In the Holland-Poirier hydrodynamical approach, the wave function plays no dynamical role. However, it may be recovered, in a nontrivial manner, by integrating the trajectories up to any given time [8]. This has proved a useful tool for making efficient and accurate numerical calculations in quantum chemistry [9,10]. Schiff and Poirier [11], while “drawing no definite conclusions,” interpret their formulation as a “kind of ‘many worlds’ theory,” albeit they have a continuum of trajectories (i.e., flow lines), not a discrete set of worlds.
Here, we take a different but related approach, with the aim of avoiding the ontological difficulty of a continuum of worlds. In particular, we explore the possibility of replacing the continuum of fluid elements in the Holland-Poirier approach by a huge but finite number of interacting “worlds.” Each world is classical in the sense of having determinate properties that are functions of its configuration. In the absence of the interaction with other worlds, each world evolves according to classical Newtonian physics. All quantum effects arise from, and only from, the interaction between worlds. We therefore call this the many interacting worlds (MIW) approach to quantum mechanics. A broadly similar idea has been independently suggested by Sebens [12], although without any explicit model being given.
The MIW approach can only become equivalent to standard quantum dynamics in the continuum limit, where the number of worlds becomes uncountably infinite. However, we show that even in the case of just two interacting worlds it is a useful toy model for modeling and explaining quantum phenomena, such as the spreading of wave packets and tunneling through a potential barrier. Regarded as a fundamental physical theory in its own right, the MIW approach may also lead to new predictions arising from the restriction to a finite number of worlds. Finally, it provides a natural discretization of the Holland-Poirier approach, which may be useful for numerical purposes. Before considering how its dynamics might be mathematically formulated and used as a numerical tool, however, we give a brief discussion of how its ontology may appeal to those who favor realist interpretations.
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In the MIW approach there is no wave function, only a very large number of classical-like worlds with definite configurations that evolve deterministically. Probabilities arise only because observers are ignorant of which world they actually occupy, and so assign an equal weighting to all worlds compatible with the macroscopic state of affairs they perceive. In a typical quantum experiment, where the outcome is indeterminate in orthodox quantum mechanics, the final configurations of the worlds in the MIW approach can be grouped into different classes based on macroscopic properties corresponding to the different possible outcomes. The orthodox quantum probabilities will then be approximately proportional to the number of worlds in each class.
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In the Everett or MW interpretation, the worlds are orthogonal components of a universal wave function [3]. The particular decomposition at any time, and the identity of worlds through time, is argued to be defined (at least well enough for practical purposes) by the quantum dynamics which generates essentially independent evolution of these quasiclassical worlds into the future (a phenomenon called effective decoherence). The inherent fuzziness of Everettian worlds is in contrast to the corresponding concepts in the MIW approach of a well-defined group of deterministically evolving configurations. In the MW interpretation, it is meaningless to ask exactly how many worlds there are at a given time, or exactly when a branching event into subcomponents occurs, leading to criticisms that there is no precise ontology [16]. Another difficult issue is that worlds are not equally “real” in the MW interpretation, but are “weighted” by the modulus squared of the corresponding superposition coefficients. As noted above, in the MIW approach, all worlds are equally weighted, so that Laplace’s theory of probability is sufficient to account for our experience and expectations.
The skeptical reader may wonder whether it is appropriate to call the entities in our MIW theory worlds at all. After all, each world corresponds to a set of positions of particles in real (3D) space. How, then, is the nature and interaction of these worlds any different from those of different gas species, say A and B, where the positions of all the A molecules constitute one world and those of the B molecules (each one partnered, nominally and uniquely, with one of the A molecules) constitute another world? The answer lies in the details of the interaction.
In the above example, any given A molecule will interact with any B molecule whenever they are close together in 3D space. Thus, a hypothetical being in the “ A world,” made of some subset of the A molecules, would experience the presence of B molecules in much the same way that it would feel the presence of other A molecules. By contrast, as will be shown later in this paper, the force between worlds in our MIW approach is non-negligible only when the two worlds are close in configuration space. It would be as if the A gas and B gas were completely oblivious to each other unless every single A molecule were close to its B partner. Such an interaction is quite unlike anything in classical physics, and it is clear that our hypothetical A-composed observer would have no experience of the B world in its everyday observations, but by careful experiment might detect a subtle and nonlocal action on the A molecules of its world. Such action, though involving very many, rather than just two, worlds, is what we propose could lie behind the subtle and nonlocal character of quantum mechanics.
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Other matters for future investigation include how spin and entanglement phenomena such as teleportation and Bell-inequality violation are modeled in the MIW approach. The latter will require studying the case of worlds with configuration spaces of at least two dimensions (corresponding to two one-dimensional systems), and will also allow analysis of the quantum measurement problem (where one system acts as an “apparatus” for the other). This may help clarify the ontology and epistemology of any fundamental new theory based upon the MIW approach to quantum mechanics.
In the context of entanglement, it is worth comparing our MIW approach with conventional many-worlds approaches. The latter are often motivated by the desire, first, to restrict reality to only the wave function, and second, to avoid the explicit nonlocality that arises from entanglement in other realist versions of quantum mechanics. Our approach is, by contrast, motivated by the desire to eliminate the wave function. It furthermore elevates the nonlocality of quantum mechanics to a kind of “supernonlocality”: particles in different worlds are nonlocally connected through the proposed MIW interaction, thus leading, indirectly, to nonlocal interactions between particles in the same world.
Turning from questions of foundations and interpretations to applied science, the MIW approach provides a promising controlled approximation for simulating quantum ground states and the time-dependent Schrödinger equation, as discussed in Secs. V and VI. In particular, we show that it is able to reproduce quantum interference phenomena, at least qualitatively. Quantitative comparisons with different initial conditions, convergence as a function of the number of worlds N, and generalizations to higher dimensions, are a matter for immediate future work.
:lol:
