Spontaneous breakdown of local conformal invariance in quantum gravity

Abstract This chapter shows how the black hole complementarity principle can be naturally implemented by treating local conformal invariance as an exact but spontaneously broken symmetry of quantum gravity. This allows a description of the black hole either in terms of the imploding particles or entirely in terms of the emerging Hawking particles. These complementary representations can be obtained from one another by a local conformal transformation; this implies that the black hole scattering matrix is equivalent to a local conformal gauge transformation. Perturbative canonical quantum gravity, coupled to a renormalizable model for matter fields, has this conformal symmetry built in, and this symmetry would be exact if the local conformal anomalies cancelled. The Einstein–Hilbert action can be regarded as breaking local conformal invariance only dynamically, not explicitly. The functional integral over the dilaton component of the metric field can be disentangled from the other integrations over the metric and the matter fields, turning the remainder of the theory into a trivially conformally invariant system. When the residual metric is treated as a flat background, this leads to a novel constraint: in combination with the dilaton contributions, the matter Lagrangian should have a vanishing beta function. The zeros of this beta function are isolated points in the landscape of quantum field theories, and so one arrives at a denumerable, or perhaps even finite, set of quantum theories for matter, where coupling constants, masses, and cosmological constant are all fixed, and computable in terms of Planck units.