4 Symmetries in quantum mechanics

Symmetry transformations - operations which preserve properties of the sys-tem - are important already in classical mechanics and classical field the-ory as they lead (via the celebrated Noether theorem) to conservation lawswhich, in turn, often simplify solving physical problems. They are equallyimportant in quantum mechanics where they allow for example to establishselection rules for quantum transitions (like the ones occurring in atoms), tounderstand spectra of Hamiltonians and sometimes even to find the exactspectra using group theoretic methods only. Symmetries become even moreimportant in quantum field theories where they usually form the very basisof their formulation (by allowing to account for the observed or postulatedconservation laws).

In this chapter we give a brief introduction to symmetries in quantumtheories. Our general considerations will be illustrated here on examplestaken from the familiar nonrelativistic quantum mechanics of a single par-ticle, but they apply equally well to nonrelativistic quantum mechanics ofmany particle developed in section 5 and to quantum field theory. In partic-ular the general results discussed here will be applied to formulate quantummechanics of relativistic particles, which is one of the possible approaches toQuantum Field Theory (presented in sections 6-9).

4.1 General considerations

In quantum mechanics physical states of a considered system are representedin the Hilbert space by rays rather than vectors. By rays we mean classes ofequivalence of (normalized) vectors which differ one from another by a phasefactor:

Ψ ∼ Ψ′ if Ψ = eiδ Ψ′,

or, in the Dirac notation,

|Ψ〉 ∼ |Ψ′〉 if |Ψ〉 = eiδ |Ψ′〉 .Observables are represented by linear Hermitian operators, i.e. operatorssuch that O = O†. The operator O† conjugate (with respect to the scalarproduct (·|·) in the Hilbert space) to a linear operator O is defined by therelation1

(

Φ|O†Ψ)

≡ (OΦ|Ψ) = (Ψ|OΦ)∗ , (4.1)

1One has to resort here to the mathematical notation Ψ, Φ, etc. instead of |ψ〉, |φ〉,etc. for state vectors because the Dirac bra-ket notation is not general enough.

56

which should hold for all2 vectors Ψ and Φ. From this definition it followsthat eigenvalues of Hermitian operators are real. (To see this, take the scalarproduct of OΨ = λΨ with Ψ, use O = O† and then apply the definition(4.1) to get λ = λ∗). Another theorem says that different state vectors, sayΨ1 and Ψ2, on which a given Hermitian operator O has different eigenvaluesλ1 and λ2, respectively, are orthogonal to each other. (To show this write:λ1(Ψ2|Ψ1) = (Ψ2|OΨ1) = (O†Ψ2|Ψ1) = (OΨ2|Ψ1) = λ2(Ψ2|Ψ1); therefore,either (Ψ2|Ψ1) = 0 or λ1 = λ2.)

If the physical system is in a state represented by a vector Ψ belongingto the ray R and Φn are some vectors belonging to the set of rays Rn, whichform a complete set of orthogonal rays (usually eigenvectors of a Hermitianoperator representing some particular observable) then the probability thatthe system will be found (as a result of a specific measurement) in the staterepresented by Rn is given by

P (R→ Rn) = |(Φn|Ψ)|2 , (4.2)

and, due to the completeness of the set of rays Rn,

∑

n

P (R→ Rn) = 1 . (4.3)

Most generally speaking, symmetry transformations are changes of thesystem which preserve probabilities of all possible measurements that can bemade on it. Symmetry transformations of a given physical system form agroup (the composition S2 · S1 of two transformations S1 and S2 is also asymmetry transformation; each symmetry transformation S has its inverseS−1; there is a trivial transformation - the identity transformation, denotedid, etc.) Thus, under the action of a symmetry transformation S

S : R → R′, Rn → R′n ,

but

P (R′ → R′n) = P (R→ Rn) ,

where R, Rn etc. are the rays representing states of the original system and R′

and R′n, etc. are the rays representing states of the transformed system (this

is the so-called active view on symmetry transformations), or the states ofthe original system but described by another observer (the so-called passive

view). In the Hilbert space symmetry transformations are represented bysome operators acting, of course, not on rays but on vectors. A fundamental

2We ignore here potential subtleties related to domains on which these operators aredefined.

57

theorem by E. Wigner says that the Hilbert space operators correspondingto symmetry transformations can be of two types only: either they are linearand unitary operators U(S), for which

U(αΨ + βΦ) = αUΨ+ β UΦ , (4.4)

and

(UΦ|UΨ) = (Φ|Ψ) , (4.5)

or they are antilinear and antiunitary operators A(S), for which

A(αΨ+ βΦ) = α∗UΨ+ β∗UΦ , (4.6)

and

(AΦ|AΨ) = (Ψ|Φ) = (Φ|Ψ)∗ . (4.7)

Defining the Hermitian conjugation A† of the antilinear operator A by

(

Φ|A†Ψ)

≡ (AΦ|Ψ)∗ = (Ψ|AΦ) , (4.8)

we obtain that for both, linear and antilinear operators, the (anti)unitarityconditions read

U † = U−1 , A† = A−1 , (4.9)

so that

(UΦ|UΨ) =(

Φ|U †U Ψ)

=(

Φ|U−1U Ψ)

= (Φ|Ψ) , (4.10)

(AΦ|AΨ) =(

Φ|A†AΨ)∗

=(

Φ|A−1AΨ)∗

= (Φ|Ψ)∗ . (4.11)

The trivial symmetry transformation - the identity id - is obviously repre-sented by the unit operator I, which is unitary. It follows, that continuoustransformations, which can be continuously deformed to the identity trans-formation, must also be represented by unitary linear operators. The onlyknown symmetry represented by the antilinear operator is the time reversaltransformation.

4.2 Continuous symmetry transformations

Suppose a (more or less) abstract group of continuous transformations isgiven and we try to realize it on states of a given physical system. The

58

group structure of the symmetry transformations must be reflected in theproperties of the operators representing them. More precisely, if

S1 : Rn → R′n , and S2 : R′

n → R′′n , (4.12)

so that

S2 · S1 : Rn → R′′n , (4.13)

then the operator U(S1) should transform vectors from Rn into vectors fromR′n and U(S2) vectors from R′

n into vectors from R′′n. Correspondingly, the

operator U(S2 ·S1) should transform vectors from Rn directly into vectorsfrom R′′

n. However, since operators act on vectors rather than on rays, onecannot exclude that

U(S2)U(S1)Ψn = e−iφn(S2,S1)U(S2 ·S1)Ψn , (4.14)

that is, there can be a phase factor in the composition law of the operatorsrepresenting the symmetry transformations. It is easy to see that the phasefactor φn(S2, S1) must be the same for all Hilbert space vectors that can besuperposed:

e−iφnm(S2,S1)U(S2 ·S1) (Ψn +Ψm) = U(S2)U(S1) (Ψn +Ψm)

= U(S2)U(S1)Ψn + U(S2)U(S1)Ψm (4.15)

= e−iφn(S2,S1) U(S2 ·S1)Ψn + e−iφm(S2,S1)U(S2 ·S1)Ψm .

Acting on both sides of this equality with U−1(S2 ·S1), we get

e−iφnm(S2,S1) (Ψn +Ψm) = e−iφn(S2,S1)Ψn + e−iφm(S2,S1)Ψm .

For linearly independent vectors Ψn and Ψm this can hold only if all thethree phase factors are equal. Therefore, in general, the composition law ofsymmetry operators takes the form

U(S2)U(S1) = e−iφ(S2,S1) U(S2 ·S1) . (4.16)

For φ(S2, S1) that cannot be absorbed by a redefinition of the operatorsU(S) (i.e. when the phase factors φ(S2, S1) form a nontrivial two-cocycle)one speaks of a projective representation.

Projective representations can always be avoided by appropriately en-larging the symmetry group (i.e. by replacing the original symmetry groupby its universal covering group) without changing its physical implications.(This will be demonstrated below on the important example of the rotationgroup.) At this point we assume that the symmetry group we are discussingis just the enlarged one.

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Continuous symmetry transformations form Lie groups that can be iden-tified with differentiable manifolds. The parameters θa varying continuouslyin some domain and parametrizing symmetry transformations S(θ) providethen the coordinate system on the group manifold. As it is customary, weassume they are such that θa = 0 corresponds to the identity transformation(id). In a given parametrization (a map on the group manifold) the compo-sition rule S2 · S1 = S, when written as S(θ2) · S(θ1) = S(θ) = S(h(θ2, θ1))defines the composition function θa = ha(θ2, θ1), which must have the follow-ing obvious property:

ha(0, θ) = ha(θ, 0) = θa . (4.17)

Operators U(θ) representing in the Hilbert space infinitesimal (close tothe identity) continuous symmetry transformations S(θ) with |θa| ≪ 1, canalways be written as

U(S(θ)) = 1− i θaQa +

1

2θaθbQ

ab +O(θ3) , (4.18)

where Qa are called symmetry generators. As follows from the symmetryθaθb = θbθa, the operators Q

ab must be symmetric in their indices, Qab = Qba.Unitarity (up to O(θ2) terms) requires the generators Qa to be Hermitianoperators. For this reason, symmetry generators are always candidates forquantum mechanical observables.

