Quantum Mechanics for Mathematicians

is the Hamiltonian for the classical fermionic oscillator. Quantizingh(see equa-
tion 31.4) will give (−i) times the Hamiltonian operator

−iH=−

i

2

∑d

j=1

(aF†jaFj−aFjaF†j) =−i

∑d

j=1

(

aF†jaFj−

1

2

)

and a Lie algebra representation ofu(1) with half-integral eigenvalues (±i^12 ).
Exponentiation will give a representation of a double cover ofU(1)⊂U(d).
Quantizinghinstead using normal ordering gives

:−iH: =−i

∑d

j=1

aF†jaFj

and a true representation ofU(1)⊂U(d), with

UA′ =iφ

∑d

j=1

aF†jaFj

satisfying
[UA′,a†F] =iφa†F, [UA′,aF] =−iφaF

Exponentiating, the action on annihilation and creation operators is

e−iφ

∑d

j=1aF

†

jaFja†Feiφ

∑d

j=1aF

†

jaFj=eiφa†F

e−iφ

∑d

j=1aF†jaFjaFeiφ

∑d

j=1aF†jaFj=e−iφaF

31.5 An example: spinors forSO(4)

We saw in chapter 6 that the spin groupSpin(4) was isomorphic toSp(1)×
Sp(1) =SU(2)×SU(2). Its action onR^4 was then given by identifyingR^4 =H
and acting by unit quaternions on the left and the right (thus the two copies of
Sp(1)). While this constructs the representation ofSpin(4) onR^4 , it does not
provide the spin representation ofSpin(4).
A conventional way of defining the spin representation is to choose an explicit
matrix representation of the Clifford algebra (in this case Cliff(4, 0 ,R)), for
instance

γ 0 =

(

0 1

1 0

)

,γ 1 =−i

(

0 σ 1

−σ 1 0

)

,γ 2 =−i

(

0 σ 2

−σ 2 0

)

,γ 3 =−i

(

0 σ 3

−σ 3 0

)

where we have written the matrices in 2 by 2 block form, and are indexing the
four dimensions from 0 to 3. One can easily check that these satisfy the Clifford
algebra relations: they anticommute with each other and

γ 02 =γ^21 =γ 22 =γ^23 = 1