Phase space of a gravitating particle and dimensional reduction at the Planck scale

Several approaches to quantizing general relativity suggest that quantum gravity at very short distances behaves effectively as a two-dimensional theory. The mechanism of this dimensional reduction is not yet understood. We attempt to explain it by studying the phase space of a test particle coupled to a gravitational field. The general relativity constraints relate the particle energy–momentum to some curvature invariants taking values in a group manifold. Some directions in the resulting momentum space turn out to be compact, which leads to a kind of "inverse Kaluza–Klein reduction" at short distances.