Causality for the Dirac Field

Question

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 commutator $\left[\Psi(x),\,\bar{\Psi}(y)\right]$ 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 $\left[\bar{\Psi}(x),\,\bar{\Psi}(y)\right]$.

Answer 1

I find this much better motivated in Weinberg's text "Quantum Theory of Fields", which starts from the idea of particles, and what we measure, rather than from a Lagrangian formulation. Essentially, what we are after is a Poincar\'e invariant and unitary S-matrix. One crucial thing that we require is a Hamiltonian density, transforming as a scalar, and commuting at spacelike separated points. This last requirement is to ensure that the time-ordering in the Dyson series is Lorentz invariant. Further, it should be built from annihilation and creation operators.

The obvious way to build such a Hamiltonian density is by packaging the annihilation and creation operators into fields in such a way that they transform under some representation of the Lorentz group; it is only here that fields make an appearance. In particular, it is rather unclear what the fields mean physically: can one measure the electron field at any particular point? I would avoid asking questions directly of the physical meaning of the fields, or their commutators or whatever, but focus instead on how their properties translate into measurable quantities, like scattering amplitudes.

With this philosophy, the commutation properties of the fields follow from the requirement that the Hamiltonian densities you build from them must commute at spacelike separated points.

I'd thoroughly recommend reading through Weinberg for the details, and in particular how the different representations give rise to the spin-statistics theorem. It's not easy going to begin with but it feels very much less ad-hoc than P&S, and gives much better motivation for why QFT looks the way it does.