Metric and Curvature in Gravitational Phase Space

At a fixed point in spacetime (say, x0), gravitational phase space consists of the space of symmetric matrices {F ab} [corresponding to the canonical momentum π(x0)] and of symmetric matrices {Gab} [corresponding to the canonical metric gab(x0)], where 1 ≤ a, b ≤ n, and, crucially, the matrix {Gab} is necessarily positive definite, i.e. ∑ uGabu b > 0 whenever ∑ (ua)2 > 0. In an alternative quantization procedure known as metrical quantization, the first and most important ingredient is the specification of a suitable metric on classical phase space. Our choice of phase space metrics, guided by a recent study of Affine Quantum Gravity, leads to gravitational phase space geometries which possess constant curvature and are all higher dimensional analogs of the Poincaré plane, which applies when n = 1. This result is important because phase spaces with such a maximal degree of symmetry lead naturally via the procedures of Metrical Quantization to acceptable Hilbert spaces of high dimension.