Dong S.-H. Wave Equations in Higher Dimensions

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Appendix C

Fundamental Properties of Lie Groups and Lie

Algebras

In this Appendix we are ready to outline some fundamental properties of Lie groups

and Lie algebras since they are very helpful for studying quantum systems with

central physical potentials.

Up to now, we have dealt exclusively with ﬁnite groups. However, the symmetry

groups for many physical systems consist of an inﬁnite rather than a ﬁnite num-

ber of elements. Thus physical problems require that we examine the theory of the

representations of groups with an inﬁnite number of elements.

The inﬁnite groups include two main types such as the inﬁnite discrete and con-

tinuous groups [138]. The elements of inﬁnite discrete group, R

, are labeled by a

subscript a which runs through the integers 1, 2,...,∞. The group manifold is a

countable set of “point” R

. As what follows, we shall pay more attention to con-

tinuous groups since the Lie groups are closely related to them.

1 Continuous Groups

We say the group is continuous if some generalized deﬁnition of “nearness” or con-

tinuity is imposed on the elements of the group manifold. For instance, the set of

transformations x



=ax +b forms a group, where two parameters a,b ∈(−∞, ∞).

We say that such a group is a two-parameter continuous group. Generally, an r-

parameter continuous group has its elements labeled by r continuously varying real

parameters a

,i ∈[1,r] so that the elements of the group are R(a

,...,a

) =R(a).

The continuity is expressed in terms of distances in the parameter space. Two group

elements R(a) and R(a



) are “near” to each other if the distance [



i=1

−a



)

]

1/2

is very small.

The requirements that the elements R(a) form a continuous group are the same

as for ﬁnite groups. There must be a set of parameter values a

such that

R(a

)R(a) =R(a)R(a

) = R(a),

R(¯a)R(a) =R(a)R(¯a) =R(0),

R(¯a) =[R(a)]

−1

,R(c)=R(b)R(a),

(C.1)

S.-H. Dong, Wave Equations in Higher Dimensions,

DOI 10.1007/978-94-007-1917-0, © Springer Science+Business Media B.V. 2011

255

256 C Fundamental Properties of Lie Groups and Lie Algebras

where R(a

) is the identity element of the group, and R(¯a) is the element inverse to

R(a).Thec are real functions of the real parameters a and b

=φ

,...,a

,...,b

), j ∈[1,r]. (C.2)

Even though the requirements are the same as for ﬁnite groups, we require that

the parameters of a product be analytic functions of the parameters of the factors;

i.e., the function in (C.1) shall possess derivatives of all orders with respect to both

variables. Also, we require that the ¯a in R(¯a) be analytic functions of the a. We then

get an r-parameter Lie group.

When we say that we have an r-parameter group, this implies that the r pa-

rameters are essential. An r-parameter Lie group of transformations is a group of

transformations



,...,x

,...,a

), i ∈[1,n], (C.3)

or symbolically



=g(x;a). (C.4)

If the parameters are not essential, then there exist parameter values a

+ ε

...,a

+ε

, where the ε’s are arbitrarily small quantities which are functions of the

with

(x;a) =g

(x;a +ε) (C.5)

for all values of the argument x. The necessary and sufﬁcient condition for the r

parameters a

to be essential is that it shall be impossible to ﬁnd r functions χ

(a)

which satisfy



k=1

(a)

∂g

(x;a)

∂a

=0, for all x and a, i ∈[1,n]. (C.6)

Starting with (C.4), we can differentiate the x



’s with respect to the x’s and obtain

a set of equations from which the ﬁnite set of parameters a can be eliminated. We

shall then be left with a ﬁnite set of partial differential equations for the x



’s which

no longer contain any arbitrary elements. Moreover, the general solution of this set

of partial differential equations will rely on just r arbitrary constants. We say that

the group is ﬁnite and continuous. Otherwise, we get an inﬁnite continuous group.

