Quantum Mechanics for Mathematicians

for allv,w∈V. Such transformations take orthonormal bases to orthonormal
bases, so one role in which they appear is as a change of basis between two
orthonormal bases.
In terms of adjoints, this condition becomes

〈Lv,Lw〉=〈v,L†Lw〉=〈v,w〉

so
L†L= 1

or equivalently
L†=L−^1
In matrix notation this first condition becomes
∑n

k=1

(L†)jkLkl=

∑n

k=1

LkjLkl=δjl

which says that the column vectors of the matrix forLare orthonormal vectors.
Using instead the equivalent condition

LL†= 1

we find that the row vectors of the matrix forLare also orthonormal. Since
such linear transformations preserving the inner product can be composed and
are invertible, they form a group, and some of the basic examples of Lie groups
are given by these groups for the cases of real and complex vector spaces.

4.6.1 Orthogonal groups

We’ll begin with the real case, where these groups are called orthogonal groups:

Definition(Orthogonal group).The orthogonal groupO(n)inndimensions
is the group of invertible transformations preserving an inner product on a real
ndimensional vector spaceV. This is isomorphic to the group ofnbynreal
invertible matricesLsatisfying

L−^1 =LT

The subgroup ofO(n)of matrices with determinant 1 (equivalently, the subgroup
preserving orientation of orthonormal bases) is calledSO(n).

Recall that for a representationπof a groupGonV, there is a dual repre-
sentation onV∗given by taking the transpose-inverse ofπ. IfGis an orthogonal
group, thenπand its dual are the same matrices, withV identified byV∗by
the inner product.
Since the determinant of the transpose of a matrix is the same as the deter-
minant of the matrix, we have

L−^1 L= 1 =⇒det(L−^1 ) det(L) = det(LT) det(L) = (det(L))^2 = 1