Applying the expansion (4.18) to the compositions rule (recall, for themoment we have assumed that the absence of possible phase factors)

U(S(θ′))U(S(θ)) = U(S(h(θ′, θ))) , (4.19)

and using the property (4.17) of the composition function ha(θ′, θ), whichimplies that its Taylor expansion has the form

ha(θ′, θ) = θ′a + θa + C bc

a θ′bθc + . . . , (4.20)

we find(

1− i θ′aQa +

1

2θ′aθ

′bQ

ab + . . .

)(

1− i θaQa +

1

2θaθbQ

ab + . . .

)

= 1− i (θ′a + θa + C bca θ′bθc)Q

a +1

2(θ′a + θa)(θ

′b + θb)Q

ab + . . . (4.21)

Comparison of the coefficients of the same combinations of the θ’s on bothsides determines Qab

Qbc = −QbQc + iC bca Qa = −QcQb + iC cb

a Qa , (4.22)

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where the second equality follows from the requirement of the symmetry ofQab in its indices. Thus, the group composition law determines the algebraicproperties of the symmetry generators Qa:

[

Qa, Qb]

= i(

C abc − C ba

c

)

Qc ≡ i f abc Qc . (4.23)

The factors f aba = −f ba

a are called the structure constants of the symmetrygroup algebra.3 The Jacobi identity

[

Qa,[

Qb, Qc]]

+[

Qc,[

Qa, Qb]]

+[

Qb, [Qc, Qa]]

= 0 , (4.24)

(satisfied identically by any three operators Qa, Qb and Qc) imposes then thefollowing relation on the structure constants

∑

a

(

f ade f bc

d + f cde f ab

d + f bde f ca

d

)

= 0 . (4.25)

Finally, we note that if the group structure is such that the compositionfunction ha(θ

′, θ) is additive: ha(θ′, θ) = θ′a + θa (i.e. f bc

a ≡ 0) - the wholesymmetry group is Abelian, or we consider only an Abelian subgroup of it -we can write

U(S(θ2))U(S(θ1)) = U(S(h(θ2, θ1))) = U(S(θ2 + θ1)) .

This allows to immediately find the operator U(S(θ)) also for finite transfor-mations:

U(S(θ)) = limN→∞

[U(S(θ/N))]N = limN→∞

(

1− iθaQ

a

N

)N

= e−iθaQa

. (4.26)

We now return to the problem of projective representations. They canappear for two different reasons: the algebraic one and the topological one.The logic is as follows. We take from “the reality” a group of symmetrytransformations which we think should be the symmetry group also in thequantum version of the problem (e.g. when considering the motion of aparticle in a rotationally symmetric potential we take for it the SO(3) groupbecause this is the symmetry group considered in this connection in classicalmechanics). To find algebraic properties of the operators Qa generating theunitary operators U , which should represent the symmetry transformationsin the quantum theory, we expand both sides of the equality

U(S(θ′))U(S(θ)) = e−iφ(S(θ′),S(θ)) U(S(h(θ′, θ))) , (4.27)

3For compact groups, like the rotation groups SO(N) or special unitary groups SU(N),the generators Qa can be chosen in such a way that the structure constants are totallyantisymmetric in all three indices. In these cases they will be simply written as fabc.

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similarly as in (4.20) and (4.21) but this time allowing for a possible phasefactor, because its presence cannot a priori be excluded by physical require-ments. Assuming that

φ(S(θ′), S(θ)) = habθ′aθb + . . . (4.28)

(because if any of the two operators on the left-hand side of the formula(4.27) is the unit operator, the phase φ should be zero) and repeating thesteps used in (4.21) which led to (4.23) we find

[

Qa, Qb]

= i(

C abc − C ba

c

)

Qc + i (hab − hba)1 ≡ i f abc Qc + i fab1 , (4.29)

with fab = hab−hba. The terms proportional to the unit operator 1 are calledthe central charges (of the algebra of generators).

The Jacobi identity (4.24) imposes on central charges certain constraintswhich usually eliminate a number of them. Furthermore, central chargesnot excluded by the Jacobi identity can or cannot be removed from thecommutation relations by redefining the generators Qa (without changingthe if ab

c parts of their commutation rules). For example, they can all beremoved in the case of the SO(3) and the Poincare groups, but not in the caseof the Galileo group (as we will show shortly). If they cannot be removed, wehave the projective representation and the group is represented projectivelyalready in the neighbourhood of the identity transformation.

Even if all central charges can be removed, the symmetry group repre-sentation can still be projective for topological reasons. This can happen ifthe group manifold is not simply connected, that is, when its first homotopy

group (called also the path group) π1(G) is nontrivial. To ascribe a Hilbertspace operator U(S) to a point S (to a symmetry transformation) on thegroup manifold in a given coordinate system θa, in which S is character-ized by θaS, one chooses some standard path θa = θa(ξ) with θa(0) = 0 andθa(1) = θaS connecting S with the identity transformation along which one

62

solves the differential equation4

d

dξU(θ(ξ)) = −iQa U(θ(ξ)) fab(θ(ξ))

dθb(ξ)

dξ, (4.30)

defining U for every point on the path (the input for ξ close to zero being theformula (4.18)). It can be shown, that if another path going from the identityto S is chosen and used to define U(S), the same operator is obtained pro-vided this new paths and the standard one can be continuously deformed toeach other. To a composition S2 · S1 of two transformations S1 and S2 therecorrespond, therefore, in principle two different operators: U(S2·S1) obtainedby integrating the defining differential equation along the standard path go-ing from the identity directly to S = S2 · S1 and the operator U(S2)U(S1),which can be thought of as being obtained by integrating the defining dif-ferential equation along the path id→ S1 → S2 · S1.

5 If the manifold is notsimply connected, the latter path may not be continuously deformable to thestandard one. If this is the case, the operator U(S(θ2)) U(S(θ1)) obtainedalong the non-standard path id→ S1 → S2 · S1 may differ by a phase factorφ(S(θ2), S(θ1)) from the operator U(S2·S1) obtained using the standard path.Furthermore, if a path id→ S1 → S2 · S1 = S is continuously deformable toanother path id→ S ′

1 → S ′2 · S ′

1 = S, then, on the basis of what has alreadybeen said, they give rise to the same operator U(S2) U(S1) = U(S ′

2) U(S′1),

that is their relative phase factors with respect to U(S2 ·S1) (obtained usingthe standard path from the identity to S) must be equal. Thus, the phasefactors that may appear in the formula (4.27) share the properties of thehomotopy group of the group manifold of the considered symmetry trans-formations. It is therefore easy to understand that the phase factors form

4It is derived as follows. The operator U(θ(ξ + dξ)) is obtained as the compositionU(δθ)U(θ(ξ)) ≈ (1− iδθaQa)U(θ(ξ)), with δθa determined by the condition

θa(ξ + dξ) ≈ θa(ξ) +dθa(ξ)

dξdξ = ha(δθ, θ(ξ)) ≈ ha(0, θ(ξ)) +

∂ha(θ, θ(ξ))

∂θb

∣

∣

∣

∣

∣

θ=0

δθb ,

where ha(θ1, θ2) is the composition function. Due to its property (4.17) one gets

δθa = fab(θ(ξ))

dθb(ξ)

dξdξ ,

where fab(θ(ξ)) is the matrix inverse to ∂ha(θ,θ(ξ))

∂θb

∣

∣

∣

θ=0. This leads to (4.30).

5This is only a convenient way of thinking. In a given coordinate system θ there aretwo standard paths: θa = θa1 (ξ1) with θa1(1) = θaS1

and θa = θa2 (ξ2) with θa2(1) = θaS2.

The path we refer to as the path id→ S1 → S2 · S1 corresponds to S2(θ2(ξ2)) · S1(θ1(ξ1))where first ξ1 is varied from 0 to 1 (keeping ξ2 = 0) and then ξ2 is increased from 0 to 1keeping ξ1 = 1 (this can of course be written more formally as a change from 0 to 1 ofa single parameter ξ). In this way infinitely many similar paths could be defined (e.g byfirst changing ξ2 and only then ξ1) but obviously all of them are continuously deformableto each other and, hence, lead to the same operator U(S2)U(S1).

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id

γ1

idγ2

A′

γ2

A

Figure 4.1: Two closed paths in the SO(3) group manifold starting from thepoint P ≡identity. The path γ1 is contractible to the trivial path. The pathγ2 is noncontractible due to the jump it makes from A to A′; the points Aand A′ in the figure represent the same point in the group manifold.

a one dimensional representation of the first homotopy group π1(G) of themanifold of the symmetry transformations (or, short, of G).

For example, the group manifold of the three-dimensional rotations isdoubly connected: π1(SO(3)) = Z2, where Z2 is the group consisting of twoelements: the identity id and −1. This can be seen by representing eachrotation by the vector whose length is equal to the rotation angle α andpointing in the direction of the rotation axis n. All such vectors fill a ball ofradius R = π. The topology of this manifold is however complicated by thefact that a rotation by α = π around n is equivalent to the rotation by α = πaround the direction −n. Therefore, the antipodal points of the surface of theball have to be identified. This leads to its double connectedness: imagine apath starting from a point P somewhere inside the ball and returning to Pwhich reaches somewhere the surface of the ball. Because the point on thesurface is equivalent to its antipodal point, the path can make a “jump” fromone side of the surface to the other side where it can again reimmerse intothe ball to reach P . Such a path cannot be continuously deformed to anotherpath going also from P to P , but which remains entirely inside the ball (cf.figure 4.1). However, a path which makes two “jumps” can be continuouslydeformed to it (see figure 4.2). Thus, π1(SO(3)) = Z2 and the phase factorse−iφ can take only two values 1 and −1.

