This is not a convenient form because it is not given in terms of number operators.
We can express i)spin in terms of number operators by introducing a new polarization basis referred to as the circular, or helicity, basis. If e1 —> x, e2 —> y, and k —> z, then the circular polarization basis, in which the states have a definite spin projection along z, is defined by e+ = spin in -(-¿-direction = —(e1 + i e2)
f (2.61) e = spin in -¿-direction = —= (c1 - i e2) .
Note the appearance of the minus sign in the definition of e+; this is a standard phase convention used in the construction of the spherical harmonics ytm for
£ = 1 and m — ± 1 from x and y. Then we define a±, the annihilation operators corresponding to these circularly polarized states, by the relation (suppress n for now)
Hence and, restoring n, a\ai — a\a2 = —i[a+a+ — aLa_]
The spin operator has now been expressed in terms of number operators for photons with a definite helicity. Note that it is a vector sum of terms which point in the +fc-direction for positive helicity and in the -¿-direction for negative helicity.
In general, the helicity of a particle is the projection of its spin along the direction of its motion, and if a massive particle has spin s, its helicity can take on any integer value between s and -s (i.e., s, s - 1, s - 2,..., -s). The direction of motion is simply one special direction in space, and a massive particle of spin s has 2s + 1 states which can always be expanded in terms of states having a definite spin projection along any chosen axis. However, Eq. (2.62) shows that photons do not have this property. It shows that the photon has spin 1, but that out of three possible states (±1 or 0), only helicity states +1 and -1 can occur. The absence of helicity zero is due to the transverse nature of the field (Coulomb gauge) which is due in turn to the absence of a photon rest mass.
The restriction of the photon helicity to its maximum and minimum possible values, ±1, illustrates a property of any massless particle. In general, if a massless particle has spin s, it may have only two helicity states: ±s. The other possible states are prohibited. This remarkable result is one of the consequences of Wigner's famous analysis of the representations of the Poincare group (which is the group which results from combining the Lorentz transformations with spacetime translations). It turns out that the representations of the Poincare group are characterized by both mass and spin and that the familiar 2s + 1 degeneracy associated with the spin s representations of the SU(2) rotation group occur only when the mass M of the particles described by the representation is non-zero. If
M = 0, the spin representations are only two dimensional, explaining why there are only two states with spin projections ±s . For more information see Ryder (1985) or Wigner's original paper [Wi 39],
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