Pilot waves

The classical alternative to quantum mechanics?

January 10, 2024

The Big Idea: Can the Mysteries of Quantum Mechanics Ride on a Classical Wave?

Quantum mechanics is the bedrock of modern physics, yet its interpretation remains a source of profound debate 1. While its predictions are astoundingly accurate, the “weirdness” – superposition, entanglement, the measurement problem – often leaves us grappling for intuition. But what if some of this quantum magic could be understood through more tangible, classical-like “pilot waves” guiding particles.

Inspired by Bohmian mechanics 2 and the visually striking hydrodynamic quantum analogues 3. Our goal is to explore the rich, complex, and potentially entirely new physics that can emerge when particles and waves engage in an intricate dance. We’re on a quest for “Wave Functions from Wakes” and a menagerie of “Quantum-esque Oddities.”

The Core Pillars of Our Exploration:

This project is a journey with four interconnected expeditions:

1. Wave Functions from Wakes: Bridging the Microscopic Ripple with the Macroscopic Wave

  • The Question: Can the abstract, complex wave function of the Schrödinger equation (our “Ensemble Descriptor”) be understood as an emergent statistical property of more “physical,” individual waves or “wakes” generated by particles (our “Individual Samplers”)?
  • The Approach:
    • Theoretical: We’ll delve into the mathematics. Starting with abstract dynamical systems – one describing an ensemble (like a probability distribution or a quantum state) and another describing individual “samplers” that might generate their own local waves – we’ll investigate the conditions under which the former can be derived from the statistical behavior of the latter. How does stochasticity (think Fokker-Planck equation 4) play a role? How can complex numbers and quantum phase emerge from real-valued individual dynamics?
    • Empirical (Computational): We’ll build simulations! We’ll model particles that create their own “wakes” and are, in turn, guided by them. By running ensembles of these simulations with varying initial conditions, can we observe emergent statistical patterns that mirror the probability densities (and perhaps even phases) of standard quantum wave functions?
  • The Vision: To find a tangible link, a mathematical bridge, between the individual, “classical-like” wave-particle interactions and the overarching, probabilistic (or complex-field) description of quantum states.

2. A Gallery of Oddities: Curating Novel Pilot-Wave Dynamics

  • The Inspiration: Just as astronomers and mathematicians collect fascinating solutions to the N-body problem 5, revealing intricate celestial ballets, we aim to discover and catalogue the weird and wonderful states and dynamics that pilot-wave systems can produce.
  • The Playground: Our simulations won’t be limited to just mimicking known quantum effects. We’ll explore:
    • Multi-particle systems with various couplings (to each other, to their waves).
    • Different “flavors” of pilot waves: How does the system behave if the wave is diffusive, reactive, or has memory?
    • Introducing classical fields: How do charged pilot-wave particles interact with electrostatic or (simplified) magnetic fields, and how does this influence both particle and wave?
  • The Output: A visually rich, interactive online gallery showcasing these novel dynamics. Think “quantum” football games, self-organizing particle swarms guided by collective waves, or wave-particle systems that exhibit unexpected forms of memory or computation. Let’s find the “strange attractors” of the pilot-wave world!

3. Riding the Computational Wave: Mastering Efficiency

  • The Challenge: Simulating wave phenomena, especially coupled wave-particle systems, can be computationally ferocious. Solving PDEs, handling particle interactions, and running large ensembles demand efficiency.
  • The Strategy: This is where art meets science. We’ll explore a spectrum of techniques:
    • Algorithmic Elegance: Investigating advanced ODE/PDE solvers (Runge-Kutta, symplectic integrators, spectral methods for wave evolution).
    • Boundary Brilliance: Clever and efficient ways to compute boundary conditions for waves (from simple reflections to more complex absorbing layers).
    • Laplacian Wizardry: Optimizing the computation of Laplacians and gradients (FFTs, convolution tricks, higher-order stencils).
    • Compiler & Hardware Hacks: Leveraging modern tools like JAX for JIT compilation, automatic differentiation (if we explore potential-based derivations), and GPU acceleration.
    • Adaptive Methods: Exploring adaptive mesh refinement or adaptive time-stepping to focus computational effort where it’s most needed.
  • The Goal: To develop a robust, efficient, and flexible computational framework that allows us to explore a vast parameter space of pilot-wave models without being crippled by performance bottlenecks.

4. Charting the Pilot-Wave Multiverse: A Systematic Exploration

  • The Unknown Territory: There isn’t just one way to construct a pilot-wave theory. The “rules” of interaction between particle and wave can vary significantly.
  • The Expedition Plan: We’ll systematically explore this “family” of possible pilot-wave systems:
    • Coupling Mechanisms: How strongly does the particle influence the wave, and vice-versa? Is the coupling local or non-local? Does it depend on particle velocity or wave amplitude?
    • Wave Generation: How is the “pilot wave” or “wake” generated? Is it a direct emission from the particle? Does it arise from a background field disturbed by the particle?
    • Wave Dynamics: What is the intrinsic nature of the wave itself (e.g., its dispersion relation, damping, non-linearities)?
    • Dimensionality and Particle Number: How do these dynamics scale and change with more particles or in different spatial dimensions?
  • The Map: We aim to categorize the resulting systems by their emergent properties. Do some configurations naturally lead to quantization? Do others exhibit chaotic behavior, stable solitons, or collective intelligence? Can we identify “universality classes” in the pilot-wave landscape?

I’m going to make some waves! Join me :)

Footnotes:

  1. Rosenblum, B., & Kuttner, F. (2011). Quantum Enigma: Physics Encounters Consciousness. Oxford University Press. ↩

  2. Bohm, D. (1952). A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I. Phys. Rev., 85, 166-179. and Bohm, D. (1952). A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. II. Phys. Rev., 85, 180-193. See also: Holland, P. R. (1995). The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press. ↩

  3. Couder, Y., & Fort, E. (2006). Single-Particle Diffraction and Interference at a Macroscopic Scale. Phys. Rev. Lett., 97, 154101. and Bush, J. W. M. (2015). Pilot-Wave Hydrodynamics. Annu. Rev. Fluid Mech., 47, 269-292. ↩

  4. Risken, H. (1996). The Fokker-Planck Equation: Methods of Solution and Applications (2nd ed.). Springer. ↩

  5. N-body problem. In Wikipedia, The Free Encyclopedia. Retrieved June 7, 2024, from https://en.wikipedia.org/wiki/N-body_problem ↩