= Solution
Let $\Sigma_t$ be a <Cauchy hypersurface> with induced metric $h_{ij}$, lapse $N$, shift $N^i$, and future unit normal $n^a$. For the normalization of the action in the question, the canonical momentum density is
$$
\Pi=2\sqrt h\,n^a\nabla_a\Phi
=\frac{2\sqrt h}{N}(\dot\Phi-N^i\partial_i\Phi).
$$
The equal-time <canonical commutation relations> are
$$
[\widehat\Phi(t,\mathbf x),\widehat\Phi(t,\mathbf y)]=0,
\qquad
[\widehat\Pi(t,\mathbf x),\widehat\Pi(t,\mathbf y)]=0,
$$
and
$$
\boxed{[\widehat\Phi(t,\mathbf x),\widehat\Pi(t,\mathbf y)]
=i\delta^{(3)}(\mathbf x-\mathbf y)}.
$$
With the conventional extra factor $1/2$ in the action, $\Pi$ loses the factor two.
For complex classical solutions, the <Klein-Gordon inner product> is
$$
(\phi_1,\phi_2)_{KG}
=i\int_{\Sigma_t}d\Sigma^a
(\phi_1^*\nabla_a\phi_2-\phi_2\nabla_a\phi_1^*).
$$
The integrand is a conserved current because both fields obey the <Klein-Gordon equation>. Applying the <divergence theorem> between two Cauchy hypersurfaces shows that the value is independent of the foliation, provided there is no boundary flux.
Solved by gpt-5.6-sol high.
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