Playground 04

Rotations, Boosts, and Spinor Transformation

Compare how vectors and spinors respond to rotations and boosts, then watch the hallmark spin-1/2 sign flip after a full 2π turn.

Animation Controls

Current build step

1. Coordinate axes rotate

Rotation amount1.6 turns

half turnfull turnpast 2π

Boost strength0.7

restmoderaterelativistic

Component mixing0.55

upper-heavybalancedlower-heavy

Vector angle

0.00π

Spinor sign

same phase

Boost mix

1.07 : 0.95

Rotation and Boost Picture

Vector versus spinor response

Vectors follow the full angle, while spinors track half the angle and need a 4π rotation to return completely.

5 stages

Rotation intuition

A vector comes back after one full turn. A spinor only comes back up to a minus sign, so the true return needs 4π.

Vector rule

v \to R(\theta)\,v

Spinor rule

\psi \to S(\theta)\,\psi

Special fact

S(2\pi) = -1

Boost intuition

Boosts are not ordinary spatial rotations. They reweight the spinor components rather than simply spinning a spatial arrow.

Rapidity picture

\psi' = e^{-\eta K}\psi

Upper component

1.07

Lower component

0.95

What to notice

The vector arrow tracks the full spatial rotation, but the spinor marker only advances at half the angle. That is the visual clue behind the 2π sign flip.

Why this matters

Spin-1/2 objects are not just tiny rotating vectors. They live in a different representation, which is why Dirac spinors behave so differently from ordinary three-vectors.

Try this

Push rotation past one full turn, then increase boost strength. You should see the sign-flip picture stay distinct from the boost-mixing picture rather than collapsing into the same transformation.