The measure matters

Understand

We adopt a framework where quantum-gravity's dynamical dimensional reduction of spacetime at short distances is described in terms of modified dispersion relations.

• We observe that by subjecting such models to a momentum-space diffeomorphism one obtains a "dual picture" with unmodified dispersion relations, but a modified measure of integration over momenta.
• We then find that the UV {\it Hausdorff} dimension of momentum space which can be inferred from this modified integration measure coincides with the short-distance {\it spectral} dimension of spacetime.
• This result sheds light into why scale-invariant fluctuations are obtained if the original model for two UV spectral dimensions is combined with Einstein gravity.