Quantum Mechanics for Mathematicians

The condition that the first row has length one gives

αα+ββ=|α|^2 +|β|^2 = 1

Using these two relations and computing the determinant (which has to be 1)
gives

αδ−βγ=−

ααγ

β

−βγ=−

γ

β

(αα+ββ) =−

γ

β

= 1

so one must have
γ=−β, δ=α

and anSU(2) matrix will have the form
(
α β
−β α

)

where (α,β)∈C^2 and
|α|^2 +|β|^2 = 1
The elements ofSU(2) are thus parametrized by two complex numbers, with
the sum of their length-squareds equal to one. IdentifyingC^2 =R^4 , these are
vectors of length one inR^4. Just asU(1) could be identified as a space with the
unit circleS^1 inC=R^2 ,SU(2) can be identified with the unit three-sphereS^3
inR^4.

3.2 Commutation relations for Pauli matrices

An important set of relations satisfied by Pauli matrices are their commutation
relations:

[σj,σk]≡σjσk−σkσj= 2i

∑^3

l=1

jklσl (3.4)

wherejklsatisfies 123 = 1, is antisymmetric under permutation of two of its
subscripts, and vanishes if two of the subscripts take the same value. More
explicitly, this says:

[σ 1 ,σ 2 ] = 2iσ 3 ,[σ 2 ,σ 3 ] = 2iσ 1 ,[σ 3 ,σ 1 ] = 2iσ 2

These relations can easily be checked by explicitly computing with the matrices.
Putting together equations 3.2 and 3.4 gives a formula for the product of two
Pauli matrices:

σjσk=δjk 1 +i

∑^3

l=1

jklσl

While physicists prefer to work with the self-adjoint Pauli matrices and their
real eigenvalues, the skew-adjoint matrices

Xj=−i

σj

2