Abstract
By developing an absurd counterfactual history, I show that many objections launched against Bohmian mechanics could also have been made against Newtonian mechanics. This paper introduces readers to Koopman–von Neumann dynamics, an operator-based Hilbert space representation of classical statistical mechanics. Lessons for quantum foundations are drawn by replaying the battles between advocates of standard quantum theory and Bohmian mechanics in a fictional classical history.
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Notes
- 1.
See Berry (1992), Chirikov, Izrailev and Shepelyanskii (1988), Della Riccia and Wiene (1966), and ’t Hooft (1997). The “classical Schrödinger equation” (1) should not be confused with another “classical Schrödinger equation” derived in the 1960’s by Schiller (1962) and Rosen (1964). This later equation defines the wavefunction on configuration space \({\varPsi }(q)\) whereas (1) applies to a wavefunction over phase space.
- 2.
Heisenberg recounting his discussions with Einstein, quoted in Becker (2018), 29.
- 3.
Bohr on physics after the Solvay conference, quoted in Becker (2018), 49.
- 4.
von Neumann describing his two dynamics, quoted in Becker (2018), 67.
- 5.
The two measurement problems are slightly different and interesting to consider. As Mauro (2002) emphasizes, the fundamental difference between Koopman–von Neumann theory and ordinary quantum theory is that in the former but not the latter the phase interacts with the modulus. Contrast a Madelung decomposition of Eq. (1) with the Schrödinger equation. Write the quantum wavefunction as \(\psi (x)=A(x)exp[\nicefrac {i}{\hbar }S(x)]\)and substitute it into the Schrödinger equation and then separate real and imaginary parts. Then as is well known one obtains
$$ \frac{\partial S}{\partial t}+\frac{1}{2m}\left( \frac{\partial S}{\partial x}\right) ^{2}+V=\frac{\hbar ^{2}}{2mA}\frac{\partial ^{2}A}{\partial x^{2}} $$$$ m\frac{\partial A}{\partial t}+\frac{\partial A}{\partial x}\frac{\partial S}{\partial x}+\frac{A}{2}\frac{\partial ^{2S}}{\partial x^{2}}=0 $$where one can see that the phase S is coupled to the modulus A. Do the same for the classical wavefunction \(\psi (x)=F(q,p)exp[\nicefrac {i}{\hbar }G(q,p)]\) when inserted into (1). Then we get
$$ i\frac{\partial F}{\partial t}=\mathcal {\hat{H}F} $$$$ i\frac{\partial G}{\partial t}=\mathcal {\hat{H}G} $$and no coupling between F and G. (Why then introduce phases at all? They become necessary if one wants the freedom of basis one gets in Hilbert space; see Mauro 2002.) As a result of this decoupling, wavefunctions without phases cannot generate them in their time evolution. Hence the measurement problem is a bit different than quantum mechanically. In the language of foundations of physics, the classical measurement problem associated with Koopman–von Neumann is like the quantum one if decoherence worked perfectly, driving the off-diagnol terms to exactly zero. That still leaves a measurement problem, the so-called “and” to “or” problem of Bell (1990) (see also Maudlin 1995). On the classical measurement problem, see Chen (2022) (section 5.4), Katagiri (2020), and McCoy (2020).
- 6.
Bohr on Bohm, cited in Becker (2018), 107.
- 7.
Pauli on Bohm, cited in Becker (2018), 107.
- 8.
Rosenfeld (1957), 4–42.
- 9.
See Cushing (1994) for many objections to Bohm along these lines, especially by Pauli. Cushing also details the political attacks on Bohm.
- 10.
See Nikolić (2008) for a less incredible counterfactual history toward the same point.
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Acknowledgements
Thanks to Jacob Barandes, Eddy Chen, Casey McCoy, the audience at the University of Lisbon’s Open Problems in Philosophy of Physics conference, and the UC San Diego philosophy of physics reading group for helpful comments. Details of von Newton’s life were drawn from von Neumann’s biography. Figures 1 and 2 were generated with the assistance of DALL\(\cdot \)E 2. Figure 1 is a blend of the faces of John von Neumann and Isaac Newton.
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Callender, C. (2024). The Prodigy That Time Forgot: The Incredible and Untold Story of John von Newton. In: Bassi, A., Goldstein, S., Tumulka, R., Zanghì, N. (eds) Physics and the Nature of Reality. Fundamental Theories of Physics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-031-45434-9_5
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