Abstract
I argue that a particle ontology naturally emerges from the basic dynamical equations of non-relativistic quantum mechanics, when the quantum continuity equation is realistically interpreted. This was recognized by J.J. Sakurai in his famous textbook “Modern Quantum Mechanics”, and then dismissed on the basis of the Heisenberg position–momentum uncertainty principle. In this paper, I show that the reasons of this rejection are based on a misunderstanding of the physical import of the uncertainty principle. As a consequence, a particle ontology can be derived from the quantum formalism without the need of additional ad hoc assumptions, and therefore it cannot be regarded as “extra-structure”.
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Notes
Cats, chairs and human beings, even if classical objects, are fundamentally quantum in nature, for they are composed of molecules and atoms at the basic level. In fact, classical mechanics is basically quantum mechanics in a specific (macroscopic and decoherence) regime (see e.g. [9, Chap. 6], [15]).
The measurement and registration of a certain eigenvalue is always mediated by a macroscopic device.
While we consider here, for simplicity, a 1-particle system, the present derivation can be straightforwardly generalized to N-particle systems.
Historically, de Broglie wrote his particle dynamics in 1926/1927 before quantum mechanics was even completed (the original idea was to unify two principles of analytical mechanics: the principle of Maupertuis and the principle of Fermat, so deriving a new particle-wave dynamics), and Bohm [3] derived the velocity formula from the quantum Hamilton–Jacobi equation. In both cases, the velocity was not added to the Schrödinger equation but mathematically derived (Bohm) or independently formulated as the new dynamics for quantum systems (de Broglie).
Hilgevoord and Uffink [8] provide a detailed reconstruction of Heisenberg’s arguments in support of this form for the uncertainty principle.
For simplicity, we suppose that the observable A has a discrete spectrum of eigenvalues. For observables having a continuous spectrum of eigenvalues, such as position, the basis expansion involves the Dirac’s \(\delta\) function. However, this mathematical subtlety does not affect the argument I am presenting here.
Note that the expression \((A,B)\vert \psi \rangle\), indicating a simultaneous measurement of the observables A and B, differs from \((AB)\vert \psi \rangle\), which usually denotes an experiment in which we perform a single measurement of the observable B and, then, after a discrete amount of time, a second measurement of the observable A.
I assume, for simplicity, that the expansion coefficients \(c_{j}\) are the same in the two cases. This is an ideal assumption: in general, they will be different for the two operators, provided that the eigenstates are the same.
The position and momentum eigenvalues of the atoms in the crystal are inferred from the total mean thermal energy of the crystal. The experimental accuracy goes beyond the uncertainty principle for temperatures lower than \(T=15 K\).
Ballentine remarks that the standard deviations \(\Delta {Q}\) and \(\Delta {P}\)cannot be properly built at all unless the level of experimental error on individual measurements \(\delta {Q}\) and \(\delta {P}\) be much smaller than the standard deviations.
According to the standard definition (see e.g. [7]), the de Broglie–Bohm theory is a hidden variable theory for the complete ontological state of the system is represented by \(\lambda =(\psi ,x)\), that is, the wave function of the system is supplemented with the variable x. The variable x takes the name of “hidden variable” since, even when we have perfect knowledge of the wave function (for example, when the wave function is a pure state), we still do not know the exact positions of the particles, but only that they are statistically distributed according to \(|\psi (x)|^{2}\).
In this essay, Einstein makes an objection to Bohm’s theory, claiming that it fails to describe the correct classical limit for a massive point-particle in an infinite potential well. He shows, in fact, that the particle remains at rest in a non-nodal point inside the well even for large mass and small de Broglie wave-length (high energy), when the classical regime is expected to arise. On this basis, he excludes Bohm’s theory as a valid option. However, the example made by Einstein does not take into account the interaction with the environment and, consequently, decoherence effects, which is nowadays the standard framework for the classical limit (historically, indeed, decoherence theory will be developed only around 1980s). When decoherence effects are taken into account, Bohm’s theory describes the correct classical regime, with the particle inside the well moving according to Newton’s theory (see e.g. [4, Chap. 8]).
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Acknowledgements
A previous draft of this paper was written in collaboration with Andrea Oldofredi. I am indebted to him for many useful discussions on the themes of this paper. This work has been supported by the Fundação para a Ciência e a Tecnologia through the fellowship FCT Junior Researcher, hosted by the Centre of Philosophy at the University of Lisbon.
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Romano, D. On the Alleged Extra-Structures of Quantum Mechanics. Found Phys 51, 29 (2021). https://doi.org/10.1007/s10701-021-00426-z
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DOI: https://doi.org/10.1007/s10701-021-00426-z