Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  1-form formulation of quantized electromagnetism

+ 6 like - 0 dislike
1121 views

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.

This post imported from StackExchange Physics at 2015-02-01 09:22 (UTC), posted by SE-user Peter Morgan
asked May 19, 2011 in Theoretical Physics by Peter Morgan (1,230 points) [ no revision ]
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.

This post imported from StackExchange Physics at 2015-02-01 09:22 (UTC), posted by SE-user David Z
There is no difference between your formalism and any of the standard ones. A is a one form in all of them.

This post imported from StackExchange Physics at 2015-02-01 09:22 (UTC), posted by SE-user Ron Maimon

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOver$\varnothing$low
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...