References - Bogoliubov Laboratory of Theoretical Physics - JINR

It is easy to see that for a (massive) particle at rest (P0,P1,P2,P3) =(M,0, 0, 0),
W0 =0,andWigenerate the group SU(2), as expected. Similarly for a spacelike state
we may have (P0,P1,P2,P3) = (0,P,0, 0), and then W0,W1,W2 are seen to generate
SO(2, 1), again as expected. These operators are, however, non-covariant. To obtain
covariant operators first define [3]
Wμν = 1
[Wμ,Wν],
M 2
which are seen to obey the commutation relation
Now define also the dual
[Wμν,Wκλ] = 1
M 2 (ɛμνκρWλ − ɛμνλρWκ)P ρ .
(W D ) κλ = 1
2 ɛκλμν Wμν.
Then the linear combinations
are found to obey
X μν = −i{W μν + i(W D ) μν }, Y μν = −i{W μν − i(W D ) μν } = i(X D ) μν
[Xμν,Xκλ] =−i(gμκXνλ − gμλXνκ + gνλXμκ − gνκXμλ),
and similarly for Y . These are the same as the commutation relations for Jμν . Xμν
and Yμν are therefore covariant spin operators. For Dirac particles they act as the spin
operators for left- and right-handed states. The covariant spin operator may therefore be
written as
Zμν = PLXμν + PRYμν. (5)
Then, putting Xi = 1
2 ɛijkX jk and Yi = 1
2 ɛijkY jk , the components of this rank 2 tensor
which transform as a 3-vector may be written in the usual way as
Zi = 1
2 (1 − γ5)Xi + 1
(1 + γ5)Yi.
2
It then turns out, perhaps as an unexpected bonus, that Zi is the Foldy-Wouthuysen
mean spin operator [3, 4].
Having now identified a covariant relativistic spin operator with the required property
that it is a rank 2 tensor, we may immediately write its covariant derivative as
Zμν:λ = Zμν,λ − Γ ρ
μλ Zρν − Γ ρ
νλ Zμρ .
With the connection coefficients derived from the metric tensor above, these give
Z12;0 = Z12,0; Z23;0 = Z23,0 + ωZ13; Z31;0 = Z31,0 − ωZ32
or
DZ3 dZ3 DZ1 dZ1 DZ2 dZ2
= ; = − ωZ2; = + ωZ1,
dt dt dt dt dt dt
as in equation (4) above. We see that the same precession formula is obtained when spin
is represented as an antisymmetric rank 2 tensor, as when it is represented by a 4-vector.
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