The free real quantum field, satisfying $[\hat\phi(x),\hat\phi(y)]=\mathrm{i}\!\Delta(x-y)$, $\hat\phi(x)^\dagger=\hat\phi(x)$, with the conventional vacuum state, which has a moment generating function
$\omega(\mathrm{e}^{\mathrm{i}\hat\phi(f)})=\mathrm{e}^{-(f^*,f)/2}$ , where
$(f,g)$ is the inner product $(f,g)=\int f^*(x)\mathsf{C}(x-y)g(y)\mathrm{d}^4x\mathrm{d}^4y$, $\mathsf{C}(x-y)-\mathsf{C}(y-x)=\mathrm{i}\!\Delta(x-y)$,
has a representation as
$$\hat\phi_r(x)^\dagger=\hat\phi_r(x)=\frac{\delta}{\delta\alpha(x)}+\pi\alpha^*(x)
+\int \mathsf{C}(z-x)\alpha(z)\mathrm{d}^4z,$$
in terms of multiplication by $\alpha(x)$ and functional differentiation $\frac{\delta}{\delta\alpha(x)}$.
Because $$\left[\frac{\delta}{\delta\alpha(x)}+\pi\alpha^*(x),
\int \mathsf{C}(z-y)\alpha(z)\mathrm{d}^4z\right]=\mathsf{C}(x-y),$$
it is straightforward to show that $\hat\phi_r(x)$ verifies the commutation relation of the free real quantum field.
For the Gaussian functional integral
$$\omega(\hat A_r)=\int\hat A_r\mathrm{e}^{-\pi\int\alpha^*(x)\alpha(x)\mathrm{d}^4x}
\prod\limits_z\mathrm{d}^4z\mathrm{d}\alpha^*(z)\mathrm{d}\alpha(z),$$
we find, as required,
$$\begin{eqnarray}
\omega(\mathrm{e}^{\mathrm{i}\hat\phi_r(f)})&=&\int\mathrm{e}^{\mathrm{i}\hat\phi_r(f)}
\mathrm{e}^{-\pi\int\alpha^*(x)\alpha(x)\mathrm{d}^4x}
\prod\limits_z\mathrm{d}^4z\mathrm{d}\alpha^*(z)\mathrm{d}\alpha(z)\cr
&=&\int\exp\left[\mathrm{i}\!\!\int\!\! \mathsf{C}(z-x)\alpha(z)\mathrm{d}^4z\right]
\exp\left[\mathrm{i}\left(\!\!\frac{\delta}{\delta\alpha(x)}+\pi\alpha^*(x)
\right)\right]\cr
&&\qquad\qquad\times\mathrm{e}^{-(f^*,f)/2}
\mathrm{e}^{-\pi\int\alpha^*(x)\alpha(x)\mathrm{d}^4x}
\prod\limits_z\mathrm{d}^4z\mathrm{d}\alpha^*(z)\mathrm{d}\alpha(z)
=\mathrm{e}^{-(f^*,f)/2}.
\end{eqnarray}$$
The last equality is a result of $\frac{\delta}{\delta\alpha(x)}+\pi\alpha^*(x)$ annihilating $\mathrm{e}^{-\pi\int\alpha^*(x)\alpha(x)\mathrm{d}^4x}$, and the Gaussian integral annihilates powers of
$\int\!\! \mathsf{C}(z-x)\alpha(z)\mathrm{d}^4z$.

This is a largely elementary transposition of the usual representation of the SHO in terms of differential operators, with a not very sophisticated organization of the relationships between points in space-time and operators, so I imagine something quite like this might be found in the literature. References please, if anyone knows of any?

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