I'm studying QFT by David Tong's lecture notes.
When he discusses causility with real scalar fields, he defines the propagator as
$$D(x-y)=\left\langle0\right|\phi(x)\phi(y)\left|0\right\rangle=\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_{\vec{p}}}e^{-ip\cdot(x-y)}$$
then he shows that the commutator of two scalar fields at arbitrary spacetime points $x,y$ is $$[\phi(x),\phi(y)]=D(x-y)-D(y-x),$$
which since the lhs involves operators, I assumed there should also be an implicit identity operator multiplying the rhs.
When he moves on to discuss the propagator of a fermion, however, he defines
$$iS_{\alpha\beta}=\{\psi_{\alpha}(x),\bar{\psi}_{\beta}(y)\},$$ as the propagator, where $\bar{\psi}=\psi^{\dagger}\gamma^{0}$ is the Dirac adjoint.
Could anyone explain to me from where this definition comes from? I find it confusing, since he didn't call the commutator of two scalar fields as the propagator before, he defined it as the vacuum expectation of the field in two different points as $D(x-y)$. Or are the propagators named differently for scalars and spinors? Why does it has to be a matrix instead of a number (meaning, why define it with the adjoint to the right of the anti-commutator instead of the other way around)? Also, why define it with an $i$ factored out?
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