Quantum Mechanics for Mathematicians

Taking real linear combinations of these two generators, the algebra one

gets is just the algebraCof complex numbers, withγ 1 playing the role of

i=

√

−1.

• Cliff(0, 2 ,R). This has generators 1,γ 1 ,γ 2 and a basis

1 , γ 1 , γ 2 , γ 1 γ 2

with

γ^21 =− 1 , γ^22 =− 1 , (γ 1 γ 2 )^2 =γ 1 γ 2 γ 1 γ 2 =−γ 12 γ^22 =− 1

This four dimensional algebra over the real numbers can be identified with

the algebraHof quaternions by taking

γ 1 ↔i, γ 2 ↔j, γ 1 γ 2 ↔k

• Cliff(1, 1 ,R). This is the algebraM(2,R) of real 2 by 2 matrices, with
  one possible identification as follows

1 ↔

(

1 0

0 1

)

, γ 1 ↔

(

0 1

1 0

)

, γ 2 ↔

(

0 − 1

1 0

)

, γ 1 γ 2 ↔

(

1 0

0 − 1

)

Note that one can construct this using theaF,a†F for the complex case

Cliff(2,C) (see 28.1) as

γ 1 =aF+a†F, γ 2 =aF−a†F

since these are represented as real matrices.

• Cliff(3, 0 ,R). This is the algebraM(2,C) of complex 2 by 2 matrices,
  with one possible identification using Pauli matrices given by

1 ↔

(

1 0

0 1

)

γ 1 ↔σ 1 =

(

0 1

1 0

)

, γ 2 ↔σ 2 =

(

0 −i

i 0

)

, γ 3 ↔σ 3 =

(

1 0

0 − 1

)

γ 1 γ 2 ↔iσ 3 =

(

i 0

0 −i

)

, γ 2 γ 3 ↔iσ 1 =

(

0 i

i 0

)

, γ 1 γ 3 ↔−iσ 2 =

(

0 − 1

1 0

)

γ 1 γ 2 γ 3 ↔

(

i 0

0 i

)

It turns out that Cliff(r,s,R) is always one or two copies of matrices of real,
complex or quaternionic elements, of dimension a power of 2, but this requires
a rather intricate algebraic argument that we will not enter into here. For the
details of this and the resulting pattern of algebras one gets, see for instance
[55]. One special case where the pattern is relatively simple is when one has
r=s. Thenn= 2ris even dimensional and one finds

Cliff(r,r,R) =M(2r,R)