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Superposition as a Relativistic Filter

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Abstract

By associating a binary signal with the relativistic worldline of a particle, a binary form of the phase of non-relativistic wavefunctions is naturally produced by time dilation. An analog of superposition also appears as a Lorentz filtering process, removing paths that are relativistically inequivalent. In a model that includes a stochastic component, the free-particle Schrödinger equation emerges from a completely relativistic context in which its origin and function is known. The result establishes the fact that the phase of wavefunctions in Schrödinger’s equation and the attendant superposition principle may both be considered remnants of time dilation. This strongly argues that quantum mechanics has its origins in special relativity.

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Notes

  1. J. S. Bell, whose deep insights into the foundations of quantum mechanics have informed generations of physicists, lamented the lack of an ‘exact’ theory underlying quantum mechanics. With incisive humour in his last publication [4], he labeled the current versions of quantum mechanics as good For All Practical Purposes (FAPP) in order to deflect criticism from those convinced of completeness. This paper argues that while non-relativistic quantum mechanics as a description is good FAPP, the origin of superposition and the roots of its strange behaviour are missing in the absence of relativity.

  2. The double slit experiment is a typical choice to display the peculiarity of ‘quantum’ superposition because it highlights the failure of the classical superposition of probabilities. It also yields quickly to ’wave superposition’ but is mute on the origin and physical reality of the waves.

  3. Note in particular the similarity of the two equations. The difference lies in the off-diagonal elements of the spatial operator. To lowest order the shift operators are unity. In (11), to lowest order the off-diagonal matrix is σ x with eigenvalues ± 1, indicating a reflection. In (12) it is − i σ y with eigenvalues ± i, indicating a rotation!

  4. In the case of the z k , α = 1 and the eigenvalues of the transfer matrix are 0 and cos(p δ). In the diffusive limit, after transformation back to position space the analog of (19) is \(Z(x,t)=\frac {1}{2}\left (\begin {array}{c}1 \\1\end {array}\right ) \frac {1}{\sqrt {4 \pi D t}}e^{-x^{2}/4Dt}\), the diffusive Green’s function. Notice that the formal analytic continuation that takes the diffusion equation to the Schrödinger equation is in this context no longer formal but specified by keeping track of parity through (9).

  5. The failure of c to appear explicitly is analogous to the failure of c to appear in the O(v 2) term in the expansion of m c 2. Phase in Schrödinger wavefunctions are not overtly linked to special relativity by an association with c for the same reason.

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Acknowledgments

The author is grateful to an anonymous referee who provided helpful comments.

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  1. Department of Mathematics, Ryerson University, Toronto, Ontario, Canada

    G. N. Ord

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Ord, G.N. Superposition as a Relativistic Filter. Int J Theor Phys 56, 2243–2256 (2017). https://doi.org/10.1007/s10773-017-3372-0

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  • DOI: https://doi.org/10.1007/s10773-017-3372-0

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