Quantum Mechanics for Mathematicians

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.