One of the common geometric symmetries is rotational symmetry. One often speaks of n-fold rotational symmetry, meaning that it is unchanged by a rotation about a certain point (for 2D) or axis (3D) through an angle that is an integer multiple of 2π/*n*. For example, the face of the two of spades, below left, has 2-fold rotation symmetry, as a 180° rotation leaves it unchanged; while the triskele symbol below right has 3-fold rotation symmetry, being unchanged by a 120° (2π/3) rotation.

So, we have *n* so that the smallest rotation under which the shape is unchanged is 2π/*n*. Note, then, that *n*=1 corresponds to a rotationally asymmetric object; it requires a full rotation to be unchanged. Lastly, let us define *n*=0 for circular (or cylindrical or spherical) symmetry, where rotation through any angle is a symmetry. What would happen, then, if we have n=½? This would be an object which would not be symmetric under a full 360° rotation, but instead requires two full rotations to be returned to its initial state. While there is no simple figure like the above to serve as a geometric example of this, one can find a physical example in the Wikipedia article on orientation entanglement. One could also analogise to a machine drive belt in the shape of a Möbius strip, as is found in many cars; the belt returns to its original configuration only after two full revolutions. With this in mind, we can also gain some idea of *n*=3/2, and other half-integer *n* values.

What, though, does this have to do with our previous parts on indistinguishable particles and particle spin? The answer is that the above provides an alternate picture of spin^{1}. We identify the rotation symmetry number *n* with the spin number *s*. Again, we must note that a point particle has no “parts” to move under a rotation, so it does not exactly fit, but the behavior of the wavefunction under rotation is similar. In particular, a particle of integer spin has a wavefunction unchanged by a full rotation, while those of half-integer spin require two full rotations to be unchanged, and have a wavefunction that changes sign under a single full rotation.

Proving that this symmetry picture of spin matches to our intrinsic angular momentum picture requires both quantum field theory and special relativity. However, doing so allows one to prove the spin-statistics theorem. Recall in our discussion of indistinguisable particles the exchange operator, under which the joint wavefunction must be an eigenstate, with eigenvalue 1 or -1, for two indistinguishable particles. The key is that this exchange operator can be constructed from (180°) rotations; the result of the swapping of two particles is determined by the way one behaves under a full rotation (see here). This gives us the spin-statistics theorem: bosons, the particles which are symmetric under exchange, are those with integer spin; while fermions, which are antisymmetric under exchange, are those with half-integer spin.

1. The above description of spin-as-rotation-symmetry is taken from my vague memories of a similar description in a general-audience quantum mechanics book I read as a child; I do not recall which work.

Tags: Boson, Fermion, Friday Physics, Indistinguishability, physics, Rotation, Spin, Spin-Statistics Theorem, symmetry

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