= Solution
\b[The displayed oscillator expansion and the displayed <action> require two separate corrections to describe one conventional positive-energy theory.] First compute what the oscillator algebra literally implies. Assume the two oscillator species commute with each other and annihilate the <vacuum>. Same-charge contractions vanish, so
$$
\boxed{\langle T\phi(x)\phi(y)\rangle_0
=\langle T\phi^*(x)\phi^*(y)\rangle_0=0.}
$$
For the two mixed contractions the printed phases give the same kernel, with $z=x-y$:
$$
\begin{aligned}
C_{\mathrm{lit}}(z)
&=\theta(z^0)\int\frac{d^3p}{(2\pi)^3\,2E_{\mathbf p}}e^{+ip\cdot z}
+\theta(-z^0)\int\frac{d^3p}{(2\pi)^3\,2E_{\mathbf p}}e^{-ip\cdot z}\\
&=\int\frac{d^4p}{(2\pi)^4}\frac{-i\,e^{-ip\cdot z}}{p^2-m^2-i0}.
\end{aligned}
$$
Thus $\langle T\phi(x)\phi^*(y)\rangle_0=\langle T\phi^*(x)\phi(y)\rangle_0=C_{\mathrm{lit}}(z)$. In $D$ dimensions replace $d^3p$ and $d^4p$ by $d^{D-1}p$ and $d^Dp$; the source's explicit three-dimensional spatial measure is the $D=4$ specialization. The covariant integral is manifestly <Lorentz invariant>. With
$$
D_F(z)=\int\frac{d^Dp}{(2\pi)^D}\frac{i\,e^{-ip\cdot z}}{p^2-m^2+i0},
$$
we have $C_{\mathrm{lit}}(z)=D_F(z)^*$. This <reversed-frequency scalar contraction> has the opposite pole prescription to the usual positive-energy <Feynman propagator>. For a single zero-<momentum> mode at positive time separation, the printed contraction is $e^{+imt}/(2m)$, while $D_F$ is $e^{-imt}/(2m)$: this is a concrete phase counterexample, rather than an immaterial change of integration variable.
There is also a <normalization of a complex scalar propagator> conflict independent of those phases. The displayed <Lagrangian density> gives the <canonical momentum> $\pi_\phi=\dot\phi^*/2$. Substituting the printed modes yields
$$
[\phi(t,\mathbf x),\pi_\phi(t,\mathbf y)]
=-\frac i2\delta^{(3)}(\mathbf x-\mathbf y),
$$
instead of the canonical $+i\delta^{(3)}$. Reversing the mode phases repairs the sign but leaves $+i\delta^{(3)}/2$, so it does not repair the normalization. The <complex scalar field> convention with unit mixed propagator normally has no $1/2$ in its quadratic action.<Quantum Field Theory by David Tong>[§2.5, equations (2.67)–(2.73): complex-field kinetic normalization and canonical commutators]{ref}
To retain the actual action in the paper, use $\phi=\phi_1+i\phi_2$ with canonically normalized real components, or equivalently replace its mode expansion by
$$
\phi(x)=\int\frac{d^{D-1}p}{(2\pi)^{D-1}\sqrt{E_{\mathbf p}}}
\left(a_{\mathbf p}e^{-ip\cdot x}+b_{\mathbf p}^\dagger e^{ip\cdot x}\right).
$$
The corresponding <path integral> with the usual <Feynman i-epsilon prescription> has
$$
\boxed{C(x,y)\equiv\langle T\phi(x)\phi^*(y)\rangle_0
=\langle T\phi^*(x)\phi(y)\rangle_0=2D_F(x-y),}
$$
with the two same-charge propagators still zero. This action-consistent kernel is used in <4/b> and <4/c>. Keeping conventional unit modes instead would require removing the quadratic factors $1/2$ from the action as well as reversing the phases. Those are distinct theories unless the field and its coupling are rescaled together.
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