This is indeed confirmed when the commutation relations (4.23) of therotation group are explicitly solved:6 one finds that the representations la-beled by half-integer j (j = 1

2, 32, . . .) are projective. Take e.g. as S1 and S2

the rotations around the z axis by some angles α1 and α2. From (4.26) it

6Solving the commutation relations of a symmetry group means (in the context ofquantum mechanics) finding explicitly matrix elements of the generators Qa of the sym-metry group in the basis of states in which the group generators belonging to the Cartansubalgebra are diagonal - see the discussion following the formula (4.55).

64

id

A

A′

B

B′

id A′

BA

B′ id

A

B

Figure 4.2: Consequtive stages of contraction to the point of a path in theSO(3) group manifold which makes two jumps.

then follows that

U(S(α)) = exp

(

− i

~αJz

)

.

Denoting |j, jz〉 the eigenstates of J2 and Jz for j = 12we have 〈1

2, jz|Jz|12 , j′z〉

= ~

2(σ3)jzj′z . For α1 and α2 such that α1 + α2 = 2π we obtain then

〈 12, jz|U(S(α2))U(S(α1))| 1

2, j′z〉 =

(

exp−2πi

2σ3

)

jzj′z

= − (I2×2)jzj′z ,

whereas S(α2)·S(α1) =id and therefore the operator ascribed to S(α2)·S(α1)along the standard path is simply the unit operator whose matrix betweenthe J2 and Jz eigenstates with j = 1

2is simply the 2 × 2 unit matrix I2×2.

The same is also true for all half integer values of j. Thus, in the case ofhalf-integer j representations (realized on states representing an odd numberof fermions, which have half-integer spis) of the rotation group SO(3) thephase factor e−iφ is indeed equal +1 or −1.

In the case of representations with j = 0, 1, . . . composing two rotationsby α1 and by α2 around the z-axis with α1 + α2 = 2π one gets

〈j, jz|U(S(α2))U(S(α1))|j, j′z〉 = (I(2j+1)×(2j+1))jz,j′z ,

so the phase factors e−iφ are always equal +1. The proof (4.15) that thephase factors cannot depend on the vector in the Hilbert space on whichoperators are acting (but are intrinsic properties of the operators) relied onthe possibility of forming a superposition of the two vectors. Because forrotations one apparently can have e−iφ = +1 or −1 for a composition of thesame two operators U(S2) and U(S1) (depending on whether they act onvectors corresponding to integer or half integer j), one is tempted to thinkthat it is the structure of the rotation group that imposes a superselection rule

which forbids forming superpositions of state vectors corresponding to integer

65

and half integer values of j.7 However, as a matter of facts, the impossibilityof forming superpositions of state vectors corresponding to integer and halfinteger spins does not follow from the topological properties of the rotationgroup. The point is (as already said) that the rotation group SO(3) can beenlarged to its universal covering group spin(3), which has the same physicalimplications, but is simply-connected (i.e. its first homotopy group is trivial)and, for this reason, has only non-projective representations. Thus, by itselfthe spin(3) ≃ SU(2) group would not impose any selection rule and we cansimply declare that the group of symmetry transformations of the quantumsystem is SU(2) and not SO(3) = SU(2)/Z2.

8 Nevertheless, we still dobelieve that preparing the system in a state which is a superposition of statevectors corresponding to integer and half integer spins is physically impossible(transitions between such states and other “normal” states are, of course,forbidden by the rotational invariance which ensures conservation the totalangular momentum).

We now demonstrate that the SU(2) group is indeed simply-connectedand constitutes the twofold covering of the rotation group SO(3). By defi-nition the SU(2) group is the group of all unitary 2 × 2 matrices with unitdeterminant (unimodular). Any such matrix M can be represented in theform

M = exp(−iH) , where H = H† and tr(H) = 0 . (4.31)

Matrices H can in turn always be written as a linear combination with realcoefficients of the three Pauli matrices σi. Hence, the SU(2) group has 3generators and its Lie algebra is isomorphic with the one of SO(3) (deter-mined by the composition law of three-dimensional rotations of a macroscopicbody). However, SU(2) has “twice as many” elements as does SO(3): givena three-dimensional vector r we can form a 2 × 2 matrix V = r · σ, whereσ is the vector formed by three Pauli matrices. Of course, V is Hermitian,tr(V ) = 0 and det(V ) = −r2. Take now a 2 × 2 matrix M belonging toSU(2). The matrix V ′

V ′ =M †VM , (4.32)

7This problem does not arise, of course, in ordinary quantum mechanics of a single par-ticle in which all vectors in the Hilbert space correspond to integer or all to half-integer j.It appears however already in nonrelativistic quantum mechanics of many particle systemswhich can describe interactions of bosons and fermions which have integer and half-integerspins, respectively. In this case the superselection rule says that a superposition of thestate of, say, two bosons and one fermion with the state of one bosons and two fermionsis impossible.

8With such a declaration nontrivial transformation properties of states formed as su-perpositions of interger and half-integer spin states under rotations by 2π are acceptablebecause then rotations by 2π are not the identity symmetry transformation.

66

is also Hermitian and traceless, so it defines a new vector r′ through theformula V ′ = r′ · σ. Since det(V ′) =det(V ) it follows that r2 = r′2. Hence,M defines a rotation R(M) such that R(M)r = r′. In fact every rotationcan be represented by a matrix M belonging to SU(2). From (4.32) it isclear, however, that the SU(2) matricesM and −M define the same rotation:R(M) = R(−M). Therefore, one says that the SU(2) group is the (universal)twofold covering of the rotation group SO(3).

The group SU(2) is simply-connected, its topology being that of a three-dimensional sphere9 (π1(Sn) = 1 for n > 2), so it has only non-projective rep-resentations. The element of SU(2) corresponding to the rotation R aroundthe z axis by an angle α is exp(− i

2σ3α). In SU(2) the range of the pa-

rameters θa, a = 1, 2, 3 is [0, 4π) because for all θa in this range differentunitary, unimodular matrices are obtained. The SO(3) group is obtainedfrom SU(2) as SU(2)/Z2 where Z2 is the (trivially) invariant subgroup10 ofSU(2) consisting of the matrices I and −I. Thus, if we declare that the truesymmetry of the quantum system is SU(2) instead of SO(3), we have onlynon-projective representations but the superselection rule still remains valid.

4.3 The Galileo group

In this subsection we apply the general formalism outlined above to nonrel-ativistic quantum theory. As examples we will consider mainly one-particlesystems, but the general considerations will be valid for many-particle sys-tems as well.

We consider first the operations which form the Galileo group of sym-metries of classical physics. These are rotations of the system in the three-dimensional space, translations in three space directions, translations in timeand transformations corresponding to boosting the considered physical sys-tem with a velocity V. An element S of the Galileo group consists thereforeof the elements

S = (R, V, a, τ) , (4.33)

9The most general form of a unitary 2× 2 matrix U † = U−1 is

U =

(

d+ ie f + ig−f + ig d− ie

)

,

where the parameters d, e, f and g have to satisfy d2 + e2 + f2 + g2 = 1 (that is,det(U) = 1), which is just the definition of a three-dimensional sphere S3 (immersed inthe four-dimensional Euclidean space).

10A group H ⊂ G is an invariant subgroup of the group G if ghg−1 ∈ H for every h ∈ Hand every g ∈ G. G/H consists of equivalence classes of elements of G defined by therelation g ∼ g′ if g′ = gh for a h ∈ H .

67

where R is an orthogonal 3× 3 matrix

RT · R = R · RT = I . (4.34)

and V and a are two three-vectors. On space and time coordinates x and tof an event the element S acts in the following way

x′ = R · x−Vt− a

t′ = t− τ. (4.35)

In the passive view on the transformation x′ and t′ are the space-time coor-dinates of the event as seen by another observer which moves with respectto the original one with velocity V, uses the coordinate axes rotated by Rand shifted by a and whose clock is late by τ with respect to the clock ofthe original observer. In the active view x′ and t′ are the coordinates of asimilar event occurring in the transformed physical system, that is, in thesystem which is shifted in space and time by −a and −τ , respectively, ro-tated by R−1 and boosted so that it moves with velocity −V with respect tothe original system.

The composition law of the group elements S can be easily deduced byconsidering the action of two successive transformations S1 = (R1,V1, a1, τ1)and S1 = (R2,V2, a2, τ2) on space and time coordinates: the transformations

x′ = R1 · x−V1 t− a1

t′ = t− τ1,

x′′ = R2 · x′ −V2 t′ − a2

t′′ = t′ − τ2,

combined give

x′′ = R2 · R1 · x− R2 ·V1 t−R2 · a1 −V2 t+V2 τ1 − a2

t′′ = t− τ2 − τ1.

