Spin, statistics, and the spinor ambiguity
The spin-statistics connection is derived in a simple manner under the postulates that the original and the exchange wave functions are just added, and that the azimuthal phase angle, which defines the orientation of the spin part of each single-particle spin-component eigenfunction in the plane normal to the spin-quantization axis, is exchanged along with the other parameters. The spin factor (−1) 2s belongs to the exchange wave function when this function is constructed so as to get the spinor ambiguity under control. This is achieved by effecting the exchange of the azimuthal angle by means of rotations, and admitting only rotations in one sense. This works in Galilean as well as in Lorentz-invariant quantum mechanics. Relativistic quantum field theory is not required. 1 Introduction The symmetrization postulate of standard quantum mechanics, I recall, postulates that any wave function or state vector of a system of identical particles must be either symmetric or antisymmetric, that is, multiplied by either +1 or −1 when the labels or parameters referring to any two particles are interchanged. There are thus two classes of systems, with different collective behaviour of the particles: systems of bosons and systems of fermions. These two classes are connected with the spins of the particles: all particles which are known to be bosons are found to have integral spin, in units of ¯ h, while all known fermions have half-integral (i.e., half-odd-integral) spin. Within quantum mechanics the connection with spin could not be proved and had to be taken as another postulate. The first proof was provided by Pauli [1], who founded it on relativistic quantum field theory. This also remained the framework for the papers which in subsequent years refined and generalized Pauli's proof [2,3]. Typically, in these papers it is postulated that no negative-energy states exist , that the metric in Hilbert space is positive definite, and that the fields either