Here we give several typical examples of Lie groups. The ﬁrst is the one-

parameter Abelian group x



=ax,a = 0. Its identity element is a = 1, and inverse

element ¯a =1/a. Its product element is c =ba. The second is the orthogonal group

in two dimensions O(2). This group is concerned with those transformations which

leave x

invariant. This invariance condition

2

=(a

x +a

+(a

x +a

=1,a

(C.7)

imposes three conditions on the four parameters. Thus we have a one-parameter

group written as









cosϕ −sinϕ

sin ϕ cos ϕ





, (C.8)

2 Inﬁnitesimal Transformations and Lie Algebra 257

where ϕ ∈[0, 2π] is the angle of rotation about the z-axis. This group is Abelian.

The angle of the resultant of two transformations is the sum of the angles of the in-

dividual transformations. Finally, we consider the orthogonal group in n dimensions

O(n). This group leaves



i=1

invariant. Thus, we impose [n +n(n −1)/2] con-

ditions on the n

parameters, which leave us with [n(n −1)/2]essential parameters.

So far we have considered only real transformations of real variables. As what

follows, let us consider a typical example with complex variables. It is the unitary

group in two dimensions U(2)













, (C.9)

where x,a are complex and detA = 0. This group is in fact a unitary group with

†

=1. Similarly, the unitary group in n dimensions U(n), i.e., r



=Ar, AA

†

=1,

imposes [n +2n(n −1)/2] conditions on the 2n

real parameters, leaving us with

essential real parameters.

2 Inﬁnitesimal Transformations and Lie Algebra

The transformation x



= f(x;a) takes all points of the space from their initial po-

sitions x to ﬁnal positions x



. It is natural to introduce the concept of inﬁnitesimal

transformations when we consider the gradual shift of the points of the space. It

can also be proved that a one-parameter continuous group is equivalent to a group

of translations and must be Abelian. Due to the inﬁnitesimal transformations, we

introduce the inﬁnitesimal operators of the group as follows



i=1

(x)

∂

∂x

. (C.10)

The operator 1 +



δa

is close to the identity operator. On the other hand, the

inﬁnitesimal operators X

have the property that their commutators satisfy

]=w

iα

∂

∂x

jβ

∂

∂x

−w

jβ

∂

∂x

iα

∂

∂x

αβ

, (C.11)

where c

αβ

are called the structure constants of the Lie group with the property c

αβ

−c

βα

Substituting (C.11) into the Jacobi identity

[[X

],X

]+[[X

],X

]+[[X

],X

]=0, (C.12)

we ﬁnd

αβ

μγ

βγ

μα

γα

μβ

=0. (C.13)

Basedonthis,wehave





1 +



δa





(x)δa

. (C.14)

258 C Fundamental Properties of Lie Groups and Lie Algebras

Consider an example in order to show how to deﬁne the inﬁnitesimal operators.

For example, the orthogonal transformations are characterized by the statement that

the transpose A

of the matrix A of the transformation is its inverse AA

=1.The

proper rotations have determinant unity so that the inﬁnitesimal rotations have a

matrix of the form A =1 +B where B has all its elements in the neighborhood of

zero. The orthogonality condition requires 1 =AA

=(1 +B)(1 +B

) ∼1 +B +

, which leads to B +B

= 0. Therefore B has a skew-symmetric matrix with

the form

B =

⎡

⎣

0 ζ −η

−ζ 0 ξ

η −ξ 0

⎤

⎦

, (C.15)

where ξ,η and ζ are constants.

The inﬁnitesimal rotations are given by

dx =ζy −ηz, dy =−ζx +ξz, dz =ηx −ξy, (C.16)

and the inﬁnitesimal operators are

∂

∂y

−y

∂

∂z

∂

∂z

−z

∂

∂x

∂

∂x

−x

∂

∂y

(C.17)

which correspond to the angular momentum operators for three coordinate direc-

tions and satisfy familiar commutation relations

]=X

, [X

]=X

, [X

]=X

. (C.18)

This means that the commutators of the inﬁnitesimal operators are linearly express-

ible in terms of the inﬁnitesimal operators. If we set iX

, then we have

]=iJ

, [J

]=iJ

, [J

]=iJ

. (C.19)

We have known that for an r-parameter transformation group there are r linearly

independent inﬁnitesimal operators X

. Linear combinations of these quantities can

be formed to give an r-dimensional vector space. If we are considering problems

concerned with the structure of the Lie group, we should take only linear com-

binations with real coefﬁcients. The r-parameter Lie group has associated with it

a real r-dimensional vector space of quantities



which is closed under a

multiplication deﬁned by (C.11). This is the Lie algebra of the Lie group. To every

Lie algebra there corresponds to a Lie group; the structure constants determine the

Lie group locally. To deduce all possible structures of Lie algebra is a formidable

mathematical problem. We do not address it here and suggest the reader refer to

Ref. [138].

