This makeshthe moment map for a simultaneous rotation in the 2j− 1 , 2 j
planes, corresponding to a matrix inso(2d) given by
L=
∑d
j=1
2 j− 1 , 2 j
As in the bosonic case, we can make the standard choice of complex structure
J=J 0 onR^2 dand get a decomposition
V∗⊗C=R^2 d⊗C=Cd⊕Cd
into eigenspaces ofJof eigenvalue±i. This is done by defining
θj=
1
√
2
(ξ 2 j− 1 −iξ 2 j), θj=
1
√
2
(ξ 2 j− 1 +iξ 2 j)
forj= 1,...,d. These satisfy the fermionic Poisson bracket relations
{θj,θk}+={θj,θk}+= 0, {θj,θk}+=δjk
(where we have extended the inner product{·,·}+toV∗⊗Cby complex lin-
earity).
In terms of theθj, the Hamiltonian is
h=−
i
2
∑d
j=1
(θjθj−θjθj) =−i
∑d
j=1
θjθj
Using the derivation property of{·,·}+one finds
{h,θj}+=−i
∑d
k=1
(θk{θk,θj}+−{θk,θj}+θk) =−iθj
and, similarly,
{h,θj}+=iθj
so one sees thathis the generator ofU(1)⊂U(d) phase rotations on the
variablesθj. The equations of motion are
d
dt
θj={θj,h}+=iθj,
d
dt
θj={θj,h}+=−iθj
with solutions
θj(t) =eitθj(0), θj(t) =e−itθj(0)
30.4 For further reading
For more details on pseudo-classical mechanics, a very readable original refer-
ence is [7]. There is a detailed discussion in the textbook [90], chapter 7.