Which Quantum Theory Must be Reconciled with Gravity? (And What Does it Mean for Black Holes?)

We consider the nature of quantum properties in non-relativistic quantum mechanics (QM) and relativistic quantum field theories, and examine the connection between formal quantization schemes and intuitive notions of wave-particle duality. Based on the map between classical Poisson brackets and their associated commutators, such schemes give rise to quantum states obeying canonical dispersion relations, obtained by substituting the de Broglie relations into the relevant (classical) energy-momentum relation. In canonical QM, this yields a dispersion relation involving h̄ but not c, whereas the canonical relativistic dispersion relation involves both. Extending this logic to the canonical quantization of the gravitational field gives rise to loop quantum gravity, and a map between classical variables containing G and c, and associated commutators involving h̄. This naturally defines a “wave-gravity duality”, suggesting that a quantum wave packet describing self-gravitating matter obeys a dispersion relation involving G, c and h̄. We propose an Ansatz for this relation, which is valid in the semi-Newtonian regime of both QM and general relativity. In this limit, space and time are absolute, but imposing vmax = c allows us to recover the standard expressions for the Compton wavelength λC and the Schwarzschild radius rS within the same ontological framework. The new dispersion relation is based on “extended” de Broglie relations, which remain valid for slow-moving bodies of any mass m. These reduce to canonical form for m mP, yielding λC from the standard uncertainty principle, whereas, for m mP, we obtain rS as the natural radius of a self-gravitating quantum object. Thus, the extended de Broglie theory naturally gives rise to a unified description of black holes and fundamental particles in the semi-Newtonian regime.