3 Structure of Compact Semisimple Lie Groups and Their Algebras 259

3 Structure of Compact Semisimple Lie Groups and Their

Algebras

When we use the term “closed” to describe Lie groups whose parameters vary over

a ﬁnite range, the group manifold itself is then said to be compact. Generally speak-

ing, a set M is compact if every inﬁnite subset of M contains a sequence which

converges to an element of M. The Lie algebra of a compact Lie group is also said

to be compact.

Cartan theorem The necessary and sufﬁcient condition for a Lie algebra to be

semisimple is that det g

μν

=0, where g

μν

μα

νβ

A necessary and sufﬁcient condition for a semisimple Lie algebra to be compact

is that the matrix g

μν

be negative deﬁnite. Finally, note that for a compact semisim-

ple Lie algebra we can choose the basis so that g

μν

=−δ

μν

. The actual analysis

of the structure of compact semisimple Lie algebras is beyond the scope of this

Appendix.

4 Irreducible Representations of Lie Groups and Lie Algebras

For simplicity, we restrict ourselves to compact Lie groups in order to prove that

every representation of a compact is equivalent to a unitary representation and can

be fully reducible to a sum of irreducible representations, all of which have ﬁnite

dimensions. The regular representation contains all irreducible representations.

Theorem For semisimple Lie groups, every representation of ﬁnite degree is fully

reducible.

Casimir theorem The operator C = g

commutes with all operators of the

representation and is a multiple of the unit operator.

For compact groups g

=−δ

in a suitable basis, so that the Casimir operator

becomes

C =



. (C.20)

For rotation group O(3) with the operators J

where μ = 1, 2, 3, the Casimir

operator is calculated as

C =



,i=1, 2, 3. (C.21)

The number of operators which are required to give a complete set is equal to the

rank of the studied algebra. This can be deﬁned as follows: for any element A,we

look for all independent solutions of the equation [A,X]=0, which always has at

260 C Fundamental Properties of Lie Groups and Lie Algebras

least the one solution X = A. We now vary A to reduce the number of independent

solutions of this equation to a minimum number l called the rank of the algebra. The

l operators of the Casimir type are required to characterize an irreducible represen-

tation.

Before ending this Appendix, let us review a few useful commutator identities

to simplify the commutation relations. We often use the following well-known rela-

tions

[AB, C]=A[B,C]+[A,C]B, [A, BC]=B[A, C]+[A,B]C (C.22)

to simplify the commutators involving the products of the operators. On the other

hand, it is shown that for arbitrary polynomial functions r, t of the indicated opera-

tors,

r(A)[t(A),B]=[t (A), r(A)B], [A,t(B)]r(B) =[Ar(B), t (B)] (C.23)

can be effectively used to move the operators in and out of the commutators. In

particular, we ﬁnd that the following formula is very useful

[A,B

m−1



i=0

[A,B]B

m−i−1

, (C.24)

which can be proved by induction on m.

In addition, we have to calculate the operator transformations of the form

−B

in scaling transformations. For this purpose, let us deﬁne

g(α) =e

−αB

αB

. (C.25)

By differentiating, we have

g(α)



−αB

[A,B]e

αB

,g(α)



−αB

[[A,B],B]e

αB

(C.26)

and in general

(α) =e

−αB

[...[[A,B],B],...,B]e

αB

, (C.27)

where the multiple commutator contains B exactly m times. Let us expand the g(α)

in Taylor series

g(α) =

∞



m=0

(0)α

. (C.28)

If setting α =1, we get the following identity

−B

=A +[A, B]+

[[A,B],B]+

[[[A,B],B],B]+···, (C.29)

which can be used to obtain some scaling transformations.

Appendix D

Angular Momentum Operators in Spherical

Coordinates

In this Appendix we give a brief review of the angular momentum operators in

spherical coordinates [462, 516] due to its wide applications in angular momentum

theory.

