Dirac demonstrates the “Dirac belt trick”, whereby two full twists in a belt are removed while keeping the ends fixed.
The inset at the bottom right shows what’s happening in the group of rotations, SO(3), drawn as a sphere. Each point of the sphere represents a rotation around the vector from the centre of the sphere to the point, by an amount equal to the length of the vector. The radius of the sphere is taken to be π radians, or 180°, and since a rotation of 180° has the same effect if you reverse the direction of rotation – or equivalently, rotate around the reversed axis – antipodal points on the surface of the sphere correspond to identical rotations.
The range of rotations producing the twists along the length of the belt are shown as a continuous path through SO(3). Once the end of the belt has been rotated through 720°, the path forms a closed loop that crosses the width of the sphere twice along the same line. The belt is then unkinked by shrinking this loop down to a single point. As this happens, the loop will appear to have split into two segments, but in fact it remains continuous, since the “breaks” lie on antipodal points of the sphere.
The inset at the top right shows the two complex numbers that comprise the spinor for an electron with its spin axis perpendicular to the belt buckle. More details.
Note that during the belt trick, the material of the belt is actually subject to stretching and shearing deformations as well as rotations and translations. For clarity, bending of the belt (as opposed to transverse twisting) is not included in the rotations shown in the inset. Also, the initial orientation of the belt is taken to lie on the border of the sphere of rotations, rather than at the centre; this just amounts to a convenient choice of reference frame, and makes no difference to any of the topology.
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