QuantumPhysics.dvi

of small incremental rotations starting from the identity. But therelationMtM=Iimplies

that detM=±1. These two cases are not continuously connected to one another, and only

the set detM = +1 contains the identity matrixM =I. Hence, the matrices for which

detM=−1 are not rotations. One example of such a matrix is

(M 0 )ij=δij− 2 δiNδjN (9.31)

Its effect onQis to reverse the sign ofqN, leaving all otherqiunchanged. This transformation

is aspace parity transformationin the directionN, and is clearly not a rotation. Any matrix

M with detM =−1 may be written asM = M 0 M′ where now detM′ = 1. Thus, any

orthogonal transformation is either a rotation inN dimensions or the product of a rotation

by a parity transformation. The group of rotations consists of matricesMsuch that we have

bothMtM =I and detM = +1, and is denoted bySO(N), the prefixSstanding for the

condition of unit determinant.

It is also very useful to examine the infinitesimal rotations inN-dim space. To do so, we

expandMaround the identityIto linear order,

M=I+̟+O(̟^2 ) (9.32)

and insist on the relationMtM=Ito this order. This requires that the matrixBbe anti-

symmetric, i.e.̟t=−̟. A real anti-symmetricN×Nmatrix hasN(N−1)/2 independent

entries, and this yields the dimension of the orthogonal groups,

dimSO(N) = dimO(N) =

1

2

N(N−1) (9.33)

Finally, infinitesimal rotations are generated by the angular momentum operators,

Lij=qipj−qjpi i,j= 1,···,N (9.34)

Since we have

[Lij,qk] = i ̄h(δikqj−δjkqi) (9.35)

the rotations are generated by the unitary operator

U= exp

 

−

i

2 ̄h

∑N

i,j=1

̟ijLij

 

 (9.36)

It is then easy to verify that

UqiU†=qi+