It is well known that the expressions for the components of the angular momen-

tum operator L =r ×p in Cartesian coordinates can be easily written as

=yp

−zp

=−i



∂

∂z

−z

∂

∂y



, (D.1)

=zp

−xp

=−i



∂

∂x

−x

∂

∂z



, (D.2)

=xp

−yp

=−i



∂

∂y

−y

∂

∂x



, (D.3)

from which we can obtain the following commutation relation

]=iε

ij k

, (D.4)

where the indices i, j, k can be x,y,orz, and the coefﬁcient ε

ij k

is named Levi-

Civita symbol (3.26). Thus we establish the commutation relations for the compo-

nents of the angular momentum of a spinless particle.

We now want to ﬁnd the form of these operators in spherical coordinates. By

using the following transformation equations

x = r sinθ cos ϕ, y =r sin θ sin ϕ, z =r cos θ, (D.5)

then we have

r =



, cosθ =

, tanϕ =

, (D.6)

where r ∈[0, ∞), θ ∈[0,π],ϕ ∈[0, 2π ].

S.-H. Dong, Wave Equations in Higher Dimensions,

261

262 D Angular Momentum Operators in Spherical Coordinates

Consequently, we have

∂r

∂x

=sin θ cos ϕ,

∂r

∂y

=sin θ sin ϕ,

∂r

∂z

=cos θ,

∂θ

∂x

cosθ cos ϕ

∂θ

∂y

cosθ sin ϕ

∂θ

∂z

=−

sin θ

∂ϕ

∂x

=−

sin ϕ

r sin θ

∂ϕ

∂y

cosϕ

r sin θ

∂ϕ

∂z

=0.

(D.7)

Making use of these relations, we are able to obtain the following expressions

=i



sin ϕ

∂

∂θ

cosθ

sinθ

cosϕ

∂

∂ϕ



=−i



cosϕ

∂

∂θ

−

cosθ

sin θ

sin ϕ

∂

∂ϕ



=−i

∂

∂ϕ

(D.8)

from which we have

=−



sinθ

∂

∂θ



sin θ

∂

∂θ



sin

∂

∂ϕ



, (D.9)

with the following properties

]=0,i=x,y,z. (D.10)

Instead of the operators L

and L

, we often use the linear combinations of the

operators L

and L

±iL

=e

±iϕ



∂

∂θ

+i cot θ

∂

∂ϕ



, (D.11)

which are called the ladder operators for the magnetic quantum number m but leave

the angular momentum quantum number l unchanged, that is,

(θ, ϕ) =



l(l +1) −m(m +1)Y

m+1

(θ, ϕ),

−

(θ, ϕ) =



l(l +1) −m(m −1)Y

m−1

(θ, ϕ),

(D.12)

where Y

(θ, ϕ) is the spherical harmonics and becomes the common eigenfunction

of L

and L

. They correspond to the eigenvalues l(l +1)

and m, respectively

(θ, ϕ) =l(l +1)

(θ, ϕ),

(θ, ϕ) =mY

(θ, ϕ).

(D.13)

If we use the properties of the ladder operators L

, then we may express the

spherical harmonics as

(θ, ϕ) =



(l +m)!

(2l)!(l −m)!

−

)

l−m

(θ, ϕ),

(θ, ϕ) =



(l −m)!

(2l)!(l +m)!

)

l+m

−l

(θ, ϕ).

(D.14)

D Angular Momentum Operators in Spherical Coordinates 263

It is well known that these ladder operators L

together with the operator L

satisfy the following commutation relations

]=±L

, [L

−

]=2L

, (D.15)

which correspond to an SU(2) group. On the other hand, it is not difﬁcult to see that

−

−L

−

. (D.16)

Finally, let us calculate the mean values and root-mean-square deviations of the

operators L

and L

in the state Y

(θ, ϕ) in terms of the ladder operators L

.By

inverting formulas (D.11), we have

−

), L

−L

−

). (D.17)

Thus we see that L

(θ, ϕ) and L

(θ, ϕ) are linear combinations of the states

m+1

(θ, ϕ) and Y

m−1

(θ, ϕ). As a result, we are able to obtain their mean values as

follows:

L

=L

=0,

L

=L

=



[l(l +1) −m

L

=L



√



l(l +1) −m

(D.18)