The systems investigated in this section can be looked at in two ways. First as prototype field theories
which have the qualitative Properties 1 – 3 tentatively identified at the end of the last Section 2.4.4 as
characteristics which an effective field theory description of the extreme UV regime of Quantum
Gravidynamics should have. Second, they can be viewed as a symmetry reduction of the gravitational
functional integral whose embedding into the full theory is left open for the time being. Technically one
starts off from the usual gravitational functional integral but restricts it from “4-geometries modulo
diffeomorphisms” to “4-geometries constant along a foliation modulo diffeomorphisms”. This means
instead of the familiar
foliation of geometries one considers a foliation in terms of two-dimensional
hypersurfaces
and performs the functional integral only over configurations that are constant as one
moves along the stack of two-surfaces. The same can be done with the functional integral over matter
configurations.
The truncation can be motivated in various ways. It is complementary to the Eikonal sector and describes gravity with collinear initial data in a sense explained later on. It takes into account the crucial ‘spin 2’ aspect, that is, linear and nonlinear gravitational waves are included in this sector and treated without further approximations. Asymptotic safety in this sector is arguably a necessary condition for asymptotic safety of the full theory. Finally, as already mentioned, the sector can serve as a test bed for the investigation of the renormalization structures needed once the extreme UV regime of has been reached.
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