Hence,

S2 · S1 = (R2 ·R1, V2 +R2 ·V1, a2 +R2 ·a1 −V2 τ1, τ2 + τ1) , (4.36)

and the transformation inverse to S = (R,V, a, τ) is

S−1 = (R−1, −R−1 ·V, −R−1 ·a−R−1 ·V τ, − τ) . (4.37)

We now parametrize the infinitesimal transformation S, or in other words,introduce a local coordinate system on the group manifold in the vicinity ofthe identity transformation. An infinitesimal rotation can be parametrizedby the three angles θi of the infinitesimal rotations around the three Cartesianaxes:

x′

y′

z′

≈

1 θz −θy−θz 1 θx

θy −θx 1

xyz

, (4.38)

68

so that the orthogonal matrix R in (4.35) takes the form

Rij = δij − ωij , where ωij = −ωji = −3

∑

k=1

ǫijkθk , (4.39)

(antisymmetry of ωij ensures orthogonality of Rij up to second order terms).For the other elements of the infinitesimal symmetry transformation one canuse the most natural parametrization by the three Cartesian components ofthe boost velocity vi, the three Cartesian components εi of the space trans-lation and the infinitesimal time shift τ . In the quantum theory there shouldexist Hermitian operators J ij ≡ ǫijkJk, Ki, P i and H , with the help of whichthe unitary operators U(S(θi, vi, εi, τ)) of the infinitesimal transformationshould take (in the chosen parametrization) the form11

U(θi, vi, εi, δ) = 1− i

~

1

2ωijJ ij +

i

~viKi +

i

~εiP i − i

~τH . (4.40)

Compared to our general notation introduced in (4.18) we have factorizedhere 1/~ and chosen the signs of the generators so that J i, P i, and H willcoincide with the operators of the total angular momentum, the momentumand the Hamiltonian, respectively.

Deriving the composition function ha from the Galileo group compositionrule (4.36) we could now find the commutation relations between the gener-ators J i, Ki, P i, and H . There is however a simpler method to find them.Since we know how the phase factors can affect the commutation rules (byintroducing central charges) we will temporarily ignore them and considerthe following operator

U(S)U(δij − ωij, vi, εi, τ)U−1(S) ,

in which S = (R,V, a, τ). This can be represented in two different ways.Firstly, we can use (4.40) to write the above composition in the form

1− i

~

1

2ωij U(S)J ijU−1(S) +

i

~vi U(S)KiU−1(S)

+i

~εi U(S)P iU−1(S)− i

~τ U(S)HU−1(S) . (4.41)

Secondly, we can apply the composition rule (4.36) and (4.37) to write

U(S)U(θi, vi, εi, τ)U−1(S) = U(R, V, a, τ ) , (4.42)

11Alternating signs of different terms in (4.40) as well as the definition of ωij in termsof θk are chosen so as to make uniform the signs in the corresponding formulae for thePoincare symmetry transformation rules (see section 6 and Appendix B).

69

where

R ≡ δij + (R · ω ·R−1)ij ≡ δij + ωij ,

V i ≡ (R · v)i − (R · ω ·R−1 · V )i ,ai ≡ (R · (ε+ vτ))i − V iδ − (R · ω · R−1 · (a + V τ))i ,

τ ≡ τ .

As the parameters of this new transformation all vanish for vanishing ωij, vi,εi and τ , they are all infinitesimal. Owing to this one can write

U(R, V, a, τ) = 1− i

~

1

2ωijJ ij +

i

~V iKi +

i

~aiP i − i

~τH . (4.43)

Comparing the coefficients of ωij, vi, εi and δ in (4.41) and in (4.43) we find

U(S)J ij U−1(S) =(

Jkl − V kK l + V lKk

−(a + V τ)kP l + (a+ V τ)lP k)

RkiRlj ,

U(S)Ki U−1(S) =(

Kj + τP j)

Rji , (4.44)

U(S)P i U−1(S) = P jRji ,

U(S)H U−1(S) = H + V iP i .

Taking now the parameters of S itself to be infinitesimal, that is, writing:Rij = δij + ωij, V i = vi, ai = εi and τ = τ , expanding both sides of theequalities (4.44) to the first order in these small parameters and compar-ing the corresponding coefficients on both sides, we arrive at the followingcommutation relations

[

J i, J j]

= i~ ǫijkJk ,[

P i, P j]

= 0 ,[

J i, P j]

= i~ ǫijkP k ,[

Ki, Kj]

= 0 ,[

J i, Kj]

= i~ ǫijkKk ,[

Ki, P j]

= 0 , (4.45)[

J i, H]

= 0 ,[

P i, H]

= 0 ,[

Ki, H]

= −i~P i .

To account for possible phases in the formula (4.16) we have to supple-ment the right hand side of each of the above commutation relations witha c-number term times the unit operator 1 - a possible central charge. Wehave, however, to check whether such central charges are allowed by the Ja-cobi identity (4.24). Let us consider the first commutator of (4.45) in themodified form

[

J i, J j]

= i~ ǫijkJk + i~ f ijJJ 1 , (4.46)

70

where f imJJ = −fmiJJ . From the Jacobi identity (4.24) with three generators Jwe get then

ǫijmfkmJJ + ǫkimf jmJJ + ǫjkmf imJJ = 0 .

Contracting this equality with δki gives

ǫjimf imJJ + ǫijmf imJJ = 0 .

This shows that the Jacobi identity does allow for a nonzero f imJJ . In contrast,the Jacobi identity with J , P and H shows that f iPH = 0 and the one withtwo J ’s and H eliminates f iJH . We consider next the modified commutationrelation

[

P i, P j]

= i~ f ijPP 1 . (4.47)

The Jacobi identity with two P ’s and one J shows then that ǫkijf ijPP = 0, i.e.that f ijPP = 0 - the central charge in the commutation rule of two P operatorsis forbidden by the Jacobi identity. Similar argument eliminates also f ijKK .However, for the central charge f ijKP the equality ǫkijf ijKP = 0 (which can beobtained in the same way) does not imply vanishing of f ijKP itself, becausef ijKP 6= −f jiKP ; the central charge f ijKP can have a nonzero symmetric part,which, taking the dimensions into account can be written as

[

Ki, P j]

= −i~M δij 1 , (4.48)

where M is some real mass parameter (the sign is chosen for latter conve-nience). Similarly, the Jacobi identities do not exclude the central chargesf ijJP , f

ijJK nor f iKH .

In the next step we try to redefine the generators to eliminate centralcharges. It is easy to see that defining

J i = J i +1

2ǫijkf jkJJ 1 , P i = P i +

1

2ǫijkf jkJP 1 , Ki = Ki +

1

2ǫijkf jkJK 1 ,

we obtain the commutation rules [J i, J j ] = i~ǫijkJk, [J i, P j] = i~ǫijkP k,[J i, Kj ] = i~ǫijkKk, and also [Ki, H ] = −i~P k, since the Jacobi identitywith J , K and H relates appropriately f ijJP with f iKH (check it!). Thus, theonly commutation rule from which the central charge cannot be eliminatedis (4.48), because there is no operator on its right hand side that could beredefined.

From now on, we will assume that the generators have been redefined andwill omit the tildas.

The consequence of the modified commutation rule (4.48) is the following.Take S2 = (1, V, 0, 0) and S1 = (1, 0, a, 0). Geometrically, that is according

71

to the rule (4.36), S2 ·S1 = (1, V, a, 0). Since Ki commute with Kj and P i

commute with P j, we can use (4.26) to write U(S2) and U(S1) separately inthe exponentiated forms and then (4.48) to obtain12

U(S2)U(S1) = ei~viKi

ei~aiP i

= ei2~

a·vM ei~(viKi+aiP i) . (4.49)

The right hand side differs by the phase factor from the operator we ascribeto S2 ·S1 with the help of (4.26). This shows that because of the presence ofthe central charge f ijKP we have to do with a projective representation.

Extension of the Galileo group which allows to avoid projective represen-tations consists of the formal inclusion in the algebra of the mass operatorM 1, which commutes with all other generators of the Galileo group andwhich acting on a state of a particle (of many particles - in the case of quan-tum mechanics of a many-particle system) gives its mass (the sum of massesof particles represented by this state). This extends the Galileo group bythe additional U(1) factor. In a quantum theory of a many-particle systemsuch a mass operator which commutes with H forbids transitions betweenstates of different masses (mass conservation). In addition, one has to imposethe superselection rule (which does not follow from the symmetry group it-self) which forbids forming superpositions of states corresponding to differentmasses.

If the Galileo group is to be the symmetry group of a given realizationof quantum mechanics, we should be able to construct the Hilbert spaceoperators J i, P i, Ki and H with the properties we have determined abovein the general way. Whether this is indeed possible depends, of course, onthe physical situation specified by giving the Hamiltonian H . For someHamiltonians (e.g. singling out a direction in the space) one is unable tofind operators satisfying all the requirements simultaneously. Usually it isthen possible to satisfy the requirements for a subgroup (consisting e.g. ofrotations and time translations only) of the full Galileo group.

We now illustrate these general considerations on examples. To specifythe action of symmetry operations on the system we shall impose physicalrequirements on transformed states. We take the active point of view onsymmetry transformations, as conceptually it is somewhat easier: to checkwhether the symmetry requirements are compatible with the dynamics wesimply ask whether the transformed system is also physically realizable, thatis, whether the transformed wave function (or the state vector) is a solutionof the original Schrodinger-like equation.

We begin with time translations. Physically the transformed state vector

12In the second step we have used the standard formula eA+B = e−1

2[A, B]eAeB valid

provided the operator C = [A, B] commutes with A and B.

