De Broglie Hypothesis and the Internal ${\mathbb{S}}^1$ Bundle Picture

1. Historical Context

• In 1924, Louis de Broglie proposed that particles have wave-like properties, analogous to how light exhibits particle-like properties.
• For light:
  • Wave-like: interference, diffraction.
  • Particle-like: photons, with energy $E=h\nu$.
• De Broglie asked: if waves can behave like particles, can particles behave like waves?

2. Core Postulate

De Broglie postulated that every particle of mass $m$ is associated with a wave, whose frequency and wavelength are:

• $E$ is the total energy of the particle.
• $p$ is the momentum.
• $\lambda$ is the de Broglie wavelength.

This relation is a postulate consistent with special relativity and verified experimentally (electron diffraction, 1927).

3. Internal Clock / Harmonic Oscillator Analogy

• A particle at rest has a "hidden clock" ticking at:$${\nu }_0=\frac{m{c}^2}{h},{\omega }_0=\frac{m{c}^2}{\hslash }$$
• This is not a literal spatial oscillation but an internal phase rotation.
• Visualized as a point moving on a circle ${\mathbb{S}}^1$, representing a phase fiber attached to the particle.

4. Relativistic Effects

• Time dilation affects the internal clock:$$\omega =\frac{{\omega }_0}{\gamma },\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$$
• The ${\mathbb{S}}^1$ circle itself is not contracted, as it is internal.
• The observed plane wave frequency in the lab frame increases with motion:$${\omega }_{\text{wave}}=\frac{E}{\hslash }=\frac{\gamma m{c}^2}{\hslash }$$

5. Phase Matching and the de Broglie Wavelength

• The external plane-wave phase is:$$\varphi =\frac{Et-\mathbf{p}\cdot \mathbf{x}}{\hslash }$$
• The “projection” of the time-dilated internal ${\mathbb{S}}^1$ clock along the particle trajectory yields:$$\lambda =\frac{h}{p}$$

This shows how the internal phase dynamics generate the spatial matter wave.

7. Summary: ${\mathbb{S}}^1$ Bundle Picture

• Every particle carries an internal phase: $\theta \in [0,2\pi )$.
• Proper-time evolution: $\theta (\tau )={\omega }_0\tau$.
• Observed from a moving frame: $\theta (t)={\omega }_0\tau (t)={\omega }_{0}t/\gamma$.
• Plane wave in spacetime encodes this phase and gives rise to interference and diffraction.
• The ${\mathbb{S}}^1$ is internal; only the phase rate is affected by motion, not the circle itself.

De Broglie’s genius: linking internal periodicity to observable matter waves, prefiguring modern fiber-bundle and gauge perspectives.