QuantumPhysics.dvi

Here,M is now anN×N matrix with complex matrix elements. The hamiltonian will

be invariant provided the quadratic formA†Ais invariant for allA, which requiresA†A→

(A′)†A′=A†M†MA=A†A, so that we must have

M†M=I (9.44)

The set of complex matrices satisfying this relation forms the group of unitary N×N

matrices, usually referred to asU(N). The identity of the group is simply the identity

matrix, while the inverse is given by the dagger : M−^1 = M†. Note that when M is

restricted to have real coefficients, we recover the orthogonaltransformations discussed in

the preceding subsection, and we thus find thatSO(N) is a subgroup ofU(N).

The corresponding infinitesimal transformations are obtained by linearizingM around

the identity matrix,

M=I+i̟+O(̟^2 ) ̟†=̟ (9.45)

The infinitesimal transformations are thus parametrized by generalN×NHermitian matrices

̟. The number of free (real) parameters of these Hermitian matrices isN^2 , so that

dimU(N) =N^2 (9.46)

The corresponding conserved charges are as follows,

Tij=a†iaj (9.47)

with commutation relations for the Lie algebra ofU(N),

[Tij,Tkℓ] =δjkTiℓ−δℓiTkj (9.48)

The generators Tij are actually not Hermitian, but suitable linear combinations may be

constructed which are Hermitian,

Tij+ = a†iaj+a†jai

Tij− = −i

(

a†iaj−a†jai

)

(9.49)

The N(N −1)/2 combinations Tij− satisfy the commutation relations of the subalgebra

SO(N)⊂U(N), and we have the relation ̄hTij−=Lij.

Finally, note that the groupU(N) is actually the product of two groups. Any unitary

matrixMCinU(N) which is proportional to the identityMC=εIfor|ε|= 1, commutes

with all elements inU(N). The elementsMCthemselves form a group, namelyU(1), and

we thus have

U(N) =U(1)×SU(N) SU(N)≡{M∈U(N), detM= 1} (9.50)