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I am trying to calculate the vacuum expectation value of the time order of a product of the derivative of a scalar field $$ \langle T \partial_{\mu} \phi \partial_{\nu} \phi \rangle. $$ Using the canonical formalism and part-integral formulation seemingly leads to different results.

Gerstein et. al once addressed this issue in "Chiral Loops Ira S. Gerstein, Roman Jackiw, Benjamin W. Lee, and Steven Weinberg Phys. Rev. D 3, 2486, 1971". It is claimed —without any proof— that via defining $$ \Delta_{\mu\, \nu}(k)\equiv \int d^4 x e^{i k x} \langle T \partial_{\mu} \phi \partial_{\nu} \phi \rangle $$ we get $$ \Delta_{\mu\, \nu}(k) = i \dfrac{k_{\mu} k_{\nu}}{k^2+i \epsilon} -i g_{\mu 0} g_{\nu 0}. $$ As discussed by the authors, using the "naive perturbation theory", we miss non-covariant contribution to the propagator.

The question is that what is wrong with conventional perturbation theory that leads to this mistake.

PS: One may find this question relevant: Time ordering and time derivative in path integral formalism and operator formalism

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