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Beyond the Born Rule in Quantum Gravity

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Abstract

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.

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Notes

  1. By de Broglie’s own account his ideas originated in a paper of 1922 on blackbody radiation, although his first paper on pilot-wave theory proper did not appear until 1923, culminating in his theory of a many-body system presented at the 1927 Solvay conference (see Ref. [2, Chap. 2]).

  2. In this paper we employ the traditional metric representation of the gravitational field. We expect similar conclusions to hold in loop quantum gravity [46].

  3. As we will see there is also a constraint on \(\Psi\) guaranteeing coordinate invariance.

  4. For a recent review see Ref. [61].

  5. For a ‘functional’ \(\Psi [\phi ]\) (mapping a function \(\phi (x)\) to a complex number \(\Psi\)) the functional derivative \(\delta \Psi /\delta \phi (x)\) at a spatial point x is defined by \(\delta \Psi =\int d^{3}x\,\left[ \delta \Psi /\delta \phi (x)\right] \delta \phi (x)\) for arbitrary infinitesimal variations \(\delta \phi (x)\).

  6. Here \(dx^{2}=(dx^{1})^{2}+(dx^{2} )^{2}+(dx^{3})^{2}\).

  7. The continuity equation (25) can also be derived from Noether’s theorem as the local conservation law associated with a global phase symmetry \(\psi \rightarrow \psi e^{i\theta }\) on configuration space [85].

  8. We are assuming the wave function has a single component \(\psi\). The method can be readily extended to spin systems with multi-component wave functions [6].

  9. For \(N_{i}\ne 0\) there are additional terms \(D_{i}N_{j} +D_{j}N_{i}\) on the right-hand side of (34).

  10. Specifically, with the ordering \((\delta /\delta g_{ij})G_{ijkl}(\delta /\delta g_{kl})\) in the kinetic term.

  11. For \(N_{i}\ne 0\) there is an additional term \(N^{i}\partial_{i}\phi\) on the right-hand side of (37).

  12. We might also append a constraint of the form (13) on P, to ensure that it is a function on the space of coordinate-independent 3-geometries. We can avoid this complication by simply working with one representation of the 3-geometry by one (coordinate-dependent) metric \(g_{ij}\).

  13. For a single particle with a static density \(\rho\) and a current \(\mathbf {j}\), this is analogous to summing the equations \(\partial_{x}j_{x}=\partial_{y}j_{y}=\partial_{z} j_{z}=0\) to yield \(\partial \rho /\partial t+\mathbf {\nabla }\cdot \mathbf {j}=0\).

  14. The same equations follow by identifying the canonical momenta (16) with a phase gradient.

  15. When \(s=0\) the exact H is constant but the coarse-grained value decreases (if there is no initial fine-grained structure) [16].

  16. This result is of course expected from the WKB form (49) (with \(\psi =\psi ^{(0)}\)).

  17. In an expanding cosmological background the ratio of \(i\hat{H}_{b}\) to \(\hat{H}_{a}\) is roughly of order \(\sim H/E\), where \(H=\dot{a}/a\) is the Hubble parameter and E is a typical energy for the field [64].

  18. For more details see Ref. [8].

  19. The creation of quantum nonequilibrium by evaporating black holes was previously suggested as a possible mechanism for resolving the information-loss puzzle [35, 63, 96, 101], but without a clear theoretical underpinning.

  20. A similar suggestion was made by Kiefer and Singh [64], who considered the effect of the Hermitian correction on atomic energy levels.

  21. In a remarkable reply to Ref. [26], Dürr and Struyve [104] invoke similar circular reasoning in their account of classical coin tossing.

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  1. Augustus College, 14 Augustus Road, London, SW19 6LN, UK

    Antony Valentini

  2. Department of Physics and Astronomy, Kinard Laboratory, Clemson University, Clemson, SC, 29634-0978, USA

    Antony Valentini

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Valentini, A. Beyond the Born Rule in Quantum Gravity. Found Phys 53, 6 (2023). https://doi.org/10.1007/s10701-022-00635-0

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