Plate Trick

In mathematics and physics, the plate trick, also known as Feynman's plate trick, Dirac's belt trick, Dirac's string trick, Balinese cup trick, spinor spanner, Bredon high-five, or quaternionic handshake, is any of several particular physical demonstrations of the mathematical theorem that SU(2) double-covers SO(3). The usual demonstration, as indicated by the name, is to hold a plate on one's flat palm, then perform two subsequent rotations of the arm holding the plate, which results in the original position. In the cup variant, supposedly inspired by a Balinese candle dance, an open cup of water or wine is held in ones hand and rotated 720 degrees, or any multiple thereof, without spilling the liquid or loosening ones grip on the cup.

Topologists usually call this the belt trick or the string trick, although some knot theorists, such as Louis Kauffman have dubbed it the quaternionic handshake. The reference to quaternions is because the action of SU(2) considered as the unit quaternions, on the 2-sphere, considered as pure unit quaternions, gives the double covering.

In mathematical physics, the plate trick is the reason why certain particles, in a four-dimensional spacetime, behave as spin 1/2 particles: when the plate or cup is taken to represent the particle, it returns to its original state only after two full rotations, not after one. This is the main property of spin 1/2 particles. The extended belt trick, visualized in the links below, shows that such spin 1/2 particles behave as fermions. See the spin-statistics theorem for more details.