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.

3.1 2 + 2 truncation of Einstein gravity + matter

3.1.1 Gravity theories

3.1.2 2-Killing vector reduction

3.1.3 Hamiltonian formulation

3.1.4 Lapse and shift in 2D gravity theories

3.1.5 Symmetries and currents

3.2 Collinear gravitons, Dirac quantization, and conformal factor

3.2.1 Collinear gravitons

3.2.2 Dirac versus covariant quantization

3.2.3 Conformal factor

3.3 Tamed non-renormalizability

3.4 Non-Gaussian fixed point and asymptotic safety

3.1.1 Gravity theories

3.1.2 2-Killing vector reduction

3.1.3 Hamiltonian formulation

3.1.4 Lapse and shift in 2D gravity theories

3.1.5 Symmetries and currents

3.2 Collinear gravitons, Dirac quantization, and conformal factor

3.2.1 Collinear gravitons

3.2.2 Dirac versus covariant quantization

3.2.3 Conformal factor

3.3 Tamed non-renormalizability

3.4 Non-Gaussian fixed point and asymptotic safety

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