This sim puts one electron on screen and lets you change who is watching it. It shows how the electron's helicity, momentum and chirality look from different frames, which is why "left-handed" means different things for a particle with mass and one without.

What's on screen

• Spin S (purple arrow) and the ball's rotation: fixed in direction and sense.
• p′ (arrow below the ball): the momentum in the observer's frame. Green points right, orange left. Its length is on a compressed scale, so keV/c and MeV/c both fit.
• Chips at the top: the boost β (left) and the helicity h (right).
• Chirality bar (bottom): the R-chiral (green) and L-chiral (orange) shares, with percentages.
• Readouts (sidebar): E′, p′, h, the electron's speed in your frame, and R : L.

Press ⌖ on a row to see where that control sits in the sim. Moving p or β restarts the animation from the left edge.

Fixed: the electron mass, mₑc² = 0.511 MeV. The spin arrow never turns. The ball's rotation rate is only for display.

Helicity is the component of the spin along the direction of motion: h = S·p̂, where p̂ is the unit vector along the momentum as the observer measures it. For an electron it has two possible values, +½ħ (spin along the momentum) or −½ħ (spin against it).

• The purple arrow is S. The arrow below the ball is p′, and the colour of both p′ and the helicity chip shows the sign of h: green for +½ħ, orange for −½ħ.
• The observer moves along the line of the electron's motion. A boost along that line does not change the spin component along it, so the sim keeps S and the ball's rotation sense fixed.
• The electron's own numbers at p = 1 MeV/c: E = √(p²c² + (mₑc²)²) = 1.123 MeV, and its speed is pc/E = 0.890 c. In the lab frame, the "speed in frame" readout shows that 0.890 c.

Boosting along p gives p′ = γ(p − βE). Factoring out E shows the whole story, with β_p = p/E the electron's own speed in the lab (c = 1):

• Below β_p you are chasing the electron and p′ points forward: h = +½ħ. Above it you have overtaken it, p′ points backward and h = −½ħ. The spin never moved; the direction it is compared with did.
• p′ passes through zero smoothly. h is the sign of p′, so it jumps. At p′ = 0 the electron is at rest in your frame and h is undefined (the sim shows this for |β′| < 0.004).
• At p = 1 MeV/c the flip is at β = 0.890, the tick under the β slider. At β = 0.9, p′ = −24.5 keV/c.

Chirality labels the two parts of the electron's field, R-chiral and L-chiral. The weak force acts on the L-chiral part only. Chirality does not depend on the frame: a state that is purely L-chiral is purely L-chiral for every observer.

For an electron with mass, helicity and chirality do not line up. A state of definite helicity is a mixture of both chiralities, and the chirality bar shows the shares:

• The mixture depends on β′, which depends on both p and β. In the lab at p = 1 MeV/c the bar reads 94.5 : 5.5. At β = 0.9 it reads 47.6 : 52.4.
• At p′ = 0 the bar reads 50 : 50. An electron at rest has no preferred handedness.
• The bar's green and orange are R and L chirality. They are a different quantity from the green and orange of the helicity chip.

• A massless particle travels at c in every frame, so no observer gets ahead of it. Its helicity never changes sign, and its chirality is the same in every frame. For such a particle helicity and chirality are the same label.
• "Left-handed neutrino" is a statement about chirality. Only for a massless particle can you read it as negative helicity.
• The sim's electron has mass, so it only approaches this limit. The flip is reachable only if β_p ≤ 0.9995, which means p ≲ 16 MeV/c. At p = 20 MeV/c, β_p = 0.99967 is past the end of the slider.

The electron still has mass, so the bar is not pinned. At p = 20 MeV/c and β = 0.9995, p′ stays positive, but the electron's speed in your frame is only 0.21 c and the bar reads 60.5 : 39.5.

• Beta decay. The weak force acts on L-chiral fields. A left-chiral electron has helicity −½ħ with probability (1 + β)/2, so electrons from β decay have a longitudinal polarisation of −v/c. These are the same (1 ± β)/2 weights as the bar, read the other way round.
• Pion decay. A pion has spin 0, so in π⁻ → ℓ⁻ ν̄ the lepton leaves with the antineutrino's helicity, +½ħ. Its L-chiral share is (1 − β)/2: 0.36 for the muon, 1.3 × 10⁻⁵ for the electron. This helicity suppression is behind the measured ratio Γ(π → eν) / Γ(π → μν) = 1.2 × 10⁻⁴.
• High-energy collisions. For E ≫ mc² helicity and chirality nearly coincide, so helicity is the standard label for particles in a collision.

Key points

• h = S·p̂ is ±½ħ. A boost along p leaves S alone and changes p′.
• p′ = γE(β_p − β): h changes sign at β = β_p = p/E, jumps, and is undefined at p′ = 0.
• Chirality is the same in every frame. A helicity state's R : L shares, (1 ± β′)/2, are not.
• For m = 0 there is no flip, and helicity and chirality coincide.

Common mix-ups

• "Helicity is intrinsic, like spin." Not for anything with mass. The spin never moved; the momentum direction did.
• "h fades smoothly through zero as you catch up." p′ does. h is its sign, so it jumps, with one undefined instant.
• "Left-chiral means negative helicity." Only when E ≫ mₑc², and for a particle, not an antiparticle.

Things to try on your own

• Set p to 0.1 MeV/c and check the marker under the β slider against p/E = 0.192.
• At p = 1 MeV/c press ride along, then step β by 0.0005 each way. Count the steps for which h reads "undefined".
• At p = 5 MeV/c, read R : L at β = 0.9. Then at β = 0, find the p that gives the same bar, and compare it with the p′ you saw.