so the self-adjointness of theσjimplies unitarity ofeiθv·σsince
(eiθv·σ)†= ((cosθ) 1 +i(sinθ)v·σ)†
= ((cosθ) 1 −i(sinθ)v·σ†)
= ((cosθ) 1 −i(sinθ)v·σ)
= (eiθv·σ)−^1The determinant ofeiθv·σcan also easily be computed
det(eiθv·σ) = det((cosθ) 1 +i(sinθ)v·σ)= det(
cosθ+i(sinθ)v 3 i(sinθ)(v 1 −iv 2 )
i(sinθ)(v 1 +iv 2 ) cosθ−i(sinθ)v 3)
= cos^2 θ+ (sin^2 θ)(v^21 +v^22 +v 32 )
= 1So, we see that by exponentiatingitimes linear combinations of the self-
adjoint Pauli matrices (which all have trace zero), we get unitary matrices of
determinant one. These are invertible, and form the group namedSU(2), the
group of unitary 2 by 2 matrices of determinant one. If we exponentiated not
justiθv·σ, buti(φ 1 +θv·σ) for some real constantφ(such matrices will not
have trace zero unlessφ= 0), we would get a unitary matrix with determinant
ei^2 φ. The group of all unitary 2 by 2 matrices is calledU(2). It contains as
subgroupsSU(2) as well as theU(1) described at the beginning of this section.
U(2) is slightly different from the product of these two subgroups, since the
group element (
− 1 0
0 − 1
)
is in both subgroups. In chapter 4 we will encounter the generalization toSU(n)
andU(n), groups of unitarynbyncomplex matrices.
To get some more insight into the structure of the groupSU(2), consider an
arbitrary 2 by 2 complex matrix
(
α β
γ δ)
Unitarity implies that the rows are orthonormal. This results from the condition
that the matrix times its conjugate-transpose is the identity
(
α β
γ δ)(
α γ
β δ)
=
(
1 0
0 1
)
Orthogonality of the two rows gives the relation
γα+δβ= 0 =⇒ δ=−γα
β