A. Zee, Quantum Field Theory in a Nutshell · Problem VIII.1.5

Field Theory Redux

Physics — QFT

Problem

The Kaluza-Klein result that we argued by symmetry considerations can of course be derived explicitly. Let me sketch the calculation for you. Consider the metric ds2=gμνdxμdxν−a2[dθ+Aμ(x)dxμ]2ds^2 = g_{\mu\nu}dx^\mu dx^\nu - a^2[d\theta + A_\mu(x)dx^\mu]^2 where θ\theta denotes an angular variable 0≤θ<2π0 \leq \theta < 2\pi. With Aμ=0A_\mu = 0, this is just the metric of a curved spacetime which has a circle of radius aa attached at every point. The transformation θ→θ+Λ(x)\theta \to \theta + \Lambda(x) leaves ds2ds^2 invariant provided that we also transform Aμ(x)→Aμ(x)−∂μΛ(x)A_\mu(x) \to A_\mu(x) - \partial_\mu\Lambda(x). Calculate the 5-dimensional scalar curvature R5R_5 and show that R5=R4−14a2FμνFμνR_5 = R_4 - \frac{1}{4}a^2F_{\mu\nu}F^{\mu\nu}. Except for the precise coefficient 14\frac{1}{4} this result follows entirely from symmetry considerations and from the fact that R5R_5 involves two derivatives on the 5-dimensional metric, as explained in the text. After some suitable rescaling this is the usual action for gravity plus electromagnetism. Note that the 5-dimensional metric has the explicit form g5AB=(gμν−a2AμAν−a2Aμ−a2Aν−a2)g_5^{AB} = \begin{pmatrix} g_{\mu\nu} - a^2A_\mu A_\nu & -a^2A_\mu \\ -a^2A_\nu & -a^2 \end{pmatrix}

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