Quantum Mechanics for Mathematicians

One can easily check that these satisfy the Clifford algebra relations for gener-
ators of Cliff(3,1): they anticommute with each other and

γ 02 =− 1 , γ^21 =γ 22 =γ^23 = 1

The quadratic Clifford algebra elements−^12 γjγkforj < ksatisfy the com-
mutation relations ofso(3,1). These are explicitly

−

1

2

γ 1 γ 2 =−

i

2

(

σ 3 0

0 σ 3

)

,−

1

2

γ 1 γ 3 =−

i

2

(

σ 2 0

0 σ 2

)

,−

1

2

γ 2 γ 3 =−

i

2

(

σ 1 0

0 σ 1

)

and

−

1

2

γ 0 γ 1 =

1

2

(

−σ 1 0

0 σ 1

)

,−

1

2

γ 0 γ 2 =

1

2

(

−σ 2 0

0 σ 2

)

,−

1

2

γ 0 γ 3 =

1

2

(

−σ 3 0

0 σ 3

)

They provide a representation (π′,C^4 ) of the Lie algebraso(3,1) with

π′(l 1 ) =−

1

2

γ 2 γ 3 , π′(l 2 ) =−

1

2

γ 1 γ 3 , π′(l 3 ) =−

1

2

γ 1 γ 2

and

π′(k 1 ) =−

1

2

γ 0 γ 1 , π′(k 2 ) =−

1

2

γ 0 γ 2 , π′(k 3 ) =−

1

2

γ 0 γ 3

Note that theπ′(lj) are skew-adjoint, since this representation of theso(3)⊂
so(3,1) sub-algebra is unitary. Theπ′(kj) are self-adjoint and this representa-
tionπ′ofso(3,1) is not unitary.
On the two commutingsl(2,C) subalgebras ofso(3,1)⊗Cwith bases (see
section 40.2)

Aj=

1

2

(lj+ikj), Bj=

1

2

(lj−ikj)

this representation is

π′(A 1 ) =−

i

2

(

σ 1 0

0 0

)

, π′(A 2 ) =−

i

2

(

σ 2 0

0 0

)

, π′(A 3 ) =−

i

2

(

σ 3 0

0 0

)

and

π′(B 1 ) =−

i

2

(

0 0

0 σ 1

)

, π′(B 2 ) =−

i

2

(

0 0

0 σ 2

)

, π′(B 3 ) =−

i

2

(

0 0

0 σ 3

)

We see explicitly that the action of the quadratic elements of the Clifford
algebra on the spinor representationC^4 is reducible, decomposing as the direct
sumSL⊕SR∗of two inequivalent representations onC^2

Ψ =

(

ψL

ψ∗R

)

with complex conjugation (interchange ofAjandBj) relating thesl(2,C) ac-
tions on the components. TheAj act just onSL, theBj just onS∗R. An