# Wave-Particle Duality and the Hamilton-Jacobi Equation

## TL;DR

This paper revisits the relativistic quantum Hamilton-Jacobi equation, showing it admits breather solutions that embody both particle and wave characteristics, providing a deterministic interpretation of de Broglie waves.

## Contribution

It introduces breather solutions to the relativistic quantum Hamilton-Jacobi equation, offering a new deterministic perspective on wave-particle duality.

## Key findings

- Breather solutions exhibit particle and wave behavior simultaneously.
- De Broglie wave gains a deterministic interpretation as a wave-like excitation.
- Discussion of quantization and double-slit experiment within this framework.

## Abstract

The Hamilton-Jacobi equation of relativistic quantum mechanics is revisited. The equation is shown to permit solutions in the form of breathers (oscillating/spinning solitons), displaying simultaneous particle-like and wave-like behavior. The de Broglie wave thus acquires a clear deterministic meaning of a wave-like excitation of the classical action function. The problem of quantization in terms of the breathing action function and the double-slit experiment are discussed.