Orientation entanglement

In mathematics and physics, the notion of orientation entanglement is sometimes[1] used to develop intuition relating to the geometry of spinors or alternatively as a concrete realization of the failure of the special orthogonal groups to be simply connected.

In other words, if we lower the coffee cup to the floor of the room, the two bands will coil around each other in one full twist of a double helix.

Clearly the geometry of spatial vectors alone is insufficient to express the orientation entanglement (the twist of the rubber bands).

Then if the cup is lowered to the floor, the two rubber bands coil around each other in two full twists of a double helix.

In the adjacent diagram, a spinor can be represented as a vector whose head is a flag lying on one side of a Möbius strip, pointing inward.

If the cup is rotated through 360°, the spinor returns to the initial position, but the flag is now underneath the strip, pointing outward.

Note that, since M is unitary, Hence SU(2) acts via rotation on the vectors X. Conversely, since any change of basis which sends trace-zero Hermitian matrices to trace-zero Hermitian matrices must be unitary, it follows that every rotation also lifts to SU(2).

Furthermore, SU(2) is easily seen to be itself simply connected by realizing it as the group of unit quaternions, a space homeomorphic to the 3-sphere.