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Added to Hadamard distribution the standard expression for the standard choice on Minkowski spacetime, as well as statement and proof of its contour integral representation (here)
I have re-done the discussion of the Hadamard propagator for the case of the Klein-Gordon operator on Minkowski spacetime (harmonized conventions, expanded details) in accord with the analogous edits at advanced and retarded propagator. Also I moved it from the “Definition”-section to the “Example”-section, where it clearly belongs. Now its here:
What I am still lacking is a direct explicit computation of the wave front set of the Hadamard propagator on Minkowski spacetime. I know how to follow my nose, but presently I still get stuck with turning the analysis of the decay property of the required Fourier transform from a vague argument to an actual proof. I am sure I am missing something basic. If anyone wants to lend a hand, I’d appreciate it.
Wish I could help. Would it be right to think that this kinds of maths is outside the mathematical comfort zone of most people here, or should any good mathematical training have provided someone with these tools?
Wish I could help. Would it be right to think that this kinds of maths is outside the mathematical comfort zone of most people here, or should any good mathematical training have provided someone with these tools?
It should be just basic analysis, nothing fancy. Where is Todd these days? :-)
Oh, I think I get it now, with a hint from Igor. I know the decomposition of the causal propagator as the sum of a singular distribution and a smooth part from (2.3.18) in Scharf’s “Finite QED”, and with that it should all be immediate.
I’ll try to write it out now. Or maybe after I wrote out the Feynman propagator first…
Igor Khavkine kindly alerted me that while saying “Hadamard propagator” for the 2-point function in Hadamard states is logical, unfortunately “Hadamard propagator” has another established meaning (vaccum expectation of the anti-commutator $\langle (\Phi(x) \Phi(y) + \Phi(y)\Phi(x))\rangle$).
Therefore to avoid clash of terminology, what I had been calling “Hadamard propagator” needs to be called instead “Hadamard 2-point function” or similar, or else Wightman propagator.
I have made the change to the latter now throughout. I hope I have cought all occurences.
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