References - Bogoliubov Laboratory of Theoretical Physics - JINR

or
Equation (1) may then be written, using the usual formula for covariant differentiation
Sμ;ν = Sμ,ν − Γ λ μνSλ
Si;0 = Si,0 − Γ λ i0 Sλ = Si,0 − Γ k i0 Sk , (3)
where the connection coefficients Γλ μν
relevant non-zero coefficients are Γ2 10 = ω, Γ1 20 = −ω, giving
or (with ω = ωn)
(2)
are calculated from the above metric tensor. The
S1;0 = S1,0 − Γ 2 10S2 = S1,0 − ωS2,
S2;0 = S2,0 + ωS1, S3;0 = S3,0, (4)
DS
Dt
= dS
dt
+ Ω ∧ S.
as in (1) above.
From a fundamental perspective, however, spin is not described by a 4-vector, but by
a rank 2 antisymmetric tensor. Indeed, the total angular momentum Jμν contains orbital
and spin parts
Jμν = Lμν + Sμν
andasiswellknownJμν are the generators of the Lorentz group. The most profound
analysis of spin, however, was made by Wigner [2] in 1939, who pointed out that to
describe spin it is insufficient to consider the (homogeneous) Lorentz group. Instead the
group of translations in space-time must be added, yielding the inhomogeneous Lorentz
group, or Poincaré group. This has generators Pμ and Jμν , and spin is generated by little
group of the Poincaré group. The structure of this group depends on the sign of P μ Pμ.
With metric (+, −, −, −) massive particles have P μ Pμ = m 2 . In the nonrelativistic limit
spin is described by the rotation group SO(3) or SU(2), and indeed this turns out to be
true for all states with P μ Pμ > 0:
P μ Pμ > 0 (timelike states) : little group is SU(2) .
Wigner showed, however, that for other values P μ Pμ the little group has a different
structure:
P μ Pμ < 0 (spacelike states) : little group is SO(2, 1) .
P μ Pμ = 0 (null states) : little group is E(2).
Thus, for example, the spin of spacelike (virtual) particles in Feynman diagrams is not
properly described by the rotation group, but by the noncompact group SO(2,1).
Wigner’s analysis identified the group structure of the little groups but did not reveal
the actual forms of the generators of these groups, leaving the question, what are
these generators? The Casimir operators of the Poincaré groupareP μ Pμ and W μ Wμ ,
describing, in effect, the mass and spin of quantum states. Here Wμ is the Pauli-Lubanski
vector
Wμ = 1
2 ɛμνκλJ νκ P λ .
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