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234 CONTINUOUS GROUPS AND REPRESENTATIONS 11.5 UNITARY GROUP IN TWO DIMENSIONS: u2 Quantum mechanics is formulated in complex space.. 236 CONTINUOUS GROUPS AND REPRESENTATIONS 11.6 SPE

GROUP INVARIANTS 233

We can write this as

For a linear transformation between (Z, g) and (z, y) we write

234 CONTINUOUS GROUPS AND REPRESENTATIONS

11.5 UNITARY GROUP IN TWO DIMENSIONS: u(2)

Quantum mechanics is formulated in complex space Hence the components

of the transformation matrix are in general complex numbers The scalar or inner product of two vectors in mdimensional complex space is defined as

( 1 1.69)

(x, Y> = x;y1 + x;?& + + x*y n n 7where x* means the complex conjugate of x Unitary transformations are linear transformations, which leaves

(1 1.70) invariant All such transformations form the unitary group U(n) An element

UNITARY GROUPIN TWO DIMENSIONS: u(2) 235

where A, B , C, and D are in general complex numbers Invariance of (x, x)

gives the unitarity condition as

IAI2 + ICI2 A'B + C*D ]

[AI2 + ICI2 = 1

[BI2 + 1Ol2 = 1

A"B + C * D = 0

(1 1.74) (1 1.75) (1 1.76)

236 CONTINUOUS GROUPS AND REPRESENTATIONS

11.6 SPECIAL UNITARY GROUP s u ( 2 )

In quantum mechanics we are particularly interested in SU(2), a subgroup of

U(2), where the group elements satisfy the condition

det u = 1

For SU(2), A and B in the transformation matrix

(11.83)

(1 1.84) satisfy

(11.86)

( 1 1.87)

(I 1.88)

This can be written as

where o, are the Pauli spin matrices:

which satisfy

(1 1.91) (1 1.92) where (i, j , k ) are cyclic permutations of (1,2,3) Condition (11.83) on the determinant u gives

+ b2 + c2 + d2 = 1

This allows us to choose (a, 6, c, d) as

(11.93)

u = cos w, b2 + c2 + d2 = sin2 w, (1 1.94)

LIE ALGEBRA OF s u ( 2 ) 237

thus Equation (11.89) becomes

u(w) = Icosw+iSsinw, where we have defined

s =a01 + pa2 + yo3

of the determinant (11.83) for SU(2); thus SU(2) can only have three in- dependent parameters These parameters can be represented by a point on the three-dimensional surface (S-3) of a unit hypersphere defined by Equation (11.93) In Equation (11.95) we represent the elements of SU(2) in terms of

w7 and (a, P, 7) 7 (1 1.98) where (a, P, y) satisfies

a 2 + p + y 2 = 1 (1 1.99)

(1 1 loo)

By changing (a, p, y) on S-3 we can vary w in

-

u(w) = I cos w+X sin w,

where we have defined

238 CONTINUOUS GROUPS AND REPRESENTATIONS

The 2 x 2 transformation matrix, u(w), transforms complex vectors

as

(1 1.105) Infinitesimal transformations of SU(2), analogous to R(3), can be written as

- v = u(w)v

v(w) = (I+XGw)v(O)

sv = Xv(0)sw v’(w) = %v(O),

(11.106) (11.107) (1 1.108) where the generator X is obtained in terms of the generators XI, %2, %3 as

If we make the correspondence

-

11E ALGEBRA OF s u ( 2 ) 239

the two algebras are identical and the groups SU(2) and R(3) are called

isomorphic Defining a unit normal vector

1-

(/3 + ia) sin w

cos w - i y sin w The transformation

(11.125)

- v = u(w)v

240 CONTINUOUS GROUPS AND REPRESENTATIONS

induces the following change in a function f(v1, ~ 2 ) :

-

f ( V ) = f Mff, P, Y)VI' (1 1.126) Taking the active view we define a n operator 0, which acts on f(v) Since both views should agree, we write

( 11.128) (11.129)

We now write the effect of the operator 0 1 , which induces infinitesimal changes in a function f(vl,v2) as

This gives the generator 0 1 as

( 11.136) Similarly, we write

LORENTZ GROUP AND ITS LIE ALGEBRA 241

The sign difference with Equation (11.112) is again due to the fact that in

the passive view axes are rotated counterclockwise, while in the active view vectors are rotated clockwise

11.8 LORENTZ GROUP AND ITS LIE ALGEBRA

The ensemble of objects [a;], which preserve the length of four-vectors in Minkowski spacetime and which satisfy the relation

!?ff0a7a6 a 0 - - 976, (11.142)

form the Lorentz group If we exclude reflections and consider only the

transformations that can be continuously generated from the identity trans- formation we have the homogeneous Lorentz group The group that

includes reflections as well as the translations is called the inhomogeneous Lorentz group or the Poincare group From now on we consider the

homogeneous Lorentz group and omit the word homogeneous

Given the coordinates of the position four-vector xa in the K frame, ele- ments of the Lorentz group, , give us the components, To, in the fi; frame

as

242 CONTINUOUS GROUPS AND REPRESENTATIONS

In matrix notation we can write this as

specifying the relative velocity of the two inertial frames

Guided by our experience with R ( 3 ) , to find the generators of the Lorentz group we start with the ansatz that A can be written in exponential form as

