# Confused with computing causality for Dirac field

In Peskin and Schroeder's QFT book, P.56 Eq.(3.95) mentions that

\begin{align} \langle 0|\bar\psi(y)_b\psi(x)_a|0\rangle = (\gamma \cdot p -m)_{ab}\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}Be^{ip(x-y)}\tag{3.95} \end{align}

When

\begin{align} \psi(x) = \int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}} \sum_s \big(a_p^su_p^s(p)e^{-ipx} + b_p ^{\dagger s} v^s(p)e^{ipx}\big) \end{align} \begin{align} \bar\psi(y) = \int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}} \sum_s \big(a_p^{\dagger s}\bar{u}_p^s(p)e^{ipy} + b_p^s\bar{v}^s(p)e^{-ipy}\big). \end{align}\tag{3.92}

The authors claim that letting $$a_p^s|0\rangle = 0$$ and $$\langle 0|a_p^{\dagger s}$$ and computing $$\langle 0|\bar{\psi}_b(y)\psi_a(x)|0\rangle$$ gives the term $$\sum_s v^s\bar v^s = (\gamma \cdot p - m)$$ in the middle of the calculation so the result is same as above.

But I found out that if I compute $$\bar \psi(y) \psi(x)$$ then the term with $$\sum_s \bar v^s v^s$$ emerge instead of $$v^s \bar v^s$$, which is equal to $$2m$$ from Eq(3.60) in the same book $$\bar u^r u^s = 2m\delta^{rs}$$.

And for another question I'm also curious about the index $$_{ab}$$ under the amplitude

\begin{align} \langle 0|\bar\psi(y)_b\psi(x)_a|0\rangle = (\gamma \cdot p -m)_{ab}\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}Be^{ip(x-y)}. \end{align}\tag{3.95}

Is it okay to write indices as $$_{ab}$$ instead of $$_{ba}$$? I know it's spinor index but not sure exactly.

Using your expressions of $$\psi,\bar\psi$$, when you calculate the propagator $$\langle \bar\psi_b \psi_a\rangle$$, you should encounter the expression: $$\sum_s \bar v^s_b(p)v^s_a(p)= (\not p-m)_{ab}$$ which is not your expression since you are summing over the $$a=b$$ (i.e. tracing out).
The order of the indices is determined by the above formula. Note that to make the bookkeeping more transparent, you could distinguish covariant/contravariant indices by lower/upper indices. For example, $$\psi^a$$ would be contravariant, $$\bar\psi_b$$ covariant, and $$(\gamma^\mu)_b^a$$ has a covariant and contravariant index. This gives some useful sanity checks to determine which index should be where.