The Field Ocean: the maths
The Klein–Gordon equation (a relativistic field) on a lattice of points, integrated step by step. The lower field has mass m; the upper is massless. Computed live.
Equations
- ∂²φ/∂t² = ∂²φ/∂x² − m²φ
- The field equation: waves on a field whose 'mass' resists being moved.
- ω² = 4 sin²(k/2) + m²
- How fast a ripple of wavenumber k oscillates on the lattice (in the continuum, ω² = k² + m²).
- vgroup = dω/dk = sin k / ω
- How fast the ripple travels. For m = 0 it's the speed of light (almost exactly, at this k); for m > 0 it's slower.
Constants
| k | 0.25 per lattice step | the travelling ripple's wavenumber |
| c | 1 lattice step per time step | units |
Worked examples ✓ checked on every change
| Massive ripple's speed as a share of light's, m = 0.36 | 56.9% | v = sin k / ω, both fields |
| …with no mass | 100% | a massless ripple travels at light speed |
What's simplified
- One space dimension; a classical field (no quantum particles), which is how the wave side of a particle behaves.
- Mass is a slider: in nature it comes from the Higgs field, fixed for each particle.
Where it breaks
At lattice-scale wavelengths (k near π), where the grid shows; the travelling ripple is kept well above that.
Sources: O. Klein (1926); W. Gordon (1926); any quantum-field-theory text (e.g. Peskin & Schroeder, 1995, ch. 2).