72

|Ψ′〉 of the system translated in time and the state vector |Ψ〉 of the originalsystem should be related in such a way that the matrix elements of operatorsO corresponding to observables satisfy

Oψ′(t′) ≡ 〈Ψ′(t′)|O|Ψ′(t′)〉 = 〈Ψ(t)|O|Ψ(t)〉 ≡ Oψ(t) , (4.50)

i.e. the result of a measurement made on the time-translated system attime t′ should coincide with the result of the same measurement made onthe original system at t. Validity of (4.50) for all superpositions of states isensured by the relation13 |Ψ′(t′)〉 = |Ψ(t)〉 for t′ = t− τ . In agreement withour general discussion, if time translations are symmetries of the physicalsystem, there should exists a unitary operator U(τ), such that

|Ψ′(t)〉 = |Ψ(t+ τ)〉 = U(τ)|Ψ(t)〉 . (4.51)

Writing the Taylor expansion of |Ψ(t+ τ)〉 we find

|Ψ′(t)〉 = |Ψ(t)〉 − i

~τ

(

i~d

dt

)

|Ψ(t)〉+ 1

2

(

− i

~τ

)2(

i~d

dt

)2

|Ψ(t)〉+ . . .

= |Ψ(t)〉 − i

~τH|Ψ(t)〉+ 1

2

(

− i

~τ

)2(

i~d

dt

)

H|Ψ(t)〉+ . . . , (4.52)

where H is the Hamiltonian. Comparing with the infinitesimal form of U(τ)we see that it is the Hamiltonian that is the candidate for the generator oftime translations. However, H is the time translation generator only if it istime independent: the composition law (4.36) tells us that time translationsform an Abelian subgroup of the Galileo group. Therefore, as follows fromthe formula (4.26), we should have U(τ) = exp(−(i/~)τH). From the lastterm in (4.52) it is however clear that this is possible only if dH/dt = 0(moreover from Section 1 we know that if the Hamiltonian depends on time,the evolution operator is given by the more complicated formula (1.9)). Al-ternatively, one can check whether infinitesimally transformed state vector|Ψ′(t)〉 satisfies the Schrodinger-like equation:

i~d

dt|Ψ′(t)〉 = i~

d

dt

(

|Ψ(t)〉 − i

~τH|Ψ(t)〉

)

= H|Ψ(t)〉 − i

~τ

(

i~d

dt

)

H|Ψ(t)〉 , (4.53)

and again, the right hand side equals

H

(

|Ψ(t)〉 − i

~τH|Ψ(t)〉

)

= H|Ψ′(t)〉 ,

13A possible phase factor in this relation can always be removed by redefining thegenerator of time translations by adding to it a constant.

73

only if H does not depend on time. Thus, if the Hamiltonian is time in-dependent translations in time are symmetries of the quantum theory andare obviously generated by the Hamiltonian itself. This result is completelygeneral and applies to any quantum theory.

In principle one can envisage situations in which H does depend on time(time translation symmetry is explicitly broken), whereas the other oper-ations (e.g. rotations and/or space translations) remain good symmetries(that is, the operators having the properties required for the generators canbe constructed). However, in such a case there are no energy eigenstatesand, therefore, consequences of the remaining symmetries are not immedi-ately clear.14 In the following we will assume that time translations are goodsymmetries of the considered physical system. As required by the commu-tation rules (4.45), the generators of space translations and/or rotations (ifthese operations are good symmetries) as well as other symmetry generatorswhich may generate symmetry groups larger than the Galileo one (see theexamples of the harmonic oscillators and the nonrelativistic Hydrogen atom)should then commute with the Hamiltonian.

This is a good place to discuss the implications for the spectrum of theHamiltonian H of a set of generators Qa which commute with it and form aclosed algebra in the sense of satisfying the commutation relations (4.23):

[

Qa, Qb]

= if abc Qc , (4.54)

with some structure constants f abc . The considerations presented below are

fairly general and will apply also in relativistic quantum field theory describ-ing interations of elementary particles. Suppose a set of such generators Qa

acting in the Hilbert space is given. Of course, if H|n〉 = En|n〉 then alsoHU(θ)|n〉 = EnU(θ)|n〉, where U(θ) is the unitary operator realizing the sym-metry transformation built from Qa and continuous parameters θa. This doesnot mean that the degeneracy of energy levels of H is infinite (parametrizedby the continuous parameters θa) but only that at each energy level there isa basis of states |n, i〉 (all satisfying H|n, i〉 = En|n, i〉) labeled by a discreteindex i (which in general is a multi-index i = (i1, . . . , ir), see below), suchthat for any θa the state U(θ)|n, i〉 can be written as a linear combinationof |n, i〉. In fact, all such basis vectors can be obtained from a given one bysuccessively acting on it with the generators Qa.

Consider now the matrices formed out of the matrix elements of the gen-erators Qa between the basis state vectors |n, i〉 of some energy level En:

T aij ≡ 〈n, i|Qa|n, j〉 . (4.55)

14But of course, symmetries whose generators Qa commute with H(t) at a particularinstant t have standard implications for the spectrum of the instantaneous eigenvalues ofH(t) at that instant.

74

The matrices T aij obviously share the algebraic properties of the generatorsQa. In particular, they satisfy the commutation relations15

T aikTbkj − T bikT

akj = if ab

c T cij , (4.56)

with the same structure constants f abc as in (4.54). Hence, they form amatrix

representation of the Lie algebra of the symmetry group generated by Qa.Mathematically speaking, at each energy level En the states |n, i〉 span thespace (i.e. they form the basis) of a matrix representation of the Lie algebraof the symmetry group (physicist used to say: they span a representation or,shortly, they form a representation). Possible matrix (i.e. finite dimensional)representations of various Lie algebras are well known from group theory. Forexample, if [H, J] = 0, rotations are good symmetries and the representationsof the Lie algebra of the rotation group we are familiar with are 2j + 1dimensional with j = 0, 1

2, 1, . . ., and the states |n, i〉 ≡ |j, jz〉 are labeled by

the eigenvalues jz of the operator Jz.

In more complicated cases one defines the rank r of the Lie algebra (as-signed also to the group generated by this algebra) as the maximal number ofits generators (or their independent linear combinations) which all commutewith one another. The maximal set of mutually commuting generators formwhat in group theory is called the Cartan subalgebra; the rank of the groupis simply the dimension of its Cartan algebra. It follows from the generalrules of quantum mechanics, that the states |n, i〉 can be chosen in such away that all mutually commuting Cartan subalgebra generators have onlydiagonal matrix elements between these states. Therefore the eigenvalues ofthe Cartan subalgebra generators are used to label the states |n, i〉 within agiven representation of the Lie algebra. In other words, i is in fact a multi-index i = (i1, . . . , ir), specifying the eigenvalues on the state |n, i1, . . . , ir〉 ofeach of these r Cartan subalgebra generators. In the case of the rotationgroup any two generators do not commute and the group rank equals 1. Jz

is then chosen as the single generator of the Cartan subalgebra. In the SU(3)group, whose rank is 2, there are two commuting generators and the stateswithin a given representation are labeled by two indices - the eigenvalues ofthe two Cartan subalgebra generators.

Finally, the fundamental theorem of Racah states that for a simple groupof rank r there are exactly r independent operators16 built from the gen-erators Qa (not necessarily as their linear or bilinear combinations), whichcommute with all the generators Qa and, because they are built out of Qa’s

15Recall, that we have assumed that the symmetry group is enlarged so that the centralcharges are absent (i.e. they are part of the algebra of Qa’s).

16Independent means that they are not powers of each other nor products of (powers)of others.

75

(assumed to commute with H), also with the Hamiltonian H . These oper-ators called the Racah operators, being constructed from Qa’s, cannot mapvectors belonging to a given representation into vectors belonging to anotherrepresentation. Moreover, since they commute with all Qa’s, they cannotmap a vector from a given representation into another vector of the samerepresentation either (commuting in particular with all Cartan subalgebragenerators, they cannot change their eigenvalues and, therefore, cannot mapone basis vectors of a given representation into another one because theseare distinguished by the eigenvalues of the Cartan subalgebra generators andall vectors can be written as superpositions of these basis vectors). Hence,all the vectors of a given representation must be eigenvectors with the sameeigenvalue of the Racah operators. For this reason the eigenvalues of theRacah operators are used to distinguish different representations (their eigen-values are different on vectors belonging to different representations) of theLie algebra of the symmetry group that are realized on various energy lev-els of the system. A simplifying circumstance is the fact that, as found byCasimir, one of the Racah operators is always given in the form of a bi-liner combination of the generators Qa. It is called the Casimir operator. Inthe case of the rotation group generated by J this Casimir operator is justJ2 = (Jx)2 + (Jy)2 + (Jz)2. Since the SU(2) group is of rank r = 1, this isthe only Racah operator and representations are completely specified by theJ2 eigenvalues.

After these general remarks we return to our considerations of the Galileogroup transformations, using now as nonrelativistic quantum mechanics of asingle particle as an example.

