac{1}{2L^3}
\frac{1}{\omega_k}\left(e^{-ik(x-y)}-e^{+ik(x-y)}\right)
\end{eqnarray*}
We can know re-write this difference in space time coordinates as:
\begin{eqnarray}
x-y &=& \langle \tau, \vec{\xi} \rangle
\end{eqnarray}
and expand out the four vector contractions:
\begin{eqnarray}
\left[\phi(x),\phi(y) \right]&=& \sum_{k,p} \frac{1}{2L^3}
\frac{1}{\omega_k}\left(e^{-i\omega_k
\tau}e^{ik\xi}-e^{+i\omega_k\tau}e^{-ik\xi}\right)
\end{eqnarray}
Turning our integral into a sum now, we write:
\begin{eqnarray}
\sum_k \frac{1}{V} \to \int \frac{d^3k}{(2\pi)^3}\\
\left[\phi(x),\phi(y) \right]&=& \int \frac{d^3k}{(2\pi)^3} \frac{1}{2
\omega_k}\left(e^{-i\omega_k
\tau}e^{ik\xi}-e^{+i\omega_k\tau}e^{-ik\xi}\right)
\end{eqnarray}
Integration over our angular coordinates is trivial, since we can set
$\vec{\xi}$ along the $k_z$ axis and get rid of $\theta$ immediately.
\begin{eqnarray}
\left[\phi(x),\phi(y) \right]&=& \int_0^\infty \int_{1}^{-1} \frac{k^2
dk du }{(2\pi)^2} \frac{1}{2 \omega_k}\left(e^{-i\omega_k \tau}e^{ik\xi
u}-e^{+i\omega_k\tau}e^{-ik\xi u}\right)
\end{eqnarray}
Notice we have set $u=\cos(\psi)$, the angle from $k_z$, above. Integrating
over $u$ gives us the typical bessel functions:
\begin{eqnarray}
\left[\phi(x),\phi(y) \right]&=& \int_0^\infty \frac{k^2 dk
}{(2\pi)^2} \frac{1}{2 \omega_k}\left(\frac{e^{-i\omega_k \tau}e^{ik\xi
u}}{ik\xi}-\frac{e^{+i\omega_k\tau}e^{-ik\xi u}}{-ik\xi} \right)
\vert^{-1}_{1}\\
\left[\phi(x),\phi(y) \right]&=& \frac{-i}{\xi}\int_0^\infty \frac{k
dk }{(2\pi)^2} \frac{1}{2 \omega_k}\left(e^{-i\omega_k \tau}e^{ik\xi
u}+e^{+i\omega_k\tau}e^{-ik\xi u}\right) \vert^{-1}_{1}
\end{eqnarray}
Taking a long hard look at the above equation, one might be able to see that
we're going to get zero if $\tau=0$ since, the evaluation at endpoints of $u$
will cancel. We can also expand out the endpoints of $u$ to get:
\begin{eqnarray}
\left[\phi(x),\phi(y) \right]&=& \frac{1}{\xi}\int_0^\infty
\frac{dk}{2\pi^2} \frac{k}{\omega_k}\sin(k\xi)\sin(\sqrt{k^2+m^2}\tau)
\end{eqnarray}
Now we see immediately that the $\tau \to 0$ limit gives us a vanishing
commutator, and the $\xi \to 0$ limit gives us something that is finite, like
\begin{eqnarray}
\left[\phi(x),\phi(y) \right]&=& \int_0^\infty \frac{dk}{2\pi^2}
\frac{k^2}{\sqrt{k^2+m^2}}\sin(\sqrt{k^2+m^2}\tau)
\end{eqnarray}
This will give an asymptotic solution of the form $e^{imt}$ which can be shown
by stationary phase arguments. And we find, therefore, that the commutator for
space-like separation is non-vanishing.
It is important to note that, during this entire discussion, we could have
framed the commutator problem as the difference of two propagators, the
probability for a particle to be created at $y$ and destroyed at $x$:
\begin{eqnarray}
\langle 0 \vert \phi(x) \phi(y)\vert 0 \rangle &=& D_{xy}
\end{eqnarray}
and therefore our commutator looks like:
\begin{eqnarray}
[\phi(x),\phi(y)] &=& D_{xy}-D_{yx}=0 \ \forall \ \eta_{\mu \nu}x^\mu
y^\nu <0
\end{eqnarray}
David Tong, in his online notes, says we can ``wrap words around this''. When
the events of creation and annihilation are space-like separated, we can
``re-order'' the events by a proper Lorentz boost. This means that the two
amplitudes for the processes $x\to y$ and $y \to x$ cancel. For a complex
scalar field, this turns into a statement about the amplitude of a particle
and an anti-particle travelling from $x \to y$, which once again cancel for
space-like separation.