# Intuition behind U(1)-gauge model of Electrodynamics in a general spacetime

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As the article Electrodynamics in general spacetime greatly explains, the U(1)-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to relate electrodynamics and complex line bundles and connections (anything to do with quantum theory, perhaps?). I know Yang-Mills theory has to do with all this idea of studying bundles and connections, but I hope it has physical meaning and not just 'geometrical convenience'. Can anyone explain me intuitively this mathematics-physics interplay?

For example, if somebody asked me "Why are particles described as complex-valued functions in Quantum Mechanics?" then I would answer "Because experiments have shown that particles should not be thought as points, but behave like having a certain probability distribution over space, which moreover has been proven to interact like waves, i.e. having constructive and destructive phenomena, and therefore it seems as there is a phase information encoded in the particle together to the probability amplitude".

Or if somebody asked "And why to use spinorial bundles when describing electrons?" the I would answer "Well, particles behave like waves, i.e. functions, but if I accept there is some spin 'vector', it becomes altogether a 'vector-valued function', whose behavior is suitably described by means of this spinorial bundle".

Now suppose our ‘electromagnetic world’ is modelled by means of a complex line bundle with base the 4-dimensional spacetime. How should be understood a section over our complex line bundle? And in this setting, which is the role of a connection?

As it may be difficult to answer this in a few words, it would be also helpful if somebody pointed out some small fragment of a book where this questions are studied. All ideas are welcomed.

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I think this is a worthwhile question to ask, but I note that Maxwell had what I take to be an answer, that EM is "about" a system of vortices. I apologize if this seems facetious, but I take it that what is meant by "physical meaning", "intuitively", and "understood" partly depends on the period, the person, and the purpose.

For a pretty good "What is a Gauge" lecture, you could look at http://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/, which I found by Googling for "gauge fixed electromag". [Thanks for the nice math.stackexchange link, BTW.]

Supposing we wanted to play a little, however, consider that in a classical context, if we have an electromagnetic potential $A_\mu(x)$ and a Dirac spinor field $U(x)$, we could note that (on Minkowski space or locally) there is a natural gauge fixing because $T^\mu(x)=\overline{U(x)}\gamma^\mu U(x)$ is a time-like 4-vector wherever $U(x)$ is non-zero, so we could fix a natural "Maxwell-Dirac axial gauge" by requiring that $T^\mu(x)\,A^{MD}_\mu(x)=0$. Given this fixing, and the mutually orthogonal and equal length 4-vectors $$Z^\mu(x)=\overline{U(x)}\gamma^\mu\gamma^5 U(x),\quad X^\mu(x)=\Re[\overline{U(x)}\gamma^\mu U^c(x)],\quad Y^\mu(x)=\Im[\overline{U(x)}\gamma^\mu U^c(x)],$$ which are orthogonal to $T^\mu(x)$, $T^\mu(x)Y_\mu(x)=T^\mu(x)Y_\mu(x)=T^\mu(x)Z_\mu(x)=0$, to each other, $X^\mu(x)Y_\mu(x)=Y^\mu(x)Z_\mu(x)=Z^\mu(x)X_\mu(x)=0$, and the same length up to a sign, $X^\mu(x)X_\mu(x)=Y^\mu(x)Y_\mu(x)=Z^\mu(x)Z_\mu(x)=-T^\mu(x)T_\mu(x)$, we could take the dynamics to be "about"  a tetrad, a 4-vector $A_\mu^{MD}$, and a U(1) phase, $$(T^\mu(x), Z^\mu(x), X^\mu(x), Y^\mu(x), A^{MD}_\mu(x), \phi(x)).$$ $X^\mu(x)$, $Y^\mu(x)$, $Z^\mu(x)$, and $A^{MD}_\mu(x)$ are all space-like because they are orthogonal to $T^\mu(x)$, but $A^{MD}_\mu(x)$ is not otherwise constrained relative to the tetrad $(T^\mu(x), Z^\mu(x), X^\mu(x), Y^\mu(x))$. Note that fixing the U(1) gauge is related to fixing the arbitrary charge conjugation phase at every point.

$\phi(x)$ is the phase of the complex number $(\overline{U(x)}U(x), \overline{U(x)}\gamma^5 U(x))$, which could be taken to have one of a number of geometrical interpretations, but I've not found any approaches that are at all compelling (even just to me).

For this system, gauge invariance is just a way of writing down a particular dynamics for a system of 4-vectors that is nonlinearly constrained by various orthogonalities. We can play this game in various different ways, for example by using the bivector forms that can be constructed using $U(x)$ or by using the 3-form that can be constructed using $T^\mu(x)$ and the electromagnetic field, $T\wedge \mathrm{d}A$, but the trouble is that it's not clear when any such construction helps with either tractability or intuition, there's nothing unique about any given choice, and extending this kind of approach to a classical electroweak theory or to QED looks kinda bad. I also find the jump from a particular geometrical construction to a visual (or other sense) intuition not obvious.

Apologies that I haven't put this in a general Lorentzian manifold, but I don't play well in that context. Please also note that I've been thinking about whether this kind of construction might be useful for a while, so I took your question as an opportunity to write up one variant and see what it looks like.

answered Nov 29, 2014 by (1,230 points)

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