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In kaluza-klein theory, there's a notion of a "ground state metric" after compactification. What is the meaning of the term "ground state metric"?

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It just means that the energy (e.g. in the GR language, the ADM energy) is minimized among all configurations with the same boundary conditions. It means that there are no gravitational or electromagnetic or other waves inside the space.

In practice, it just means that the geometry is a Cartesian product $M^4\times Y$ where $Y$ is the manifold of compact dimensions, and this classical geometry may be used as the "vev" defining the (second-quantized vacuum) vacuum in calculations in quantum field theory.

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