# Is there a meaning to the E,B analogues of other gauge fields?

From the gauge field $A_\mu$ and the QED lagrangian we can derive maxwell's equations in terms of electric and magnetic fields. Are there any situations where similar derivations using the other gauge fields like $W^{\pm}_{\mu}$, $Z_\mu$ or the gluon fields, e.g. $curl W$ could become useful?

edit: reformulated for clarity.

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## 2 Answers

The decomposition into electric and magnetic fields is valid in a specific reference frame, those fields mix under Lorentz transformations. Based on that, you may expect that this separation will be more useful in the non-relativistic (low energy) limit. There is also the historical fact: Maxwell equations were written down before special relativity was discovered. Since we still, by and large, make the odd choice of teaching things in historical (rather than logical) order, you've probably seen Maxwell equations in components before seeing them in their more compact and symmetric form.

But, for fields relevant only in very high energies (the electroweak gauge bosons), which were also discovered after SR, the relativistic form of the equations is the default. You can certainly write things in components, but this is not a very useful notation.

There are some exceptions, some corners in high energy physics where this decomposition can give you some intuition. For example if you look at the QFT text by Peskin and Schroeder you'd find some intuitive picture on why the $\beta$ function of non-Abelian gauge theories is negative. This picture is not gauge invariant, but it relies on some "magnetic" as opposed to "electric" effects. But these exceptions are rare and not always all that useful.

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Fine answer, Moshe, +1. One trivial point that should be made for W and Z boson gauge fields: they're massive, so the wavelength of these weakomagnetic waves is $10^{-18}$ meters or so, so one can't imagine that there are "macroscopic" regions with nearly uniform values of the fields. Also, in QCD, the chromomagnetic fields are not gauge-invariant because they carry an adjoint index, so one has to construct invariants such as $Tr F_{\mu\nu} F^{\mu\nu}$ to make physical sense. – Luboš Motl Feb 28 '11 at 18:49
this figure in P&S actually sparked my question. accepted. – luksen Feb 28 '11 at 20:09

In the analysis of constraints and canonical/Dirac quantization of gauge fields it is "natural" to choose a time direction. This means that the electric and magnetic components of the field are separated and seem to play different roles. Of course, at the end, Lorentz invariance can be recovered.

For an example of this, see Matschull's paper: Dirac's Canonical Quantization Programme.

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