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In Minkowski spacetime, the commutator of the Klein-Gordon field operator with itself at different spacetime points evaluates to the advanced minus retarded Green's function (i.e., the "causal propagator") of the classical theory,

\begin{align} [\phi (x),\phi (y)]=\langle 0|[\phi (x),\phi (y)]|0\rangle =G_A(x-y)-G_R(x-y), \tag{*}\label{*} \end{align} where $x,y$ are 4-vectors and I use the convention that the K-G Green's functions are defined by $(\partial^2+m^2)G(x-y)=-i\delta^{(4)}(x-y)$.

Now, the canonical equal-time commutation relation for this theory are

\begin{align} [\phi (x^0,\vec{x}),\partial_{y^0} \phi (y^0,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y}). \end{align}

It appears to me that the canonical commutation relations imply differentiating \eqref{*} with respect to $y^0$ and subsequently setting $x^0=y^0$ should, therefore, produce the 3-dimensional delta function: i.e.,

\begin{align} \partial_{y^0} [\phi (x)\phi (y)]|_{x^0=y^0}=\partial_{y^0} \left(G_A(x-y)-G_R(x-y)\right)|_{x^0=y^0}=i\delta^{(3)}(\vec{x}-\vec{y})\tag{**}\label{**} \end{align}

Q: Does \eqref{**}, in fact, hold? If so, how does it follow from the definition/properties of the advanced and retarded Green's functions?

Update: I think the \eqref{**} can be justified using the commutator directly. Note first

\begin{align} \partial_{y^0}\langle 0|[\phi(x)\phi(y)]0\rangle&=\partial_{y^0}\int\frac{d^3 k}{(2\pi)^3}\frac{1}{2\omega_{\vec{k}}}\left\{e^{-i\omega_\vec{k}(x^0-y^0)+i\vec{k}\cdot(\vec{x}-\vec{y})}-e^{i\omega_\vec{k}(x^0-y^0)+i\vec{k}\cdot(\vec{x}-\vec{y})}\right\}\\ &=\frac{i}{2}\int\frac{d^3 k}{(2\pi)^3}\left\{e^{-i\omega_\vec{k}(x^0-y^0)+i\vec{k}\cdot(\vec{x}-\vec{y})}+e^{i\omega_\vec{k}(x^0-y^0)+i\vec{k}\cdot(\vec{x}-\vec{y})}\right\}. \end{align} Subsequently setting $x^0=y^0$ then yields a delta function identity, \begin{align} \partial_{y^0}\langle 0|[\phi(x)\phi(y)]0\rangle|_{y^0=x^0}=\int \frac{d^3k}{(2\pi)^3}e^{i\vec{k}\cdot(\vec{x}-\vec{y})}=\delta^{(3)}(\vec{x}-\vec{y}). \end{align}

It's still somewhat mysterious to me why the time derivative of the advanced minus retarded Green's functions should lead to such a relation. Or how general such a relation is--i.e., if it would hold for other QFTs.

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