The relationship can be deduced from the formulae in QM (57.9), which connect the components of a vector wave function with those of the equivalent spinor of rank two.
The various cases of the polarization of the photon are in a certain relationship to the possible values of its helicity. It is described by a “spin” wave function which is a vector ein the ξη-plane the two components of this vector are transformed into combinations of each other by any rotation about the ζ-axis and by any reflection in a plane passing through that axis. ‡The state of a photon having a definite momentum in fact corresponds to one type of these doubly degenerate states. †If we also impose the condition of symmetry under reflections in planes passing through the ζ-axis, the states differing in the sign of λ will be mutually degenerate, and when λ ≠ 0 there is therefore twofold degeneracy. the component of its angular momentum along the ζ-axis, which we denote by λ. When there is axial symmetry, only the helicityof the particle is conserved, i.e. In such a case there is clearly no symmetry with respect to the whole group of rotations in three dimensions, but only axial symmetry about the preferred axis. For such a particle, there is always a distinctive direction in space, the direction of the momentum vector k (the ζ-axis). If the mass of the particle is zero, however, there is no rest frame, since it moves with the velocity of light in every frame of reference. In particular, a particle having a vector (three-component) wave function has spin 1. The property which describes the symmetry of the particle with respect to this group is its spin s this determines the degree of degeneracy, the number of different wave functions which are transformed into linear combinations of one another being 2 s+ 1. with respect to the entire spherical symmetry group) must be considered. Symmetry with respect to all possible rotations about the centre (i.e.
The intrinsic symmetry properties of the particle, as such, will evidently appear in this particular frame of reference. This property is closely related to the fact that the mass of the photon is zero.Ī freely moving particle with non-zero mass always has a rest frame.
The possibility that the photon has two different polarizations (for a given momentum) is equivalent to the statement that each eigenvalue of the momentum is doubly degenerate.
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