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 $\hat A$ to be an (operator-valued distributional) 1-form, we can write a smeared operator $\hat A_H$ in terms of the inner product on the exterior algebra as $\bigl<\hat A,H\bigr>$ (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). $\hat A$ projects into annihilation and creation parts, $\hat A^+$ and $\hat A^-$ respectively, $\hat A=\hat A^+ +\hat A^-$, for which the action on the vacuum is defined by $\hat A^+\left|0\right>=0$, and we have the commutation relations $\Bigl[\bigl<\hat A^+,H\bigr>,\bigl<\hat A^-,J\bigr>\Bigr]=\left<H,E(J)\right>$, where $\widetilde{E(J)}(k)=2\pi\delta(k^2)\theta(k_0)\tilde J(k)$ projects to the positive frequency forward light-cone.

The commutation relations are not positive semi-definite for arbitrary test functions $H$, $\Bigl[\bigl<\hat A^+,H^*\bigr>,\bigl<\hat A^-,H\bigr>\Bigr]\not\ge 0$, which is fixed by the Gupta-bleuler condition, which can be stated in this formalism as $\delta\hat A^+\left|\psi\right>=0$, for all states $\left|\psi\right>$, not just for the vacuum state.

By the Hodge decomposition theorem, we can write the test function $H$ uniquely as $H=d\phi+\delta F+\omega$, where $\phi$ is a 0-form, $F$ is a 2-form, and $\omega$ is a harmonic 1-form, so we can write $$\bigl<\hat A^+\!,H\bigr>\left|\psi\right>\! =\bigl<\hat A^+\!,d\phi+\delta F+\omega\bigr>\left|\psi\right>\! =\Bigl(\!\bigl<\delta\hat A^+,\phi\bigr>+\bigl<\hat A^+\!,\delta F+\omega\bigr>\!\Bigr) \left|\psi\right>\! =\bigl<\hat A^+\!,\delta F+\omega\bigr>\left|\psi\right>.$$ The harmonic 1-form $\omega$ has to be zero to satisfy the Schwartz space condition, and by the left action of $\hat A^-$ on arbitrary states we have the same projection of the arbitrary test function $H$ to $\delta F$, so all we are left with is the electromagnetic field observable $\hat\Phi_F=\hat A_{\delta F}=\bigl<\hat A,\delta F\bigr>$. Unsurprisingly, in the free field case, in the absence of charges, the electromagnetic potential observable is exactly equivalent to just the electromagnetic field observable, for which we can verify that $\Bigl[\bigl<\hat A^+,\delta F^*\bigr>,\bigl<\hat A^-,\delta F\bigr>\Bigr]\ge 0$, using which we can use the GNS construction of a free field Hilbert space.

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.

  • $\begingroup$ Just stumbled across this again... I'm not familiar with the literature but I'll take a look around and see if I can find anything, since this is a really good question and I think for the good of the site, we need to be able to come up with answers for these sorts of things. $\endgroup$ – David Z Jul 18 '11 at 3:41
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    $\begingroup$ There is no difference between your formalism and any of the standard ones. A is a one form in all of them. $\endgroup$ – Ron Maimon Sep 3 '11 at 6:46

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