Hi everyone,(adsbygoogle = window.adsbygoogle || []).push({});

I have a question related about the relation between potentials, connections and curvature in gauge theories. In Newtonian physics, the common starting point is Newton's law, which determines the motion in terms of the derivative of the potential, i.e. sth. like

[tex] \ddot x_{\alpha} \propto \partial_{\alpha}\phi.[/tex]

In electrodynamics, [tex]\phi[/tex] is contained in the vector potential [tex]A_{\mu}[/tex]. Now in QED, the latter gives rise to a connection which enters e.g. the covariant derivative, [tex]D_{\mu}=\partial_{\mu}+iqA_{\mu}[/tex]. The failure of these covariant derivatives to commute gives rise to the electromagnetic field tensor [tex]F_{\mu\nu}=\partial_{[\mu}A_{\nu]}[/tex], which is also referred to as the electromagnetic curvature.

Now in GR, what resembles the potential, is the metric, since in the Newtownian limit, the Newtownian potential is contained in it. However, it is not this potential which enters the corresponding covariant derivative. Instead, it's aderivativeof the potential, which gives the Christoffel symbols, schematically, [tex]\partial g \propto \Gamma[/tex]. Now this connection again enters covariant derivatives and their commutator yields the Riemann curvature tensor, roughly speaking.

So eventhough the overall structure is similar, the crucial differnce is while in QED, the potential enters the cov. derivative D, in GR it is thederivativeof the potential entering D. Both times, D enters the equation of motion e.g. for matter fields in the same way and both times, the corresponding field strength/curvature is constructed in a similar fashion. Yet this difference - why?

I posed my question very sketchy, I hope I got across what I'm asking. Looking forward to your responses!

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# Potentials, connections and curvature

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