Loops, Knots, Gauge Theories and Quantum Gravity 2026, 31 - Quantum Gravity Hamiltonian

A traditional Hamiltonian formulation for GR sees spacetime split into time and space component to recover a time-axis, which makes available the idea of time-evolution and the typical idea of a Hamiltonian. The canonical form breaks the spacetime covariance of the theory through the time direction, but in the end the specific direction chosen in that step is irrelevant, so the covariance is restored. It turns out, the direction is chosen for construction only. A spacetime with metric g_ab and some topology with space-like surface wrt. g_ab that is assumed to be a Cauchy surface, has a foliation that yields a time-like future directed vector t^a, which describes this flow of time. A unit vector field n^a normal to the foliation determines a unique positive definite spatial metric on a 3D slice q_ab = g_ab + n_an_b, and t^a has the decomposition Nn^a + N^a (Eq. 7.4) with the lapse N and N^a the shift vector. From these quantities, the curvature K_ab = q^c_aq^d_b∇_cn_d (Eq. 7.7) derives easily, which coincides with the Killing vector. The Einstein action then takes on the familiar form S = ∫ dt L. For the mechanics, the 3-metric q can be used as a canonical variable, whose conjugate momentum is

(Eq. 7.12). This formalism has no time derivative of the lapse or the shift, so the conjugate momenta are zero, which introduces as many constraints to the systems as the theory has conjugate momenta. Additionally, four expressions of the generalized variables vanish. This has to remain true on each hypersurface.

(Eq. 7.17 - 18). These equations need to be true when choosing the coordinates for the gravitational field. They have the same character as the Gauss law for electromagnetism, which emerges from the U(1) invariance of the Maxwell equations (7.2.3). From the canonical form follows easily the canonical quantization from the general quantization scheme (7.2.4)