In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps to cite. Taking the electromagnetic field ˆA to be an (operator-valued distributional) 1-form, we can write a smeared operator ˆAH in terms of the inner product on the exterior algebra as ⟨ˆA,H⟩ (taking the test function H also to be a 1-form, but satisfying Schwartz space-like smoothness conditions in real space and in fourier space instead of being a distribution). ˆA projects into annihilation and creation parts, ˆA+ and ˆA− respectively, ˆA=ˆA++ˆA−, for which the action on the vacuum is defined by ˆA+|0⟩=0, and we have the commutation relations $\Bigl[\bigl<\hat A^+,H\bigr>,\bigl<\hat A^-,J\bigr>\Bigr]=\left
The commutation relations are not positive semi-definite for arbitrary test functions H, [⟨ˆA+,H∗⟩,⟨ˆA−,H⟩]≱0, which is fixed by the Gupta-bleuler condition, which can be stated in this formalism as δˆA+|ψ⟩=0, for all states |ψ⟩, not just for the vacuum state.
By the Hodge decomposition theorem, we can write the test function H uniquely as H=dϕ+δF+ω, where ϕ is a 0-form, F is a 2-form, and ω is a harmonic 1-form, so we can write ⟨ˆA+,H⟩|ψ⟩=⟨ˆA+,dϕ+δF+ω⟩|ψ⟩=(⟨δˆA+,ϕ⟩+⟨ˆA+,δF+ω⟩)|ψ⟩=⟨ˆA+,δF+ω⟩|ψ⟩.
So this is a reference request. Is there any literature that uses this kind of mathematical formalism for the quantized electromagnetic field? Even vaguely the same! My sense is that AQFT has moved to more abstract methods, while more practical interacting QFT has become historically committed to index notations that are little changed from 50 years ago, even though the methods of interacting quantum fields have changed in many other ways, and that mathematicians who take on the structures of QFT, as Folland does in http://www.amazon.com/Quantum-Theory-Mathematical-Surveys-Monographs/dp/0821847058, make strenuous efforts not to let their notation and methods stray too far from the mainline.
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