Quantum Phenomena Modeled by Interactions between Many Classical Worlds

We investigate whether quantum theory can be understood as the continuum limit of a mechanical theory, in which there is a huge, but finite, number of classical “worlds,” and quantum effects arise solely from a universal interaction between these worlds, without reference to any wave function. Here, a “world” means an entire universe with well-defined properties, determined by the classical configuration of its particles and fields. In our approach, each world evolves deterministically, probabilities arise due to ignorance as to which world a given observer occupies, and we argue that in the limit of infinitely many worlds the wave function can be recovered (as a secondary object) from the motion of these worlds. We introduce a simple model of such a “many interacting worlds” approach and show that it can reproduce some generic quantum phenomena—such as Ehrenfest’s theorem, wave packet spreading, barrier tunneling, and zero-point energy—as a direct consequence of mutual repulsion between worlds. Finally, we perform numerical simulations using our approach. We demonstrate, first, that it can be used to calculate quantum ground states, and second, that it is capable of reproducing, at least qualitatively, the double-slit interference phenomenon.

Popular Summary

Quantum mechanics provides our most fundamental description of nature, but there is a long-standing and passionate debate among physicists about what all the math “really” means. We provide an answer based on a very simple picture: The world we experience is just one of an enormous number of essentially classical worlds, and all quantum phenomena arise from a universal force of repulsion that prevents worlds from having identical physical configurations. Probabilities arise only because of our ignorance as to which world an observer occupies. This picture is all that is needed to explain bizarre quantum effects such as particles that tunnel through solid barriers and wave behavior in double-slit experiments.

Our “many-interacting-worlds” approach hinges on the assumption that interactions between deterministically evolving worlds cause all quantum effects. Each world is simply the position of particles in three-dimensional space, and each would evolve according to Newton’s laws if there were no interworld interactions. A surprising feature of our approach is that the formulation contains nothing that corresponds to the mysterious quantum wave function, except in the formal mathematical limit in which the number of worlds becomes infinitely large. Conversely, Newtonian mechanics corresponds to the opposite limit of just one world. Thus, our approach incorporates both classical and quantum theory. We perform numerical simulations and show that our approach can reproduce interference with a double slit. As few as two interacting worlds can result in quantumlike effects, such as tunneling through a barrier.

Our approach, which provides a new mental picture of quantum effects, will be useful in planning experiments to test and exploit quantum phenomena such as entanglement. Our findings include new algorithms for simulating such phenomena and may even suggest new ways to extend standard quantum mechanics (e.g., to include gravitation). Thus, while Richard Feynman may have had a point when he said “I think I can safely say that nobody understands quantum mechanics,” there is still much to be gained by trying to do so.

Figure 1

Two one-dimensional worlds, with equal masses $m$ and initial speeds $v_0$, approach a potential energy barrier of height $V_0$. Because of mutual repulsion the leading world can gain sufficient kinetic energy to pass the barrier.

Figure 2

Oscillator ground state for $N=11$ worlds. The steps of the stepped blue curve occur at the values $q={\xi }_1,{\xi }_2,\dots ,{\xi }_{11}$, corresponding to the stationary world configurations $x_1,x_2,\dots ,x_{11}$ via Eq. (41). The height of the step between ${\xi }_n$ and ${\xi }_{n+1}$ is $P_N({\xi }_n)≔N^{-1}({\xi }_{n+1}-{\xi }_n{)}^{-1}$, which from Eq. (21) is expected to approximate the quantum ground-state distribution for a one-dimensional oscillator, $P_{{\mathrm{\Psi }}_0}(\xi )=(2\pi {)}^{-1/2}{e}^{-{\xi }^2/2}$, for large $N$. The latter distribution is plotted for comparison (smooth magenta curve). All quantities are dimensionless.

Figure 3

Convergence of the ensemble of world-particles $x_n=(x_1,\dots ,x_N)$ for $N=11$ towards the ground-state configuration as a function of the step number in the iteration algorithm described in the text. Here, dimensionless positions are used, ${\xi }_n=\sqrt{2m\omega /\hslash }{x}_n$, as per Eq. (41), and the temporal step size is $\mathrm{\Delta }t=5\times 10^{-2}{\omega }^{-1}$. As the plot shows, convergence is complete by step number 6000, at which the distribution of worlds is close to the Gaussian quantum ground state.

Figure 4

Average energy (solid blue line) for the ensemble of worlds, as a function of iteration step. Details as in Fig. 3. Note that convergence to the exact value for MIW is $(1-1/N)\hslash \omega /2$ (red dotted line), which coincides with the quantum mechanical value of $\hslash \omega /2$ (green dashed line) for $N\to \infty$.

Figure 5

A density plot of $|{\mathrm{\Psi }}_t{|}^2$ for the exact quantum evolution of two identical initial Gaussian wave packets in units of the initial spread (i.e., $\sigma =1$); time is given in units of $\hslash /(2m)$.

Figure 6

Trajectory and (smoothed) density plots of the computed MIW (top) and dBB (bottom) world trajectories for two identical initial Gaussian wave packets in units of the initial spread (i.e., $\sigma =1$) and $N=41$ worlds; time is given in units of $\hslash /(2m)$.

Figure 7

Smoothed histogram plots of the computed MIW (top) and dBB (bottom) world trajectories for two identical initial Gaussian wave packets in units of the initial spread (i.e., $\sigma =1$) and $N=41$ worlds; time is given in units of $\hslash /(2m)$.