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This is a reference request for a relationship in quantum field theory between the electromagnetic potential and the electromagnetic field when they are presented in test function form. $U(1)$ gauge invariance becomes a particularly simple constraint on test functions for smeared electromagnetic potential operators to be gauge invariant observables. This is such a simple constraint that I think it must be out there, but I have never seen this in text books or in the literature, presumably because we mostly do not work with test function spaces in QFT; instead we use operator-valued distributions directly, where, however, gauge fixing is a perpetual nuisance.

For the electromagnetic potential operator-valued distribution smeared by a test function $f^\rho(x)$ on Minkowski space, $\hat A_f=\int_M \hat A_\rho(x)f^{\rho*}(x)\mathrm{d}^4x$, to be an observable that is invariant under $U(1)$ gauge transformations $\hat A_\rho(x)\rightarrow\hat A_\rho(x)-\partial_\rho\alpha(x)$, we require that $\int_M \partial_\rho\alpha(x)f^{\rho*}(x)\mathrm{d}^4x$ must be zero for all scalar functions $\alpha(x)$.

Integrating by parts over a region $\Omega$ in Minkowski space, we obtain, in terms of differential forms, $$\int_\Omega d\alpha\wedge(\star f^*)=\int_{\partial\Omega}\alpha\wedge(\star f^*)-\int_\Omega \alpha\wedge(d\!\star\! f^*),$$ which will be zero for large enough $\Omega$, and hence for the whole of Minkowski space, for any smooth test function $f^\rho(x)$ that has compact support and is divergence-free, $d\!\star\! f=0$. [If we constrain the gauge transformation function $\alpha(x)$ not to increase faster than polynomially with increasing distance in any direction, it will be enough for the test function $f^\rho(x)$ to be Schwartz and divergence-free.]

So we have proved:

Theorem: The smeared electromagnetic potential $\hat A_f$ is a $U(1)$ gauge invariant observable if the test function $f^\rho(x)$ is smooth, of compact support, and divergence-free.

The divergence-free condition on $f^\rho(x)$ ensures that the commutator for creation and annihilation operators associated with the electromagnetic potential $\hat A_f=\mathbf{\scriptstyle a}^{\,}_{f^*}+\mathbf{\scriptstyle a}^{\dagger}_f$ , $$[\mathbf{\scriptstyle a}^{\,}_f,\mathbf{\scriptstyle a}^\dagger_g]=-\hbar\int \tilde f^*_\rho(k)\tilde g^\rho(k)2\pi\delta(k_\nu k^\nu)\theta(k_0)\frac{\mathrm{d}^4k}{(2\pi)^4},$$ is positive semi-definite (which is necessary for us to be able to construct a vacuum sector Hilbert space), and that because $\delta f=\delta g=0$ we can construct, on Minkowski space, $f=\delta F$, $g=\delta G$, where $F$ and $G$ are bivector potentials for the electromagnetic potential test functions $f$ and $g$.

In terms of $F$ and $G$, we can write $a^{\,}_F=\mathbf{\scriptstyle a}^{\,}_{\delta F}$ , $a_G^\dagger=\mathbf{\scriptstyle a}^\dagger_{\delta G}$ , which satisfy the electromagnetic field commutator $$[a^{\,}_F,a_G^\dagger]=-\hbar\int k^\alpha\tilde F_{\alpha\mu}^*(k) k^\beta\tilde G_\beta{}^\mu(k)2\pi\delta(k_\nu k^\nu)\theta(k_0)\frac{\mathrm{d}^4k}{(2\pi)^4}.$$ Consequently, turning around the usual relationship because we are working with test functions instead of directly with quantum fields, we can regard test functions for the electromagnetic field as potentials for test functions for the electromagnetic potential.

Because of the restriction that electromagnetic potential test functions must have compact support (or that gauge transformations must be constrained if electromagnetic potential test functions are taken to be Schwartz), electromagnetic potential observables are less general than electromagnetic field observables if electromagnetic field test functions are taken to be Schwartz (as is most commonly assumed), or equivalent if electromagnetic field test functions are taken to be smooth and to have compact support.

So, references?

EDIT (October 24th 2011): Noting the Answer from user388027, and my comment, a decent reference for what constraints are conventionally imposed on gauge transformations would be welcome. I would particularly hope for a rationale for the constraints from whatever theoretical standpoint is taken by the reference.

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I'm a little confused as to what result you want to find a reference for. As far as I can see, the test functions for the EM potential and for the EM Field can be identified as long as the Poincaré lemma holds. This is in turn the same as having trivial de Rham cohomology, with differential forms restricted to the appropriate functional space (Schwarz or compactly supported). Are you claiming that in one of these cases the de Rham cohomology of Minkowski space is non-trivial, so that the two kinds of test functions cannot be identified? –  Igor Khavkine Oct 20 '11 at 18:34
    
@Igor It seems elementary that a smeared EM potential will be gauge invariant if the test function is divergence-free and has compact support, but I've never seen this in a textbook or paper. The question is: ?Where can I find it? Compact support is needed for gauge invariance when smearing the EM potential, otherwise the boundary term would be nontrivial for some faster-than-polynomially-increasing gauge tfns. Compact support isn't needed when smearing the EM field, Schwartz is OK, because the EM field is already gauge invariant. Or, one can restrict gauge tfns not to increase too fast. –  Peter Morgan Oct 21 '11 at 1:15

2 Answers 2

I think you don't want to allow every smooth function to $G$ to be a gauge transformation. In particular, you should not treat the constant maps to $G$ as gauge transformations. This symmetry group is the one that gives rise to charge conservation, which has real physical consequences.

I think the convention is to take the gauge transformations to be those which approach the identity at infinity. (Allow non-trivial growth at infinity and disallow the constants and your gauge transformations won't form a group.)

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Right, that is Useful, +1. You now have two up-votes, so you get at least 50 reps, with my blessing, unless someone comes up with something better. Note that from the point of view of my Question, the constraint on gauge transformations that you think is the convention (and I also have vague memories of, and is not unreasonable) makes Schwartz test functions on the EM potential equivalent to test functions on the EM field. Makes sense, but this is a constraint on the gauge transformations. If you or someone else can give a good reference that states that, I'll give the bounty for that. –  Peter Morgan Oct 24 '11 at 0:08

Maybe it is rather comment, but I anyway may not post that in such a way w/o necessary reputation points.

I may guess, that maybe some tools or intuition from classical probability theory you are using could hardly be merged into rather standard mathematical theory used for description of gauge fields. I mean, that it looks as a try to insert generalized functions into theory of connections, but it may ruin that by some reasons. So the problem with references.

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