The first one is the symmetry with respect to space translations in quan-tum mechanics of a single particle. Physically the wave function ψ′ (the statevector |ψ′〉) of the translated system and the wave function ψ (|ψ〉) of theoriginal system should be related by |ψ′(r′)| ≡ |〈r′|ψ′〉| = |〈r|ψ〉| ≡ |ψ(r)|,where r′ = r−a. Since, for translation to be symmetry operations this musthold for all possible superpositions of states, we must have ψ′(r′) = ψ(r).Therefore, the unitary operator U(a) realizing translations in the Hilbertspace: |ψ′〉 = U(a)|ψ〉, must be such that

ψ′(r) = 〈r|ψ′〉 = 〈r|U(a)|ψ〉 = U(a)ψ(r) = ψ(r+ a) , (4.57)

where the last U(a) here is understood to be the operator in the positionrepresentation. Proceeding as in the case of time translations we find

U(a) ≈ 1 +i

~a·(−i~∇) = 1 +

i

~a·P ≈ exp

(

i

~a·P

)

, (4.58)

where the fact that [P i, P j] = 0 enabled us to use (4.26) to get U(a) forfinite transformations. This shows that it is the ordinary momentum oper-

76

ator which plays the role of space translations generator. However, if timetranslations are also symmetries of the system then, as we have found, thestructure of the Galileo group requires that H and P commute, which can bethe case only if H is the Hamiltonian of a free particle.17 (This needs not bethe case for systems of many particles interacting one with another by forcesdepending only on ri − rj.) Generalizing to arbitrary systems, we concludethat for time-independent Hamiltonians, space translations are symmetriesof the quantum theory only if the momentum operator P commutes with theHamiltonian. If it does, then the state vector of the transformed system is

|Ψ′(t)〉 = U(a)|Ψ(t)〉 , (4.59)

with the translation operator given by

U(a) = exp

(

i

~a·P

)

, (4.60)

with P the operator of the total momentum of the system.

One can try next to represent boosts. Physically the wave functions ofthe boosted system and of the original one should be related by

|〈r|ψ′(t)〉| ≡ |ψ′(t, r)| = |ψ(t, r+Vt)| = |〈r+Vt|ψ(t)〉| , (4.61)

so that if the boost are symmetries of the system a unitary operator U(V)should exist, such that |ψ′〉 = U(V)|ψ〉 with the property (in the positionrepresentation)

U(V)ψ(t, r) = ψ(t, r+Vt) , (4.62)

which for infinitesimal transformations can be approximated by U(V) ≈1 + (i/~)V iKi. By analogy with (4.57) it is clear that the generators Ki

should have a part of the form Ki ∼ tP i, which ensures such a shift of thespace argument of ψ. However, this is not enough, as one has to satisfy alsothe commutation relations of Ki with H and P i given in (4.45) and (4.48). Itturns out that this is possible only if H is the Hamiltonian of a free particle,18

in which case

Ki = −Mxi + tP i , (4.63)

17In the case of quantum mechanics of a single neutral particle with spin and a nonzeromagnetic moment µ (as e.g. neutron), P commutes with H also if the particle interactswith a uniform, constant external magnetic field through the interaction Hint = −µ·B.

18Or of a neutral particle with nonzero magnetic moment in a constant uniform magneticfield.

77

and its commutation rule with H = P2/2M requires that the mass parameterin (4.48) coincides with the mass of the particle. It is then easy to check that

ψ′(t, r) = exp

(

i

~V i(−Mxi + tP i)

)

ψ(t, r)

= exp

(

−iMV2

2~t

)

exp

(

− i

~MV ixi

)

ψ(t, r+Vt) , (4.64)

satisfies the free Schrodinger equation (check it!).

Finally we consider rotations which most frequently are symmetry trans-formations of the system, even if translations and boosts are not. Considerfirst a spinless nonrelativistic particle. Its wave functions has only one com-ponent (with the usual probabilistic interpretation). The wave function ofthe original and rotated systems should, on physical grounds, be related by|ψ′(r′)| ≡ |〈r′|ψ′〉| = |〈R ·r|ψ′〉| = |〈r|ψ〉| ≡ |ψ(r)|. This equality holds forarbitrary superpositions of the state vectors provided

ψ′(r) ≡ 〈r|ψ′〉 = 〈R−1·r|ψ〉 ≡ ψ(R−1·r) , (4.65)

which in turn must be ensured by the the rotation operator U(R) with hasthe properties such that |ψ′〉 = U(R)|ψ〉 or, in the position representation,that ψ′(r) = U(R)ψ(r). The matrix R of an infinitesimal rotation is of theform (4.38). Defining the vector θ ≡ (θx, θy, θz) we have

R−1 ·r ≈ r+ θ × r , (4.66)

and the right hand side of (4.65) becomes

ψ(r) + (θ × r)·∇ψ(r) + . . . = ψ(r) +i

~(θ × r)·Pψ(r) + . . . (4.67)

or, in the Dirac notation, 〈r|(

1 + i~(θ × r)·P

)

|ψ〉. With a straightforward

vector algebra we get therefore

ψ′(r) ≈(

1 +i

~θ ·L

)

ψ(r) , (4.68)

(where L = r×P is the familiar orbital angular momentum operator), thatis,

|ψ′〉 = U(R)|ψ〉 ≈(

1 +i

~θ ·L

)

|ψ〉 . (4.69)

Obviously, if rotations are symmetries of the physical system the operatorsL and, hence, also U(R), commute with the Hamiltonian H .

78

If the particle has a nonzero spin, its wave function has more compo-nents labeled by an index σ. In the appropriate basis in the internal space|ψσ(r)|2 ≡ |〈r, σ|ψ〉|2 is the probability density of finding the particle at r withspin projection in the z-direction equal σ~, i.e. of finding it in the state |r, σ〉(strictly speaking this interpretation is valid only in nonrelativistic quan-tum mechanics). On physical grounds, and generalizing (4.65), if rotationsare symmetries of the system, the wave function components ψσ transformunder rotations one into another forming a linear, finite-dimensional (andunitary - because the rotation group is compact) matrix representation ofthe rotation group. In other words, one must then have

〈r, σ|ψ′〉 =j

∑

σ′=−j

Dσσ′(R) 〈R−1·r, σ′|ψ〉 or

ψ′σ(r) =

j∑

σ′=−j

Dσσ′(R)ψσ′(R−1·r) . (4.70)

Expanding (4.70) to the first order in θi and using the formula (4.66) togetherwith the form

D(s)σσ′(θ) ≈ δσσ′ +

i

~θiS

(s)iσσ′ + . . . (4.71)

of theD(s)σσ matrices (for spin s) valid for infinitesimal rotationsR parametrized

by three angles θi one finds in this case that the Hilbert space generators Jof the rotation (sub)group

U(θ) ≈ 1 +i

~θiJ i , (4.72)

such that U(θ)|ψ〉 = |ψ′〉 must have the form

J i = Li + Si , (4.73)

with Li acting on the space variable r and Si acting on the discrete variablesσ. Thus, [Li, Sj] = 0 and in order to satisfy the commutation relation[J i, J j] = i~ ǫijkJk one must also have

[Si, Sj] = i~ ǫijkSk . (4.74)

J i’s are simply the operators of the total angular momentum split (in nonrel-ativistic quantum mechanics) into the orbital angular momentum operatorsLi and the spin operators Si. The latter, being matrices in the spin indicesσ must form a finite dimensional, unitary representation of the commutationrelations (4.74) which are the commutation relations of the SU(2) group. Allsuch representations are well known. They are labeled by s = 0, 1

2, 1, 3

2, 2 . . .

79

and have dimension 2s + 1. In standard textbooks one finds the matricesSiσσ′ of the group generators in these representations by solving explicitly thecommutation relations (4.74). One then finds that for s = 1

2the matrices Si

are the familiar Pauli matrices Siσσ′ =12σiσσ′ and for s = 1

Sx =~√2

0 1 01 0 10 1 0

, Sy =~√2

0 −i 0i 0 −i0 i 0

, Sz = ~

1 0 00 0 00 0 −1

,

and so on.

In the next section we will discuss the realization of the rotation groupin the Hilbert space in more details.

4.4 The rotation group

In many applications it is the rotation group which is a good symmetryof the nonrelativistic systems. Moreover, rotations form also an importantsubgroup of the Lorentz group relevant for relativistic systems. It is thereforeworthwhile to complete with some details its discussion begun in section 4.2and to derive some useful formulae.

We first chose the parametrization proposed in connection with the dis-cussion of the double-connectedness of the rotation group in subsection 4.2.In this parametrization the rotation is specified by giving the direction n ofthe rotation axis and the rotation angle 0 ≤ ψ ≤ π. In turn, the directionn is specified by giving its polar angles 0 ≤ ϑ ≤ π and 0 ≤ ϕ ≤ 2π. Thisparametrization (the map on the group manifold) is not globally defined: forψ = 0 the angles ϑ and ϕ are unspecified. Alternatively, one can specifythe three Cartesian coordinates (ψx, ψy, ψz) of the rotation vector ψ ≡ ψ n.These coordinates are restricted to the domain |ψ| ≤ π. This map is well de-fined everywhere, except the for points such that |ψ| = π because, as alreadyexplained, ψ and −ψ define then the same rotation.