We now multiply Equation (11.147) from the left by gpl and from the right

by A-' to write

g - l z g [AA-'I = g-'gA-', ( 11.150) which gives

-

LORENTZ GROUP AND ITS LIE ALGEBRA 243

Since for the Minkowski metric

-Using Equation (11.153) and the relations g2 = I , A = eL, and A-l = ecL

we can also write

gAg = egLg = e P L ; (11.154) thus

This equation shows that g L is an antisymmetric matrix Considering that g

is the Minkowski metric and L is traceless, we can write the general form of

L = 1

Introducing six independent parameters (PI, 0 2 , j33) and (01, 02, 0 3 ) , this can also be written as

L = PiV1 + P 2 V 2 + P 3 V 3 + & X I + 0 2 x 2 + 0 3 x 3 , (11.158) where

0 0 0 0

- 1 0 0 0 (11.159) and

0 0 0 -1 0 0 1 1 , x 2 = [:;: :I.”- 0 - 1 0 0

0 0 0 0 (11.160)

244 CONTINUOUS GROUPS AND REPRESENTATIONS

x’

Fig 11.3 Boost and boost plus rotation

Note that (XI, Xz, X3) are the generators of the infinitesimal rotations about the d-, z2-, z3-axes [Eq (10.84)], respectively, and (Vl,Vz,V3) are the generators of the infinitesimal Lorentz transformations or boosts from one inertial observer to another moving with respect to each other with velocities

(Pl,pz, p3) along the d-, z2-, z3-axes, respectively These six generators

satisfy the commutation relations

(11.161) (11.162) (11.163) The first of these three commutators is just the commutation relation for the rotation group R(3); thus the rotation group is also a subgroup of the Lorentz group The second commutator shows that Vi transforms under rotation like a vector The third commutator indicates that boosts in general do not commute, but more important than this, two successive boosts is equal to a boost plus a rotation (Fig 11.3), that is,

Thus boosts alone do not form a group An important kinematic consequence

of this is known as the Thomas precession

We now define two unit bvectors:

(11.165)

LORENTZ GROUP AND ITS LIE ALGEBRA 245

and

and introduce the parameters

(1 1.166)

8 = vf8:+8;+8,2 and

246 CONTINUOUS GROUPS AND REPRESEN JAJIONS

Using the parametrization

( 11.177)

coshx sinhx 0 0

A b o o s t ( f i l ) = [ sinhx ; coshx ; 0 ; 0 ;],

which is reminiscent of the rotation matrices [Eqs (10.42-43)] with hyper-

bolic functions instead of the trigonometric Notice that in accordance with our previous treatment in Section 11.2, the generator V1 can also be obtained from

Vl = A L ( P 1 = 0) (11.178) The other generators can also be obtained similarly

1 1.9 G RO U P REP R ES E N TAT 10 N S

As defined in Section 11.1, a group with its general element shown with g is an abstract concept It gains practical meaning only when G is assigned physical operations, D(g), t o its elements that act in some space of objects called the

representation space These objects could be functions, vectors, and in general tensors As in the rotation group R(3) group representations can be accomplished by assigning matrices to each element of G , which correspond to rotation matrices acting on vectors Given a particular representation D(g), another representation can be constructed by a similarity transformation as

Representations that are connected by a similarity transformation are called

equivalent representations Given two representations D(') (9) and D(2) (9)

we can construct another representation:

GROUP REPRESENTATIONS 247

reducible representation If D(g) cannot be split into the sums of smaller representations by similarity transformations, it is called an irreducible rep- resentation Irreducible representations are very important and they form

the building blocks of representation theory A matrix that commutes with every element of an irreducible representation is a multiple of the unit ma- trix We now present without proof an important lemma due to Schur for the criterion of irreducibility of a group representation

11.9.1 Schur's Lemma

Let D(')(g) and D(')(g) be two irreducible representations with dimensions

n1 and n 2 , and suppose that a matrix A exists such that

AD(')(g) = D(2)(g)A (11.181) for all g in G Then either A = 0, or n1 = n 2 and det A # 0, and the two representations D(') ( g ) and d 2 ) (9) are equivalent

By a similarity transformation if D(g) can be written as

is transformed into another element of that space by the action of the group el- ements D(')(g) For example, for the rotation group R(3) a three-dimensional representation is given by the rotation matrices and the representation space

is the Cartesian vectors In other words, rotation of Cartesian vectors always results in another Cartesian vector

11.9.2 Group Character

The characterization of representations by explicitly giving the matrices that

represent the group elements is not possible, because by a similarity transfor- mation one could obtain a different set of matrices Thus we need to identify properties that remain invariant under similarity transformations One such

