Abstract
The paper explains why the de Broglie–Bohm theory reduces to Newtonian mechanics in the macroscopic classical limit. The quantum-to-classical transition is based on three steps: (i) interaction with the environment produces effectively factorized states, leading to the formation of effective wave functions and hence decoherence; (ii) the effective wave functions selected by the environment—the pointer states of decoherence theory—will be well-localized wave packets, typically Gaussian states; (iii) the quantum potential of a Gaussian state becomes negligible under standard classicality conditions; therefore, the effective wave function will move according to Newtonian mechanics in the correct classical limit. As a result, a Bohmian system in interaction with the environment will be described by an effective Gaussian state and—when the system is macroscopic—it will move according to Newtonian mechanics.
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Notes
As long as they are isolated, even macroscopic systems maintain their quantum behavior, such as the VIRGO interferometer for the detection of gravitational waves.
Typical decoherence times for a dust grain in collisional decoherence range from \(10^{-18}-10^{-31}s\). See, for example, Schlosshauer ([2], Sect. 3.4, p. 135)
I have analyzed the limits of decoherence theory in the standard context in [4].
This consideration is independent from the specific interpretation of the wave function that we adopt in Bohm’s theory. Different interpretations will simply draw the line between the formalism and the ontology at different points of the proposed strategy.
In the macroscopic regime, the number of particles composing the system or the environment are typically characterized by the Avogadro’s number: \(10^{23}\).
A master equation is the dynamical equation that describes the evolution of the subsystem (represented by a reduced density matrix) in interaction with the environment.
The quantum potential (and the quantum force) may be seen as a novel physical potential (and force) or just as part of the mathematical description for the dynamics of the Bohmian particles. My position on this topic has been developed in [7], but the strategy presented in this paper is independent from the specific interpretation of the quantum potential (and more generally, of the wave function) that one takes in Bohm’s theory.
Under these conditions, the Bohmian trajectories will be approximately and not exactly Newtonian trajectories, but with a level of approximation that is impossible to detect in the macroscopic regime.
Note however that the amplitude R(x, t) and phase S(x, t) of the wave function are not independent terms, they are coupled together through the continuity equation (4).
In the 1953 Einstein’s example of the particle in a box, the Bohmian particle is at rest even if the system has a finite total energy but both the kinetic energy and the classical potential energy are zero inside the box. The energy is indeed absorbed by the quantum potential: \(E_{tot}=Q\). See, on this point, ([10], p. 184) I will analyze this example in the next section.
Two different approaches have been proposed in the literature to explain why the initial particle configuration of a Bohmian system is distributed according to \(|\psi |^2\): the typicality approach by Dürr et al. [11] and the relaxation dynamical approach by Valentini [12]. A comparative review of the two approaches has been made by Norsen [13]. See also Drezet [14], for a recent proposal to justify the Born’s rule using a decoherence framework.
For example: the quantum force is what makes the particles’ trajectories deviate from straight lines in the two-slit experiment, even if there is no classical force acting on the Bohmian particles between the slits and the final screen.
Some of these examples are discussed e.g. in ([15], ch. 6).
And Bohm’s theory, briefly discussed by Einstein in the original article.
According to Einstein, those conditions characterized completely the classical limit. The importance of the entanglement with the environment and the resulting decoherence process was not known at that time: the first important works on decoherence were published only some decades later.
This is one of the few paradoxes in quantum mechanics that involves only the classical limit without reference to the collapse of the wave function or the measurement problem. In a way, we could just rename it as the Einstein’s particle-in-a-box paradox.
Decoherence theory will be formulated only some decades later with respect to Einstein’s example.
This claim cannot be taken too literally in the standard interpretation, as we cannot assign a wave function to an open system–subsystem of an entangled state—but only a reduced density matrix. However, it can be taken quite literally in Bohm’s theory, where an open system is likely to be described by an effective wave function, especially in the macroscopic regime where there is strong interaction with the environment and the system itself has a large numebr of degrees of freedom.
That is: the selected initial state of a system in standard quantum mechanics is already the result of a measurement process.
See Rovelli [18] for a recent analysis of the preparation of initial states in Bohm’s theory based on the effective factorization process.
Note that, in this process, the dynamics of the two particles is correlated: if particle X enters the component A(C), particle Y will necessarily enter the component B(D).
The empty components are discarded for practical purposes but, if they happen to overlap with the effective wave function in the future evolution of the system, then they become relevant again and produce interference. For this reason, [19] preferred to call them “inactive” rather than empty components.
The process of effective factorization has been originally described by Bohm and Hiley [19] and the notion of effective wave function appears in Sect. 7 (p. 344)). A detailed analysis of the effective factorization is also given by Dürr et al. [11], Dürr and Teufel ([20], ch. 9)), Holland ([15], ch. 8).
On this point, see e.g. Rovelli [18], in particular Eqs. (5) and (6).
