= Solution
The corrected expression in <3/d> is a <vacuum two-point function>, or <Wick contraction>, of the free <Dirac field>. It equals the negative spinor transpose of the usual <Dirac propagator> with reversed spacetime arguments. It supplies the internal fermion lines in <Feynman diagrams>, rather than a position-space detection probability.
For $x^0>y^0$, $\psi(y)$ creates an <antiparticle> and $\bar\psi(x)$ annihilates it. The $v\bar v$ term describes positive-energy <antiparticle> propagation from $y$ to $x$. For $y^0>x^0$, $\bar\psi(x)$ creates a <particle> and $\psi(y)$ annihilates it. The $u\bar u$ term describes positive-energy <particle> propagation from $x$ to $y$, with the minus sign required by exchanging the odd operators in <fermionic time ordering>. Neither term describes a physical negative-energy excitation.
The <Heaviside step functions> select these chronological alternatives. The factors $e^{\mp ip\cdot z}$ are on-shell plane waves with $p^0=E_{\mathbf p}>0$, and $d\Pi_p$ is the positive-energy <Lorentz-invariant phase-space measure>. The sum over $s$ includes the two <spin> states. With standard relativistic spinor normalization, the <fermion spin sums> are
$$
\sum_s u^s(\mathbf p)\bar u^s(\mathbf p)=\not{p}+\mu,\qquad
\sum_s v^s(\mathbf p)\bar v^s(\mathbf p)=\not{p}-\mu.
$$
They encode the <spinor> structure and solve the corresponding on-shell <Dirac equations>. In particular, the corrected answer can also be written
$$
C_{ab}(x,y)=\int\frac{d^4p}{(2\pi)^4}\frac{i(\not{p}-\mu)_{ba}}{p^2-\mu^2+i0}\,e^{-ip\cdot(x-y)}.
$$
The <Feynman i-epsilon prescription> specifies the temporal boundary conditions of the <Green function>; its positive- and negative-frequency residues reproduce the two terms and their relative minus sign. With $S_F$ defined in <3/d>,
$$
(i\not{\partial}-\mu)S_F(w)=i\delta^{(4)}(w)I.
$$
The contact term is fixed by the equal-time <canonical anticommutation relations>. Choosing a value for $\theta(0)$ does not change the unequal-time sign demonstrated above.
Finally, <microcausality> requires field <anticommutators> to vanish at spacelike separation; the time-ordered <vacuum two-point function> itself need not vanish there. These facts are compatible with relativistic causality. The plus-plus expression printed in the paper lacks the <fermionic sign> needed for this standard <Dirac propagator> interpretation.
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