248 CONTINUOUS GROUPS AND REPRESENTATIONS

property is the trace of a matrix We now define character ~ ( ~ ) ( g ) as the trace

of the matrices d i ) ( g ) , that is,

tors The superscript 3 indicates the degrees of freedom, in this case the

three independent components of a vector D(5)(g) is the representation cor- responding to the transformation matrices for the symmetric second-rank Cartesian tensors In this case the dimension of the representation comes from the fact that a second-rank symmetric tensor has six independent com- ponents; removing the trace leaves five In general a symmetric tensor of rank

n has (2n+ 1) independent components; thus the associated representation is

(2n + 1)-dimensional

SPHERlCAL HARMONICS AND REPRESENTATIONS OF R(3) 249

11.11 SPHERICAL HARMONICS A N D REPRESENTATIONS OF

R(3)

An elegant and also useful way of obtaining representations of R(3) is to con-

struct them through the transformation properties of the spherical harmon- ics The trivial representation D(’)(g) simply consists of the transformation

of Yo0 onto itself d 3 ) ( g ) describes the transformations of ql=1)m(Q,4) The three spherical harmonics (Yl- 1 , Y ~ o , Y11) under rotations transform into lin- ear combinations of each other In general, the transformation properties of the (2l+ 1) components of x,(e, 4) generate the irreducible representations

D(21+1)(g) of R(3)

11.11.1

In quantum mechanics angular momentum, L, is a differential operator acting

on a wave function q ( x , y, 2) It is obtained from the classical expression for the angular momentum,

Angular Momentum in Quantum Mechanics

250 CONTINUOUS GROUPS AND REPRESENTATIONS

Fig 11.4 Counterclockwise rotation of the physical system by 6, about fi

11.11.2 Rotation of the Physical System

We have seen that the effect of the operator [Eq.(11.40)]

-

( 1 I 196) ,-x he,

is to rotate a function clockwise about an axis pointing in the fi direction by

On In quantum mechanics we adhere to the right-handed screw convention, that is, when we curl the fingers of our right hand about the axis of rotation and in the direction of rotation, our thumb points along fi Hence we work with the operator

R =e-iL 66, ( 11.198) For a rotation about the z-axis this gives

R9(r,6,4) = [e-iLzd] @ ( T , 8,4) (11.199)

R@(z,Y,z) = e-iL6en@(z,y,z) (11.2oo) For a general rotation about an axis in the fi direction by 8, we write

SPHERICAL HARMONICS AND REPRESENTATIONS OF R(3) 251

11.11.3

Using the Euler angles we can write the rotation operator

Rotation Operator in Terms of the Euler Angles

of the sequence of rotations, which starts with a counterclockwise rotation by

Q about the z-axis of the initial state of the system:

e-i7Lz, ~ (z2,y2, z 2 ) t (z’, Y’, 2’) (1 1.205)

11.11.4

One of the disadvantages of the rotation operator expressed as

Rotation Operator in Terms of the Original Coordinates

we need to express R as rotations entirely in terms of the original coordinate

in expressing the operator R in terms of the ( z ~ , yn, zn) coordinates Action

of R on the coordinates induces the following change in @(z, y, z ) :

252 CONTINUOUS GROUPS AND REPRESENTATIONS

Fig 11.5 Transformation to the (z,, g,, 2,)-axis

Similarly for another point we write

Operating on Equation (11.210) with R we get

Using Equation (11.207), this becomes

We now operate on this with R to get

SPHERICAL HARMONICS AND REPRESENTATIONS OF R(3) 253

We now observe that (This may take a while to convince oneself We recom- mend that the reader first plot all the axes in Equations (11.203) to (11.205) and then, operate on a radial vector drawn from the origin with (11.217) Finally, trace the orbit of the tip separately for each rotation while preserving the order of rotations.)

e-iYLz, = e-iPLy, e-i7L,1 eiPLyl ( 11 217)

to write

R ,e-iPLgl e-i’YLzl [eiPLyl e-iPLgl] e-iaLz ( 11 218) The operator inside the square brackets is the identity operator; thus

R ,e-iPLgl e - i 7 L z 1 e - i a L ~ (1 1.219)

We now note the transformation

e- iPL,, = e- i a L , e iPLY e i a L , (1 1.220)

e-iaL,e-iPLye-i7L, \[I ( ,Y,Z) = Wz’,y’,z’)

In spherical polar coordinates this becomes

R(a, P7 Y ) W , 0, $1 = w-, 8,’ 4’) (1 1.225) Expressing the components of the angular momentum operator in spherical polar coordinates:

x + zy = r sin &+a6 and

z = rcosf?,

(1 1.226)

254 CONTINUOUS GROUPS AND REPRESENTATIONS

Using Equations (11.227-11.229) we can now write Equation (11.224) as

which is now ready for applications to spherical harmonics &(O, 4)

SPHERICAL HARMONICS AND REPRESENTATIONS OF R(3) 255

where dQ = sin Qdsdq5 In the study of angular momentum in quantum physics

we frequently need expansions of expressions like

&z(Q, 4) = f(Q7 4>Xm(Q7 4) (1 1.243)

256 CONTINUOUS GROUPS AND REPRESENTATIONS

and

(11.244)

a a

Glrn(e,4) = o(- a6 ' -,Q,4)Krn(Q,4), a4

where O($, &, 8,4) is some differential operator For &,(6,4) we can write

For Gtrn(6,4) we can write