We investigate whether small perturbations can cause relaxation to quantum equilibrium over very long timescales. We consider in particular a two-dimensional harmonic oscillator, which can serve as a model of a field mode on expanding space. We assume an initial wave function with small perturbations to the ground state. We present evidence that the trajectories are highly confined so as to preclude relaxation to equilibrium even over very long timescales. Cosmological implications are briefly discussed.

We have recently developed a new understanding of probability in quantum gravity. In this paper we provide an overview of this new approach and its implications. Adopting the de Broglie–Bohm pilot-wave formulation of quantum physics, we argue that there is no Born rule at the fundamental level of quantum gravity with a non-normalisable Wheeler–DeWitt wave functional \(\Psi\). Instead the universe is in a perpetual state of quantum nonequilibrium with a probability density \(P\ne \left| \Psi \right| ^{2}\). Dynamical relaxation to the (...) Born rule can occur only after the early universe has emerged into a semiclassical or Schrödinger approximation, with a time-dependent and normalisable wave functional \(\psi\), for non-gravitational systems on a classical spacetime background. In that regime the probability density \(\rho\) can relax towards \(\left| \psi \right| ^{2}\) (on a coarse-grained level). Thus the pilot-wave theory of gravitation supports the hypothesis of primordial quantum nonequilibrium, with relaxation to the Born rule taking place soon after the big bang. We also show that quantum-gravitational corrections to the Schrödinger approximation allow quantum nonequilibrium \(\rho \ne \left| \psi \right| ^{2}\) to be created from a prior equilibrium ( \(\rho =\left| \psi \right| ^{2}\) ) state. Such effects are very tiny and difficult to observe in practice. (shrink)