Mathematical Principles of Theoretical Physics

158 CHAPTER 3. MATHEMATICAL FOUNDATIONS

andψhas a continuous inverseψ−^1 :Rn→U.
The groupSU(N)consists of allN-th order unitary matrices with unit determinant:

SU(N) ={A|Ais an N-th order matrix,A†A=I,detA= 1 }.

Each matrixA⊂SU(N)can be written as

A=







a 11 ··· a 1 N

..

.

..

.

aN 1 ··· aNN





,

andakl=xkl+iykl∈C( 1 ≤k,l≤N). Thus the matrixAcan be regarded as a pointpAin
R^2 N
2
:

(3.5.10) pA= (x 11 ,y 11 ,···,x 1 N,y 1 N,···,xN 1 ,yN 1 ,···,xNN,yNN)∈R^2 N

2

.

Therefore, we have thatSU(N)can be regarded as a subspace ofR^2 N
2
.
ByA†A=Iand detA=1, the entriesakl( 1 ≤k,l≤N)ofAsatisfy

(3.5.11)

aija∗jk=δik for 1≤i,k≤N,

det(aij) = 1 ,

which areN^2 +1 equations, as constraints for the pointpAin (3.5.10). HenceSU(N)can be

regarded as subspace ofR^2 N

2
has dimensionN^2 −1.
MathematicallySU(N)is a manifold. In fact, at any pointpAof (3.5.10), each equation
of (3.5.11) represents a hypersurface nearpAinR^2 N
2
:

Σik:aija∗jk=δik, for 1≤i,k≤N,

Σ 1 : det(aij) = 1 ,

and theN^2 +1 hypersurfaceΣikandΣ 1 transversally interact inR^2 N
2
to constitute anN^2 − 1
dimensional surfaceΓ(pA)near eachpA∈R^2 N
2
, and the sum of allΣ(pA)is theSU(N)space:

SU(N) =

⋃

pA

Γ(pA).

Hence,SU(N)possesses the manifold structure.

2.Tangent space TASU(N).SinceSU(N)is anN^2 −1 dimensional manifold, at each point
A∈SU(N)there is a tangent space, denoted byTASU(N), which is anN^2 −1 dimensional
linear space, as shown in Figure3.2.
Now we derive some properties of tangent vectorsτonTASU(N). To this end, letγ(t)⊂
SU(N)be a curve passing through the pointA∈SU(N)withτA∈TASU(N)as its tangent
vector atA, as shown in Figure3.2. Letγ( 0 ) =A. Then the curveγ(t)satisfies the following
equation

(3.5.12)

dγ(t)

dt

=τt τt∈TSU(N)withτA=τ|t= 0 ,

γ( 0 ) =A.