Scalar Field on a Lattice | mysimulator.uk

Field Parameters

Wave speed c 1.00

Mass² term 0.50

Damping γ 0.05

Actions

Field energy

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Peak amplitude

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Localized quanta

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Lattice size

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Click the field surface to inject a localized perturbation at that point.

How it works

The mesh is a scalar field φ defined only at discrete lattice sites — the essence of classical field theory on a lattice. Every site obeys the same local update rule, derived from the Klein-Gordon field equation, using only its immediate neighbours:

∂²φ/∂t² = c²∇²φ − m²φ − γ·∂φ/∂t

Wave speed c — how fast a disturbance at one lattice site propagates to its neighbours; raise it and ripples race outward and reflect off the fixed (φ=0) boundary faster.
Mass² term — the Klein-Gordon mass term. At m²=0 the field is a pure massless wave equation (like light); raising m² adds a restoring force at every site independent of its neighbours, which suppresses long-wavelength ripples and favours short-lived, localized "lumps" — the classical analogue of why massive quantum fields describe particles with rest mass while massless fields describe radiation.
Damping γ — dissipates energy from the field over time, so a struck field eventually settles rather than ringing forever.

Toggle "Show quanta" to mark every lattice region where |φ| rises above threshold with a small sphere. This is the field-vs-particle picture in miniature: nothing exists at those points except the continuous field itself, yet wherever it develops a large, localized bump, it behaves — for purposes of counting, colliding, and moving as a lump — like a particle. In real quantum field theory this is exactly how particles arise: as quantized excitations of an underlying field, not as separately-existing objects.

φ < 0 (trough) φ ≈ 0 φ > 0 (crest) localized quantum

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