Quantum gravity and quantum probability

Abstract

We argue that in quantum gravity there is no Born rule. The quantum-gravity regime, described by a non-normalisable Wheeler-DeWitt wave functional $\Psi$, must be in quantum nonequilibrium with a probability distribution $P\neq\left\vert \Psi\right\vert ^{2}$ (initially and always). A Born rule can emerge only in the semiclassical regime of quantum systems on a classical spacetime background, with normalisable Schr\"{o}dinger wave functions $\psi$. Conditioning on the underlying quantum-gravitational ensemble yields a nonequilibrium distribution $\rho\neq\left\vert \psi\right\vert ^{2}$ at the beginning of the semiclassical regime, with quantum relaxation $\rho\rightarrow\left\vert \psi\right\vert ^{2}$ taking place only afterwards. Quantum gravity naturally creates an early nonequilibrium universe. We also show how small corrections to the Schr\"{o}dinger equation yield an intermediate regime in which the Born rule is unstable: an initial distribution $\rho=\left\vert \psi\right\vert ^{2}$ can evolve to a final distribution $\rho\neq\left\vert \psi\right\vert ^{2}$. These results arise naturally in the de Broglie-Bohm pilot-wave formulation of quantum gravity. We show that quantum instability during inflation generates a large-scale deficit $\sim1/k^{3}$ in the primordial power spectrum at wavenumber $k$, though the effect is too small to observe. Similarly we find an unobservably large timescale for quantum instability in a radiation-dominated universe. Quantum instability may be important in black-hole evaporation, with a final burst of Hawking radiation that violates the Born rule. Deviations from the Born rule can also be generated for atomic systems in the gravitational field of the earth, though the effects are unlikely to be observable. The most promising scenario for the detection of Born-rule violations appears to be in radiation from exploding primordial black holes.