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A Non-local Reality: Is There a Phase Uncertainty in Quantum Mechanics?

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Abstract

A century after the advent of quantum mechanics and general relativity, both theories enjoy incredible empirical success, constituting the cornerstones of modern physics. Yet, paradoxically, they suffer from deep-rooted, so-far intractable, conflicts. Motivations for violations of the notion of relativistic locality include the Bell’s inequalities for hidden variable theories, the cosmological horizon problem, and Lorentz-violating approaches to quantum geometrodynamics, such as Horava–Lifshitz gravity. Here, we explore a recent proposal for a “real ensemble” non-local description of quantum mechanics, in which “particles” can copy each others’ observable values AND phases, independent of their spatial separation. We first specify the exact theory, ensuring that it is consistent and has (ordinary) quantum mechanics as a fixed point, where all particles with the same values for a given observable have the same phases. We then study the stability of this fixed point numerically, and analytically, for simple models. We provide evidence that most systems (in our study) are locally stable to small deviations from quantum mechanics, and furthermore, the phase variance per value of the observable, as well as systematic deviations from quantum mechanics, decay as \(\sim \)(energy \(\times \) time)\(^{-2n}\), where \(n \ge 1\). Interestingly, this convergence is controlled by the absolute value of energy (and not energy difference), suggesting a possible connection to gravitational physics. Finally, we discuss different issues related to this theory, as well as potential novel applications for the spectrum of primordial cosmological perturbations and the cosmological constant problem.

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Notes

  1. Beables generally refer to properties innate to the system rather than determined by observation. This term is used in hidden variable theories to give a name to the properties of the system which actually exists and are subject to the laws of evolution of the system. It typically includes both the observables and the hidden variables.

  2. In order to match quantum mechanics, one needs to include a factor of 2 that was missing from Equation 30 from [9]. This correction, however, will not effect any later conclusions, and it was fixed in the published version. The other change in this from Smolin’s notation is the replacement of \(\delta \) with \(\beta \) to prevent confusion with the Kronecker \(\delta \)-function.

  3. Note that there could be no exponentially growing mode, as the total probability is conserved.

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Acknowledgments

We would like to thank Lee Smolin for inspiring this work, and many discussions along the way. We also thank Lucien Hardy, and Steve Weinstein for comments on the draft. This work was supported by the Natural Science and Engineering Research Council of Canada, the University of Waterloo and by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.

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Authors and Affiliations

  1. Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

    Elizabeth S. Gould & Niayesh Afshordi

  2. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

    Elizabeth S. Gould & Niayesh Afshordi

Authors
  1. Elizabeth S. Gould
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  2. Niayesh Afshordi
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Correspondence to Elizabeth S. Gould.

Appendix: A Various Plots of Numerical Simulations

Appendix: A Various Plots of Numerical Simulations

Here are given additional plots of the numerical simulations for those that wish to confirm certain aspects covered in the paper (See Tables 3, 4, 5).

Table 3 Plots of the evolution of spin-\(\frac{1}{2}\) systems in the non-equilibrium real ensemble model for the case with a moderate initial separation of phases
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Table 4 Plots of the evolution of spin-\(\frac{1}{2}\) systems in the non-equilibrium real ensemble model for the case of large initial separation of phases
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Table 5 Plots of the evolution of spin-\(\frac{1}{2}\) systems in the non-equilibrium real ensemble model for the case with a Hamiltonian proportional to the identity
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Gould, E.S., Afshordi, N. A Non-local Reality: Is There a Phase Uncertainty in Quantum Mechanics?. Found Phys 45, 1620–1644 (2015). https://doi.org/10.1007/s10701-015-9948-8

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