Beschreibung
Can you hear the Planck mass?
Upon compactification of higher-dimensional theories of gravity, a spectrum of spin 2 particles (Kaluza-Klein modes) is generated, whose masses depend on the details of warp factor and of the internal geometry. For the Laplacian of an n-Riemannian manifold X, the Weyl law states that the k-th eigenvalue is asymptotically proportional to (k/V)^{2/n}, where V is the volume of X. In this talk, I will show how this result can be derived via physical considerations by demanding that the gravitational potential for a compactification on X behaves in the expected (4+n)-dimensional way at short distances. In simple product compactifications, when particle motion on X is ergodic, for large k the eigenfunctions oscillate around a constant, and the argument is relatively straightforward. The Weyl law thus allows to reconstruct the four-dimensional Planck mass from the asymptotics of the masses of the spin 2 KK modes. For warped compactifications, a puzzle appears: the Weyl law still depends on the ordinary volume V, while the Planck mass famously depends on a weighted volume obtained as an integral of the warp factor. I will show how this apparent tension is resolved by arguing that in the ergodic case the eigenfunctions oscillate now around a power of the warping function rather than around a constant, leading to the notion of "weighted quantum ergodicity". This has implications for the problem of gravity localization, which I will discuss if time permits.
