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I'm reading landau electrodynamics . For the investigation of the dynamics of a charged particle in an electromagnetic field , the author introduces a four vector potential field on spacetime . The action is $\frac{-e}{c^{2}}\int A^{\mu }dx_{\mu }$ I want to do the same for higher rank tensors $T^{\mu,\rho,..}$ contracted with other higher rank tensors . I think that derivations will be very similar to the four vector field but more complicated . Are there any interesting consequences of these theories ?

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2 Answers 2

First, let's look at the Lagrangian you show in a bit more detail. This is the action for an electron moving in a fixed electromagnetic field, where the field is described by this vector potential, and the integral is taken over the path of the electron. Usually an action is a scalar (Lagrangian) integrated over time. You can actually see that your expression is equivalent to this, where the Lagrangian is given by $L = -e\, A^\mu\, \frac{dx_\mu}{dt}$, which comes from the Lorentz force law in terms of the vector potential. (Here, $dx_\mu/dt$ is the velocity of the electron.) So you can rewrite your expression as $\int L\, dt$, which is the standard form of the action.

Now, to answer your question, as long as you turn your tensor into a scalar with sufficient contractions, you can make it into a Lagrangian. (Whether or not that Lagrangian actually represents anything physical is a different question.) For example, the Lagrangian of the E&M field itself is usually written as $-\frac{1}{4}F^{\mu\nu}\, F_{\mu\nu}$. (Actually that makes it a Lagrangian density, but people usually drop the "density" and/or the integration over space for brevity.) Integrate this quantity over time, and you have the action of the electromagnetic field. There's the Lagrangian for general relativity: $\sqrt{-g}\, R_{\mu\nu}g^{\mu\nu}$. In GR, a particle moving through a fixed background spacetime follows a geodesic of the metric. It's not too hard to show that the geodesic equation is equivalent to writing \begin{equation} L = \left\lvert g_{\mu\nu} \frac{dx^\mu}{dt} \frac{dx^\nu}{dt} \right\rvert^{1/2} \end{equation} (Heuristically, you could think of playing the same trick of "canceling out" the factor of $dt$ in the action, as was done in your example. But it's harder to make that rigorous in this case.)

Examples abound. So, yes, there are interesting consequences to some such theories -- though not all such contractions are interesting.

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Thanks , Mike ,In the case of , say, non-abelian gauge field or GR what would be the lagrangian that's defined by evaluating the tensor field on each point on the worldline of the particle and contracting with $dx^{\mu}$ ? $F^{\mu,\rho}F_{\mu\rho}$ yields the dynamics of the field not the particle in a field . I want a lagrangian that gives you the dynamics of a particle in GR , nonabelian gaauge theory and other theories –  Bill May 24 '13 at 6:49
    
I've added the example for a particle in GR to my answer. In quantum field theory, you usually treat a "particle" in terms of the corresponding field – the best example being the Standard Model. It would be possible to write down a Lagrangian for a particle moving in a fixed background, but I don't know of any useful examples off the top of my head. –  Mike May 24 '13 at 13:36

There are a lot of theories which are very interesting when you contract Lorentz indices to make a scalar quantity. For example, in QED, one usually considers the E&M field strength tensor $F_{\mu \nu} F^{\mu \nu}$, which describes the dynamics of the photon field at the quantum level. In the case of topological insulators (and also in QCD), one can consider another element which satisfies all of the physical requirements (Lorentz invariance and such), the so-called "theta term":

$L = \theta \; \epsilon^{\mu \nu \rho \sigma} F_{\mu \nu} F_{\rho \sigma}$

The physics described by this term is very interesting: it involves the topology of the space on which you are working, it has to do with quantum anomalies due to the axial symmetries of your system (non-conservation of fermion numbers), and so forth.

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