In this parametrization the most natural “standard” path along whichone defines the Hilbert space operator U(ψ) corresponding to a given rota-tion R(ψ) is the path ψ(s) = sψ with 0 ≤ s ≤ 1. It defines a one-parameterAbelian subgroup of rotations, so that integration of the appropriate differ-ential equation reduces to the formula (4.26) and one gets

U(ψ) = exp

(

− i

~ψ ·J

)

, (4.75)

where J i are the Hilbert space rotation group generator. In agreement withour discussion (around eq. (4.54)), if rotations are symmetries of the quantum

80

system, one can find in the Hilbert space the states |n, j,m〉 (the label ndistinguishing e.g. the Hamiltonian eigenvalues; it will be omitted in thefollowing), which under rotations transform one into another

U(ψ)|j,m〉 ≡ e−i~ψiJi|j,m〉 =

+j∑

m′=−j

|j,m′〉D(j)m′m(ψ) . (4.76)

Closing this relation from the left with 〈j,m′| (assuming that these statesare normalized to unity) we get the rotation matrices

D(j)m′m(ψ) = 〈j,m′|e− i

~ψiJi|j,m〉 , (4.77)

in the ψ-parametrization. Using matrix elements of the generators J i (theyare obtained in standard textbook by solving the algebra of J i’s in the basisin which Jz is diagonal) given by

(

Jz(j))

m′m≡ 〈j,m′|Jz|j,m〉 = m~ δm′,m ,

(

Jx(j) ± iJy(j)

)

m′m≡ 〈j,m′|Jx ± iJy|j,m〉 (4.78)

= ~

√

(j ∓m)(j ±m+ 1) δm′m±1

= ~

√

j(j + 1)−m(m± 1) δm′m±1 ,

the matrices D(j)m′m(ψ) can be obtained by exponentiating the matrix ele-

ments(

ψiJ i(j)

)

m′mof the generators J i contracted with the components of

the rotation vector ψ. For j = 12one finds e.g. in this way (ψ ≡ |ψ|)

D(1/2)m′m (ψ) = cos

ψ

2− in·σ sin

ψ

2. (4.79)

Another parametrization of the rotation group is provided by the Eulerangles α, β and γ. A given rotation R(α, β, γ) is in this parametrizationobtained by composing first a rotation by the angle γ around the z axis, thena rotation by β around the new y axis and finally a rotation by α around thenew z axis. The ranges of these angles are: 0 ≤ γ, α < 2π and 0 ≤ β ≤ π.Also this parametrization is not globally defined: for β = 0, when the newand the old z-axes coincide, the rotation depends only on γ + α (modulo2π). The rotation inverse to R(α, β, γ) is given by R(π−γ, β, π−α). In thisparametrization it is natural to chose for the “standard path”, along whichone ascribes to a given rotation R(α, β, γ) the corresponding Hilbert operatorU(α, β, γ), the path id→ Rz(γ) → Ry(β) · Rz(γ) → Rz(α) · Ry(β) · Rz(γ).The operator U is then given by

U(α, β, γ) = exp

(

− i

~αJz

)

exp

(

− i

~βJy

)

exp

(

− i

~γJz

)

. (4.80)

81

It is important to note however, that (as can be easily verified)

U(α, β, γ)U(π − γ, β, π − α) = e−i2πJz

. (4.81)

which is the unit operator only on the states with integer j.

The rotation matrices D(j)m′m(α, β, γ) take in this parametrization the form

D(j)m′m(α, β, γ) ≡ 〈j,m′|e− i

~αJz

e−i~βJy

e−i~γJz |j,m〉

= e−im′α d

(j)m′m(β) e

−imγ , (4.82)

where the matrices d(j)m′m(β) are defined by the formula

d(j)m′m(β) ≡ 〈j,m′|e− i

~βJy |j,m〉 . (4.83)

As in the ψ-parametrization, since the matrix elements of Jy betweenthe |j,m〉 states are known, the matrices d

(j)m′m(β) can be obtained just by

exponentiating the matrices(

βJy(j)

)

m′m. For j = 1

2with Jy(j) = ~

2σy the

matrix d(1/2)m′m (β) is easy to find and reads

d(1/2)(β) =

(

cos β2

sin β2

− sin β2

cos β2

)

. (4.84)

For higher j the matrices d(j)m′m(β) can be found straightforwardly using the

trick presented in Appendix A, where the properties of the d(j)m′m(β) and

D(j)m′m(α, β, γ) matrices are also collected.

An important concept (allowing to formulate e.g. the orthogonality re-

lations satisfied by the D(j)m′m(α, β, γ) matrices) is the integration over the

group of a function f(g) of the group element g. For compact groups G, likethe rotation group, the integral which is both left- and right- invariant canbe defined. This means the following property

∫

dµ(g) f(g) =

∫

dµ(g) f(g′g) =

∫

dµ(g) f(gg′) , (4.85)

where dµ(g) is the appropriate measure and g′ is an arbitrary element of G.In a given parametrization of the group elements by continuous parametersθa, a = 1, . . . , n, the integral is given by

∫

dµ(g) f(g) =

∫

dnθ ρ(θ)f(θ) , (4.86)

82

where ρ(θ) is some density determined by the requirement that the integralis left- and right- invariant. For example, left-invariance means that thefollowing equality must hold

∫

dnθ ρ(θ)f(θ(θ′, θ)) =

∫

dnθ ρ(θ) f(θ) , (4.87)

with θa = ha(θ′, θ), where ha is the composition function introduced in sub-section 4.2. After changing in the right integral the integration variables toθ, (4.87) is equivalent to

ρ(θ) = ρ(h(θ′, θ))

∣

∣

∣

∣

∂ha(θ′, θ)

∂θb

∣

∣

∣

∣

. (4.88)

The right-hand side must be of course independent of θ′. Assuming that adensity ρ fulfilling this condition exists, one can determine it by consideringonly infinitesimal values of the parameters θa → 0. Evaluating (4.88) forθa = 0 gives then

ρ(θ′) = ρ(0)

∣

∣

∣

∣

∂ha(θ′, θ)

∂θb

∣

∣

∣

∣

−1

θ=0

. (4.89)

This fixes ρ up to an inessential multiplicative constant ρ(0).

In the parametrization of the rotations by the vector ψ = (ψx, ψy, ψz)one finds in this way (normalizing the measure so that in this particularparametrization ρ(0) = 1)

dψxdψydψz ρ(ψ) = dψxdψydψz2(1− cos |ψ|)

|ψ|2≡ d|ψ| dΩψ 2(1− cos |ψ|) . (4.90)

The volume of the rotation group SO(3) is then

Vol(SO(3)) =

∫ 2π

0

dϕ

∫ π

0

dϑ sinϑ

∫ π

0

d|ψ| 2(1− cos |ψ|) = 8π2 . (4.91)

For the parametrization by the Euler angles one finds

dα dβ dγ ρ(α, β, γ) = dα dβ dγ sin β , (4.92)

and the rotation group volume is given by

∫ 2π

0

dα

∫ π

0

dβ

∫ 2π

0

dγ sin β = 8π2 . (4.93)

83

For a function defined genuinely on the SU(2) group instead of SO(3), whichmeans that it takes different values on those SU(2) elements which corre-spond to the same element of SO(3),19 the range of integration over ψ in thefirst parametrization is (0, 2π); the SU(2) group volume is 16π2.

The matrices D(j)m′m(α, β, γ) can be viewed as a collection of (2j+1)2 func-

tions defined on the group and can therefore be integrated over it. Moreover,the product of two such matrices D

(j1)m′

1m1

and D(j2)m′

2m2

with j1 and j2 both in-

teger or both half integer (the case which in view of the superselection rule isthe only interesting one) is a single-valued function and can be, therefore, in-tegrated over SO(3). Their product satisfies then the following orthogonalityrelation∫ 2π

0

dα

∫ π

0

dβ

∫ 2π

0

dγ sin β D(j2)∗m′

2m2

(α, β, γ)D(j1)m′

1m1

(α, β, γ)

=8π2

2j + 1δj1j2δm1m2

δm′

1m′

2, (4.94)

which exemplifies the general orthogonality relation satisfied by matrices offinite dimensional unitary representations of a compact group: the coefficientof the delta functions is always given by the volume of the group divided bythe dimension of the representation. The completeness relation reads

1

8π2

∑

j

+j∑

m=−j

+j∑

m′=−j

(2j + 1)D(j)∗m′m(α, β, γ)D

(j)m′m(α, β, γ)

= δ(α− α)δ(cos β − cos β)δ(γ − γ) . (4.95)

Since the dependence of theD(j) functions on the angles α and γ is trivial,the integrals over these angles can be taken explicitly giving the relation

∫ π

0

dβ sin β d(j′)∗m′m(β) d

(j)m′m(β) =

2

2j + 1δj

′j . (4.96)

The appropriate inverse relation is

∑

j

(j +1

2) d

(j)∗m′m(β) d

(j)m′m(β) = δ(cos β − cos β) . (4.97)

4.5 Composing angular momenta

We recall first some general features of group representations. Consider twothree-dimensional vectors V i and W i. Under an arbitrary rotation R the

19In some textbooks such functions are termed double-valued because from the point ofview of SO(3) they assume two different values on every its element.