Effectively factorized components with no Bohmian particles.
See e.g. Bohm and Hiley ([9], Sect. 7.6).
See e.g. by Dürr et al. ([11], pp. 24–25).
See e.g. Romano ([21], p. 13).
There is a difference though between the two: the apparatus interacts with the system at a given time and all the apparatus’ degrees of freedom are involved on that interaction at that single time, whereas the environment is generally described as a tensor product of many particles which interact one at a time with the system. However, these two descriptions eventually converge to the same result, with the difference that the decoherence process induced by the environment is continuous and progressive whereas that one induced by the measurement apparatus is instantaneous and discrete (this corresponds to the collapse of the wave function in the standard interpretation).
See e.g. Schlosshauer ([5], pp. 8–10).
Zurek et al. [6]
This is an assumption in decoherence theory, but it is possible to show that this condition is fulfilled in the long run (i.e. after many interactions between the system and the external particles composing the environment) even if the different relative environmental states are not orthogonal.
Position is a privileged quantity in Bohm’s theory. For example: the velocity of the Bohmian particles is given by the gradient of the phase of the wave-function in the position representation.
The condition of superorthogonality between environmental states or, equivalently, the requirement of disjoint supports, is the implementation of the orthogonality condition in the position basis.
The condition for the effective factorization—disjoint supports on configuration space—is stronger than the standard decoherence condition—orthogonality of states. Nevertheless, we expect the condition of disjoint supports to be approximately or exactly satisfied when the number of degrees of freedom of the environment is very large and the system–environment interaction very strong, as it happens at the macroscopic regime.
Collisional decoherence is decoherence induced by scattering of environmental particles on the system of interest. It is thus a type of environmental decoherence and one of the most important models of decoherence for the quantum-to-classical transition, together with the Quantum Brownian Motion.
This limit applies when the wavelength of the environmental particles is larger than the spatial distance between the subsystem components.
See Schlosshauer ([2], pp. 128–132) for the derivation of the master equation.
Note: the distance \(|x-x'|\) has a threshold after which the long-wavelength limit is not valid anymore and has to be replaced by the short-wavelength limit, which applies when the wavelength of the environmental particles is shorter than the spatial distance \(|x-x'|\). Decoherence in this latter limit is much more effective than decoherence in the long-wavelength limit.
This is a standard claim in decoherence theory. However, see Romano [4] for a critical assessment of this claim.
There is an analogy that can be drawn between pointer states in quantum mechanics and material points in classical mechanics. Since the pointer states are those states that get least entangled with the environment, they follow a quasi reversible dynamics and so they are considered to mimic the reversible dynamics of a material point in classical mechanics. In standard quantum mechanics, a pointer states is thus the closest quantum analogue of a material point in classical mechanics.
This permits to describe an approximately reversible dynamics for the pointer states that could be irremediably lost in the strong coupling limit, where the interaction Hamiltonian dominates over the other terms and the subsystem dynamics becomes quickly irreversible.
I have analyzed this point in more detail in ([4], Sect. 1.2).
A different approach is taken by Sorgel and Hornberger [25]: in their strategy, the pointer states are represented by soliton-like solutions of the system–environment entangled wave function. Even if different from Zurek’s approach, also this strategy amounts to individuate some stable dynamical structures within the wave function and reify them as states. So, it does not seem to alleviate the problem discussed here.
For example: in the case of the QBM analyzed by Zurek et al. [6], the ES-EWF of the system is the ground state of the quantum Harmonic oscillator.
Within a range of approximation that is not empirically detectable at the macroscopic scale.
We may think of extreme cases in which, even if the quantum potential has a very small value, its spatial derivative oscillates very rapidly and produces of a non-negligible quantum force. Such cases cannot be excluded a priori, but for the moment we can (quite safely) assume that when the quantum potential is approximately zero the influence of the quantum force will be also negligible.
Ballentine ([26], ch. 14).
Note that the quantum potential in (49) goes to zero even if we maintain \(\hbar\) as a constant and let the limit varying on the value of the mass, which approaches \(m \rightarrow \infty\) for macroscopic systems.
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Acknowledgements
I wish to thank Mario Hubert for his helpful feedbacks and comments on previous drafts of this paper. I also want to thank Valia Allori, Guido Bacciagaluppi, Andrea Oldofredi, Patricia Palacios and Antonio Vassallo with whom I discussed the topic presented in this work on many occasions over the years. 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 of the University of Lisbon.
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Manuscript for the special issue edited by A. Drezet: Pilot-wave and beyond: Louis de Broglie and David Bohm’s quest for a quantum ontology, Foundations of Physics.
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Romano, D. A Decoherence-Based Approach to the Classical Limit in Bohm’s Theory. Found Phys 53, 41 (2023). https://doi.org/10.1007/s10701-023-00679-w
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DOI: https://doi.org/10.1007/s10701-023-00679-w