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Presumably there is a field theory of electromagnetism that classically gives trivial equations of motion, but when quantized shows interesting topological phenomena. I am talking about the Lagrangian density $$\mathcal{L}=\epsilon^{\mu\nu\lambda\sigma} F_{\mu\nu}F_{\lambda\sigma}\propto \vec{E}\cdot\vec{B}$$ where $F_{\mu\nu}$ is the field strength tensor.

Unfortunately, I don't know the name of this theory or what keywords to search for, so I am unable to dig up any references that treat this theory.

It may well be that this question is a duplicate. In that case, a link to an old question would suffice.

So my "question" is: Please give me a reference where I can read about this.

Thank you for your time.

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Are you asking about classical field theory? i.e, the classical theory of electromagnetic fields? –  Kitchi Feb 18 '13 at 12:55
In my E+M course, we called this term "eff mew new", to which the response was usually "hey, buddy! eff you too!" –  KDN Feb 18 '13 at 13:12

1 Answer 1

up vote 7 down vote accepted

Your term in the Lagrangian density is usually not given a special name; it is called the "F wedge F" term from the $p$-form notation $F\wedge F$ (the symbol is written as $\backslash{\rm wedge}$) and represents a tensor multiplication of antisymmetric tensor followed by a new antisymmetrization of all the indices (up to some normalization that depends on conventions a bit). Google search for "F wedge F" in quotation marks is useful.

However, in your "question", you are making a guess about the literature. You think that "F wedge F" is universally interesting and has some localized papers about it. But as expected from the fact that you don't really have any reason for your ad hoc guess, this guess isn't quite right. A better guess would be obtained if you analyzed the possible term in more detail.

If you did so, you would find out that in electromagnetism i.e. $U(1)$ gauge theory, the term is unphysical even quantum mechanically because it may be written as the total derivative $${\mathcal L}\sim c\cdot \int\partial_\kappa \epsilon^{\kappa\lambda\mu\nu} A_{\lambda}F_{\mu\nu}. $$ By Gauss' theorem, this may be written as the $S^3$ boundary integral of the 3-form $A\wedge F$, known as the Chern-Simons form. If you imagine a very large (infinite) spacetime region, the gauge field has to be pure gauge at infinity, $A_\mu\sim \partial_\mu\lambda$, and you may verify that the 3-dimensional integral over the surface vanishes.

This changes in non-Abelian gauge theory because the gauge parameter $\lambda$ takes values in the 3-dimensional $SU(2)$ (or higher-dimensional) manifold and the $\lambda\in SU(2)$ may "wrap" the actual 3-sphere in the spacetime. That's why terms proportional to $$\int {\rm Tr}(F\wedge F) $$ in the action of a non-Abelian gauge theory such as QCD may be nonzero. With a proper normalization factor (essentially the volume of the 3-sphere), this term (once integrated over the spacetime) is integer-valued. The integer measures how many times the values of $\lambda$, when studied over the $S^3$ boundary in the spacetime, wrap the $S^3$ sphere inside the gauge group while realizing that $SU(2)$ group manifold is topologically another $S^3$.

Because the term is nonzero (and integer), you may change the theory by adding $\theta N$ where $N$ is the properly normalized integral of $F\wedge F$. If that's so, the theory depends on $\theta$ so that $\theta\sim \theta+2\pi k$ gives the same theory (changes of the action by multiples of $2\pi$ don't matter quantum mechanically). The parameter $\theta$ is known as the "theta angle" or rarely "vacuum angle" (a Google search word) and the configurations of the gauge field with $N=\pm 1$ (or more generally any nonzero $N$) are known as "instantons" (they have the wrapping of one 3-sphere on another, localized solution with some "knot" in the middle of the Euclidean spacetime).

If the coefficient $\theta$ is promoted to a scalar field, we get the Peccei-Quinn theory which was designed to explain why the original $\theta$ in QCD is – experimentally – almost exactly zero even though it could be of order one (the "strong CP-problem", another keyword).

The 4-dimensional theory in which you would only have the "F wedge F" term would be ill-behaved because the partition sum would just be multiplied by an infinite factor (only the instanton number of a configuration matters, but you must still integrate over all configurations). However, there exist other theories defined by a purely topological action of similar kinds. Keywords are "topological field theories" and "Chern-Simons theory" and "topological terms". The literature (and details and consequences) is/are huge; it seems strange to mention just a few papers.

Even though $SU(2)$ in the electroweak theory is large enough to allow instantons etc., there is no $\theta$-angle in the electroweak theory. More precisely, it may be absorbed into the phases of the left-handed fermions (which changes the action by a multiple of the "F wedge F" term due to the chiral anomaly).

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Thank you very much! You have given me a lot of stuff to read and hints to further reading! –  Espen Nielsen Feb 18 '13 at 15:57

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