Dirac Dispersion Lab

Choose a mass m, momentum p and positive or negative frequency branch, then select Draw. The electron, muon and proton examples use the rounded teaching values 1, 206.77 and 1836.15 times the electron mass; Massless uses zero. The examples change the mass and keep the momentum. The units are fixed: c = ħ = mₑ = 1, with momentum in mₑc and energy in mₑc². Mass and momentum fields accept finite decimal numbers, including a decimal comma. The allowed mass is 0 or 10⁻⁹…10⁶; momentum is 0 or has magnitude 10⁻⁹…4×10⁶. The nonzero lower bound is a numerical input limit, not a physical exclusion.

For branch sign s = ±1, the free relation is E_s(p) = s√(p² + m²). Its local tangent has group velocity v_g = dE_s/dp = p/E_s. The line from the origin to the selected point has phase velocity v_p = E_s/p. Both retain their signs. At p = 0, phase velocity and its line are undefined. When m > 0, group velocity at rest is zero. At m = p = 0, the massless curves ±|p| have a cusp: no unique tangent or group velocity exists. For m = 0 and p ≠ 0, both speed magnitudes are 1.

For m > 0, γ = √(p² + m²)/m; for m = 0 it is not applicable. Rest energy is m and the separation of the two vertices is 2m. This gap is not a general photon pair-production threshold: energy and momentum conservation and the interaction process matter. The tool calculates no pair creation or interaction. Phase velocity can have magnitude greater than c; it does not describe signal transport.

The positive and negative curves are eigenvalue/frequency branches of the free equation. The optional shaded half-plane denotes negative frequencies. In the quantized free field, both particle and antiparticle excitations have positive energy. Interpreting a negative-frequency mode as an antiparticle also reverses the momentum assignment; the slopes here refer to the plotted mode. The Dirac sea is a historical interpretation, not an additional filled-particle simulation in this graphic.

The optional massless references are E = ±p. For m > 0 the nonrelativistic approximations are ±(m + p²/(2m)), valid for |p| much smaller than m; only their visible portions are shown. They do not apply at m = 0. The horizontal half-range is p_max = min(4×10⁶, max(3, 4m, 1.25|p|)); the energy half-range is 1.12√(p_max² + m²). Numeric parameter or branch changes invalidate the result and require Draw again. After drawing, the momentum slider and plot taps update the probe on the same axes. A plot-tap coordinate with magnitude below 10⁻⁹ selects zero to remove coordinate-rounding residue; the number field never silently clamps values.

The optional envelope is A(x,t) = exp[−½((x − v_g t)/0.45)²], displayed over −6 ≤ x ≤ 6 and 0 ≤ t ≤ 4. Its center is exactly x = v_g t, in the natural time and length units associated with c = 1. Its fixed Gaussian width is schematic: there is no carrier wave, spinor, spreading or complete dispersive evolution. Start and Pause control this illustration; Time to 0 resets its clock while retaining the dispersion plot. The animation stops at t = 4 without wrapping. At the massless cusp it is unavailable.

Changing the probe or an option pauses the envelope and resets its time to zero. Hiding the page or leaving the tool pauses it without catching up later. Valid parameters and view settings can be stored locally. Returning within the app can retain the computed plot and probe, paused; a reload restores settings and waits for Draw.

Sources: David Tong, The Dirac Equation, §4.7, for plane-wave solutions and positive/negative frequencies; Quantizing the Dirac Field, §5.2–5.3, for positive-energy particle and antiparticle excitations and the historical hole interpretation.