Quantum Mechanics for Mathematicians

This algebra is a four dimensional algebra overC, with basis

1 , aF, a†F, a†FaF

since higher powers of the operators vanish, and the anticommutation relation
betweenaFanda†Fcan be used to normal order and put factors ofaF on the
right. We saw in chapter 27 that this algebra is isomorphic with the algebra
M(2,C) of 2 by 2 complex matrices, using

1 ↔

(

1 0

0 1

)

, aF↔

(

0 0

1 0

)

, a†F↔

(

0 1

0 0

)

, a†FaF↔

(

1 0

0 0

)

(28.1)

We will see in chapter 30 that there is also a way of identifying this algebra
with “differential operators in fermionic variables”, analogous to what happens
in the bosonic (Weyl algebra) case.
Recall that the bosonic annihilation and creation operators were originally
defined in terms of thePandQoperators by

aB=

1

√

2

(Q+iP), a†B=

1

√

2

(Q−iP)

Looking for the fermionic analogs of the operatorsQandP, we use a slightly
different normalization, and set

aF=

1

2

(γ 1 +iγ 2 ), a†F=

1

2

(γ 1 −iγ 2 )

so

γ 1 =aF+a†F, γ 2 =

1

i

(aF−a†F)

and the CAR imply that the operatorsγjsatisfy the anticommutation relations

[γ 1 ,γ 1 ]+= [aF+a†F,aF+a†F]+= 2

[γ 2 ,γ 2 ]+=−[aF−a†F,aF−a†F]+= 2

[γ 1 ,γ 2 ]+=

1

i

[aF+a†F,aF−a†F]+= 0

From this we see that

• One could alternatively have defined Cliff(2,C) as the algebra generated
  by 1,γ 1 ,γ 2 , subject to the relations

[γj,γk]+= 2δjk

• Using just the generators 1 andγ 1 , one gets an algebra Cliff(1,C), gener-
  ated by 1,γ 1 , with the relation

γ 12 = 1

This is a two dimensional complex algebra, isomorphic toC⊕C.