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The Madelung Picture as a Foundation of Geometric Quantum Theory

Foundations of Physics volume 47, pages 1317–1367 (2017)Cite this article

Abstract

Despite its age, quantum theory still suffers from serious conceptual difficulties. To create clarity, mathematical physicists have been attempting to formulate quantum theory geometrically and to find a rigorous method of quantization, but this has not resolved the problem. In this article we argue that a quantum theory recursing to quantization algorithms is necessarily incomplete. To provide an alternative approach, we show that the Schrödinger equation is a consequence of three partial differential equations governing the time evolution of a given probability density. These equations, discovered by Madelung, naturally ground the Schrödinger theory in Newtonian mechanics and Kolmogorovian probability theory. A variety of far-reaching consequences for the projection postulate, the correspondence principle, the measurement problem, the uncertainty principle, and the modeling of particle creation and annihilation are immediate. We also give a speculative interpretation of the equations following Bohm, Vigier and Tsekov, by claiming that quantum mechanical behavior is possibly caused by gravitational background noise.

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Fig. 1

Notes

  1. “In a recent paper Heisenberg puts forward a new theory, which suggests that it is not the equations of classical mechanics that are in any way at fault, but that the mathematical operations by which physical results are deduced from them require modification.” [1]

  2. Here we use the usual notation for vector calculus on \(\mathbb {R}^3\) with standard metric \(\delta \).

  3. “Es besteht somit Aussicht auf dieser Basis die Quantentheorie der Atome zu erledigen.” [28, p. 326]; translation by author: “There is hence a prospect to complete the quantum theory of atoms on this basis.”

  4. We believe that the lack of physicality is a consequence of neglecting spin in the Schrödinger theory. We refer to [40, 41] for an elaboration on this point of view.

  5. For the same reason as in the case of Newtonian spacetimes, it is sensible to assume spacetimes to be also space-oriented.

  6. Physically, a test particle is an almost point-like mass (relatively speaking), whose influence on the spacetime geometry can be neglected in the physical model of consideration.

  7. For a discussion on this interpretation and why alternative ones should be excluded, see e.g. [47, §4.2].

  8. Note that only the \((E<0)\)-solutions are admissible, as the other ones are not \(L^2\)-integrable (c.f. [58, §36]).

  9. Note that this can already not be the case for the momentum operator \(\hat{\vec {p}}\), but can only hold true for its “components” \(\hat{p}_i\).

  10. Usually this is called the Hamiltonian operator, but we take the Hamiltonian to be (3.6g). Considering \(\i \hbar \, \partial /\partial t\) as the energy operator is more natural from the relativistic point of view.

  11. ‘\(\Psi _t\) vanishes on \(\partial \Omega _t\)’ means that for any sequence \(({\vec {x}}_n)_{n \in \mathbb {N}}\) in \(\Omega _t\) converging to \({{\vec {x}}} \in \partial \Omega _t \subset \mathbb {R}^3\), we have \(\lim \limits _{n \rightarrow \infty } \Psi _t \left( {{\vec {x}}}_n\right) = 0\).

  12. On a technical note, to assure convergence of (5.2), we also require that the function

    $$\begin{aligned} N \rightarrow \mathbb {R}:{{\vec {x}}} \rightarrow \sup _{t \in I} |\left( \frac{\partial \rho }{\partial t} \bigl (t, \vec \Phi _t \left( {{\vec {x}}}\right) \bigr ) + \left( \nabla \cdot \left( \rho {\vec {X}} \right) \right) \bigl (t, \vec \Phi _t \left( {{\vec {x}}}\right) \bigr ) \right) \, \det \biggl ( \biggl (\frac{\partial \vec \Phi _t}{\partial {{\vec {x}}}} \biggr ) \left( {{\vec {x}}} \right) \biggr ) | \end{aligned}$$

    is bounded and integrable over N. This is trivially true if the continuity equation holds.

  13. It should be noted that there have already been attempts to find a stochastic formulation of the Madelung equations within the so-called theory of ’stochastic mechanics’, developed mainly by Nelson. See e.g. [70] for a review.

  14. This is another instance where the von Neumann approach to probability (see Sect. 4) leads to questionable results: Why should one change the probability theory in the large mass-approximation?

  15. This means that Alice does not observe any gravitational lensing or deviation from straight-line motion of macroscopic, unaccelerated objects.

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Acknowledgements

I would like to thank the following people who have made this work possible in various ways: Ina, Heiko and Marcel Reddiger, Helmut and Erika Winter, Erich and Rita Reddiger, Katie, Timothy and Camden Pankratz, Whitney Janzen-Pankratz, Viktor Befort, Viola Elsenhans, Adam Murray, Marcus Bugner, Alexander Gietelink Oldenziel, Christof Tinnes, Knut Schnürpel, Thomas Kühn, Arwed Schiller, Gerd Rudolph, Dennis Dieks, Gleb Arutyunov, Wolfgang Hasse and Maaneli Derakhshani. Moreover, I am grateful to Adam Murray and Markus Fenske for their help in correcting the manuscript and the two anonymous referees for their constructive comments. Thanks goes also to the dear fellow who wrote the Wikipedia article on the Madelung equations, without which I might have not stumbled upon them and realized their importance.

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    Maik Reddiger

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Reddiger, M. The Madelung Picture as a Foundation of Geometric Quantum Theory. Found Phys 47, 1317–1367 (2017). https://doi.org/10.1007/s10701-017-0112-5

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Keywords

  • Geometric quantization
  • Interpretation of quantum mechanics
  • Geometric quantum theory
  • Madelung equations
  • Classical limit