# Causality for the Dirac Field

In Peskin & Schroeder page 54, they are trying to show how far they can take the idea of a commutator for the Dirac field instead of anti-commutator. To this end they are examining causality, which they choose to test by looking at the quantity $\left[\Psi(x),\,\bar{\Psi}(y)\right]$ at non-equal times. This quantity indeed turns out to be zero outside the light-cone (that is, for $g(x-y,\,x-y)<0$ where $g$ is the metric), but in a somewhat pathological way (both the $a$ particles and the $b$ particles which propagate from $y$ to $x$ cancel each other, and the probability for propagation from $x$ to $y$ is identically zero).

My question is: why are we looking at $\left[\Psi(x),\,\bar{\Psi}(y)\right]$? What does this quantity symbolize physically? In chapter two, when examining causality for the Klein-Gordon field, we looked at $\left[\Phi(x),\,\Phi(y)\right]$. If the commutator of two operators is zero, then we may diagonalize both simultaneously with the same basis. So $\left[\Phi(x),\,\Phi(y)\right]=0$ meant that we may diagonalize $\Phi(x)$ and $\Phi(y)$ simultaneously and measuring one wouldn't interfere with measuring another (and when this is indeed zero outside the light cone we deem the theory to be compatible with our notion of causality).

Alternatively we also looked at $<0|\Phi(x)\Phi(y)|0>$ whose square was the probability of a particle to propagate from $y$ to $x$, which we would also want to be zero outside the lightcone.

If we interpret $|x>\propto\bar{\Psi}(x)|0>$, I would expect us to check causality for the Dirac field rather by either computing $<0|\Psi(x)\bar{\Psi}(y)|0>\propto<x|y>$ outside the lightcone (whose square would be the probability for a particle to propagate from $y$ to $x$) or $\left[\Psi(x),\,\Psi(y)\right]$ or $\left[\bar{\Psi}(x),\,\bar{\Psi}(y)\right]$.

• Thank you! I had a similar question for a long time. Which sections from the text would you recommend me to read if I want to understand the measurable quantities properly? For example, nowhere in Peskin and Schroeder's book do they define properly what the field $\phi$ exactly is, and what the basis actually means. It becomes implicit only later, when we see that $a_{\bar{p}}^{\dagger}|0\rangle$ is a one-particle momentum eigenstate with 3-momentum $\bar{p}$. – Abhinav Dec 13 '13 at 18:40