= Solution
Measure the <statistical Hamiltonian> in units of $k_BT$ so that the weight is $e^{-H_0}$, and take $\alpha>0$, $m^2>0$ for convergence. Use the <Fourier transform> convention
$$
\phi(x)=\int_p e^{ip\cdot x}\widetilde\phi(p),\qquad
\int_p=\int\frac{d^Dp}{(2\pi)^D},\qquad
\widetilde\phi(-p)=\widetilde\phi(p)^*.
$$
The last identity is the reality condition, so modes at $p$ and $-p$ are not independent complex fields. Orthogonality of the Fourier exponentials gives $\int d^Dx\,\phi^2=\int_p|\widetilde\phi(p)|^2$, while a derivative multiplies a Fourier mode by $ip$. Hence
$$
\boxed{H_0=\frac12\int_p\widetilde\phi(p)\widetilde\Delta(p)\widetilde\phi(p)^*,\qquad
\widetilde\Delta(p)=\alpha^{-1}p^2+m^2.}
$$
The kernel is positive for the stated parameters. Keeping an explicit inverse temperature instead would replace this kernel in the statistical weight by $\beta_{\rm th}\widetilde\Delta$; the following source convention absorbs that factor.
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