Roy R. Gould

Am. J. Phys., Vol 63, No. 2,
February 1995

Twenty years after he posed the question,^[1] Feynman himself sketched an elementary argument ^[2] for why "particles with half-integral spin are Fermi particles whose amplitudes add with the minus sign." The argument invokes a remarkable property of three dimensional space: when an object is rotated 360 degrees, it returns to a state that looks the same geometrically but that is topologically distinct with respect to its surroundings -- a twist has been introduced. A second full rotation (a total of 720 degrees) brings the object back to its original state. See Ref. 3 for a clear visualization of this. For a quick demonstration, try the Philippine wine dance: ^[4] Place a wine glass on your palm, and rotate your hand 360 degrees in a plane parallel to the floor. The ensuing twist in your forearm can, remarkably, be undone by continuing to rotate in the same direction!) In nature, this bizarre topology is exhibited by electrons and all particles with half-integral spin ("fermions'). Rotate a fermion by 360 degrees - using a magnetic field and the fermion's magnetic moment as a "handle" - and it will not be the same beast as before the rotation: Experimentally, its interference properties will have changed; mathematically, its phase will have shifted by 180 degrees and its wave function will have changed sign. Now, interchanging two objects is topologically the same as rotating either one of them 360 degrees. (To see this, first grasp the two ends of a belt, one end in each hand; then interchange the postion of your hands. You have introduced a "twist" which is topologicallly equivalent to having rotated one end of the belt by 360 degrees.) Thus, when fermions are interchanged, one must keep track of this "implied rotation" and the phase shift, sign change, and destruction interference to which it gives rise. For example, if A(1)B(2) describes "electron 1 in state A and electron 2 in state B," then the state with electrons interchanged must be -A(2)B(1) and their superposition is

A(1)B(2) - A(2)B(1)

The existence of spin 1/2 follows from the marraige of relativity and quantum theory. ^[5] But it is topology the underlies the Fermi statistics, and therefore the Pauli exclusion principle - and by extension the existence of atoms and ourselves.