= Kaluza-Klein theory
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{title2=$ds^2=g_{\mu\nu}dx^\mu dx^\nu+R^2(dy+A_\mu dx^\mu)^2$}
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= Kaluza-Klein
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Kaluza-Klein theory obtains lower-dimensional fields from a higher-dimensional metric and other fields on compact dimensions. In the displayed fixed-radius circle ansatz, the off-diagonal metric supplies a vector. The coordinate change $y\mapsto y+\lambda(x)$ is compensated by $A\mapsto A-d\lambda$, producing an Abelian <gauge symmetry>. Allowing the radius to vary adds a scalar modulus. <Kaluza-Klein modes> also include nonzero internal momenta, giving a tower of massive fields.
// Target: quantum-field-theory.bigb
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