Confused with computing causality for Dirac field

Question

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.

Answer 1

This is why it is easier to see what’s happening with indices rather than in matrix form.

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.

Hope this helps