84

vectors are transformed into V ′i andW ′i given by V ′i =∑

j RijV j andW ′i =

∑

j RijW j with Rij the corresponding orthogonal 3×3 matrix. This obviously

determines also the transformation law of the nine components T ij = V iW j

of the tensor formed out of these two vectors:

T ′ij =∑

(kl)

[M](ij),(kl)T kl ,

with the elements of the 9×9 matrix [M](ij),(kl) given by [M](ij),(kl) = RikRjl.It is however possible to make a linear change of the basis in the 9 dimensionalspace of the tensor components:

T kl =∑

(mn)

[O](kl),(mn) Tmn , T ′ij =∑

(mn)

[O](ij),(mn) T ′mn ,

so that

T ′ij =∑

(kl)

[M](ij),(kl)T kl ,

with

[M](ij),(kl) =∑

(mn)

∑

(rs)

[O](mn),(ij) [M](mn),(rs) [O](rs),(kl) . (4.98)

The change of the basis can be chosen in such a way that for all rotations the9 × 9 matrices [M] (which have all the properties necessary for a represen-tation of the rotation group) take on the block-diagonal structure with one1×1 block, one 3×3 block and one 5×5 block. Therefore the 9-dimensionalrepresentation of the rotation group spanned by the 9 components V iW j

is reducible and decomposes into three irreducible representations: one 1-dimensional, one 3-dimensional and one 5-dimensional. The elements of theorthogonal matrix O of the change of basis are what in the context of theSU(2) group covering the rotation group is called the Clebsch-Gordan coef-ficients. The block-diagonal structure of the matrices [M] means that afterthe linear change of the basis the nine components of the tensor are groupedinto a one component which does not change under rotations, 3 componentswhich transform among themselves and 5 other components which also trans-form among themselves. (In the case of the tensor V iW j these 1, 3 and 5component objects correspond, of course, to the scalar V ·W, the antisym-metric tensor Aij = V iW j−V jW i which has 3 independent components andthe symmetric traceless tensor Sij = V iW j + V jW i − 2

3(V·W)δij which has

5 independent components). This illustrates a general rule: a representation(of any group) formed by taking the tensor product of two (irreducible) rep-resentations is in general reducible and can be decomposed into irreducibleones by an appropriate change of the basis.

85

Basically the same can be repeated in the Dirac notation for the eigen-states of the angular momentum operators. In this case we have state vectorsof the general form |n, j1, m1, j2, m2〉 depending on two20 different sets of la-bels, j1, m1 and j2, m2, of which the first set forms a complete representationof the su(2) Lie algebra generated by the operators J1 and the second set hasthe same property with respect to the algebra generated by J2. An exampleis provided by quantum mechanics of a particle with a nonzero spin (see theformulae (4.70)) whose state can be written as a superposition

|ψ〉 =∑

n

∑

l=0

+l∑

ml=−l

+s∑

σ=−s

cnlmlσ|n, l,ml, s, σ〉 , (4.99)

of the states |n, l,ml, s, σ〉 (with fixed s corresponding to the particle’s spin)such that

ei~ψiLi |n, l,ml, s, σ〉 =

+l∑

m′

l=−l

|n, l,m′l, s, σ〉D(l)

m′

lml(ψ) ,

ei~ψiSi|n, l,ml, s, σ〉 =

+s∑

σ′=−s

|n, l,ml, s, σ′〉D(s)

σ′σ(ψ) , (4.100)

with the matrices D(j)m′

lml

introduced in subsection 4.4. In quantum mechanics

of many particles, nonrelativistic (see section 5) or relativistic, the two sets ofstates may represent e.g. angular momentum states of two different spinlessparticles. More complicated cases of angular momenta of more then twoparticles and/or of two or more particles with nonzero spins can be treatedby applying successively the rules formulated in this subsection for two setsof states.

Obviously, in the example of a single particle with spin J1 = L and J2 = S

act on different sets of labels. Mathematically the states |j1, m1; j2, m2〉 havethe structure of the tensor product

|j1, m1; j2, m2〉 ≡ |j1, m1〉 ⊗ |j2, m2〉 , (4.101)

and the operators J1 act on |j1, m1〉 while J2 act on |j2, m2〉. The basis ofstates (4.101) called the product basis (of the subspace of the Hilbert spacecorresponding to fixed label n) is formed by the states which under the SU(2)transformations behave as an SU(2) representations constructed as the tensorproduct of two irreducible SU(2) representations: one characterized by j1 and

20n labels the state vectors with respect e.g. to their Hamiltonian eigenvalues; as we willbe now considering state vectors with the same value of n this label will be not explicitlyindicated.

86

the second one characterized by j2. As in the example with two vectors, it ispossible to change the basis in the (sub)space spanned by the vectors (4.101),so that this representation of SU(2) explicitly decomposes into a direct sum ofirreducible representations, which (as all irreducible SU(2) representations)are characterized by integers j. This change of basis is written (in the Diracnotation) as

|j1, m1; j2, m2〉 =∑

j

j∑

m=−j

|j,m〉〈j,m|j1, m1; j2, m2〉 , (4.102)

where |j,m〉 are the eigenstates of J2 = (J1 + J2)2 with the eigenvalues

j(j + 1)~2 and of Jz = Jz2 + Jz1 with the eigenvalues m~.

The scalar products on the right hand side are just the (complex conjugateof the) Clebsch-Gordan coefficients:

Cj2j1(m2, m1|j,m) ≡ 〈j1, m1; j2, m2|j,m〉 . (4.103)

The following reasoning tells us which representations j appear in the sum(4.102) for a given pair j2 and j1, i.e. determines for which j the Clebsch-Gordan coefficients are nonzero. As will become clear, in the decompositionof into irreducible representations of a tensor product of any two SU(2)representations each representation with a given j can appear at most once.Acting with the operator Jz = Jz2 + Jz1 on both sides of (4.102) we get

(Jz2 + Jz1 )|j2, m2; j1, m1〉 =∑

j

+j∑

m=−j

Jz|j,m〉〈j,m|j2, m2; j1, m1〉 .

Action of Jzz+Jz2 (of Jz) on the left (right) hand side reduces to multiplication

by m2+m1 (m) and expressing next the state |j2, m2; j1, m1〉 on the left handside using (4.102) we arrive at the relation

∑

j

+j∑

m=−j

(m−m2 −m1)|j,m〉〈j,m|j2, m2; j1, m1〉 = 0 , (4.104)

which, in view of the completeness of the |j,m〉 states, means that in (4.102)only the terms with m = m2 +m1 can appear. In other words, the Clebsch-Gordan coefficients (4.103) vanish for m 6= m2 + m1. This expresses theobvious fact that the z-axis projection of the total angular momentum of thesystem is the sum of the z-axis projections of the angular momenta of itscomponents. This enables us to find the representations |j,m〉 by comparingthe numbers of states with the same value of m = m2+m1 in the two bases.

As the maximal values of m1 and m2 on the left hand side of (4.102)equal j1 and j2, respectively, it follows that the maximal value of m on the

87

right hand side of (4.102) is j2 + j1. Hence, the maximal j in the sum mustalso be equal j2 + j1. Consider now m = j2 + j1 − 1. In the original productbasis (4.101) there are two states corresponding to this value of the z-axisprojection of the total angular momentum: |j2, j2−1; j1, j1〉 and |j2, j2; j1, j1−1〉. After changing the basis one their linear combination becomes the state|j = j2 + j1, m = j2 + j1 − 1〉 and combines with the state |j = j2 + j1, m =j2 + j1〉 to form the j = j2+ j1 representation. The other linear combinationmust be the state |j = j2 + j1 − 1, m = j2 + j1 − 1〉 and it is the state ofmaximal z-axis spin projection of the representation with j = j2 + j1 − 1.It is easy to see, that this process continues so long as lowering m by oneunit increases the number of states with the new value of m: at each step anew representation with j lower by one unit appears. In order to see whenthis process stops, or, in other words, what is the minimal value of j, wecan simply compare the total number of states. In the original product basis(4.101) there are (2j2+1)(2j1+1) states. This number must equal the totalnumber of the |j,m〉 states in the irreducible representations found in thedecomposition of the direct product. Thus

(2j2 + 1)(2j1 + 1) =

j=j2+j1∑

j=?

(2j + 1) , (4.105)

(we already know the maximal value of j). Since on the right hand sidej = j2 + j1, j2 + j1 − 1, . . ., it is easy to find that the lowest possiblej = |j2 − j1|. Hence, when two angular momenta j2 and j1 are composed,one gets all irreducible representations |j,m〉 with

|j2 − j1| ≤ j ≤ j2 + j1 , (4.106)

and j changing in unit steps. It follows, that the sum over j in (4.102)extends to such values of j only, or, equivalently, that

Cj2j1(m2, m1|j,m) 6= 0 only for

m = m1 +m2

|j1 − j2| ≤ j ≤ j1 + j2. (4.107)

Using the unitarity of the Clebsch-Gordan coefficients (being the matrixof the change of basis, they must be unitary)

j1+j2∑

j=|j1−j2|

+j∑

m=−j

Cj2j1(m2, m1|j,m)C∗j2j1

(m′2, m

′1|j,m) = δm1m′

1δm2m′

2,

+j2∑

m2=−j2

+j1∑

m1=−j1

Cj2j1(m2, m1|j,m)C∗j2j1

(m2, m1|j′, m′) = δjj′δmm′ , (4.108)

88

the relation (4.102) can be inverted to give

|j,m〉 =+j1∑

m1=−j1

+j2∑

m2=−j2

|j2, m2; j1, m1〉Cj2j1(m2, m1|j,m) . (4.109)

Acting on both sides of (4.109) with J± = (Jx2 + Jx1 )± i(Jy2 + Jy1 ) and usingagain the completeness of the states |j1, m1; j2, m2〉 we can obtain usefulrelations between the Clebsch-Gordan coefficients:

√

j(j + 1)−m(m± 1)Cj2j1(m2, m1|j,m± 1)

=√

j2(j2 + 1)−m2(m2 ∓ 1)Cj2j1(m2 ∓ 1, m1|j,m) (4.110)

+√

j1(j1 + 1)−m1(m1 ∓ 1)Cj2j1(m2, m1 ∓ 1|j,m) .

From the construction it is also obvious that the Clebsch-Gordan coefficientscan be chosen so that

Cj2j1(j2, j1|j = j1 + j2, m = j1 + j2) = 1 . (4.111)

This condition plus the recurrence relations (4.110) allow to determine com-pletely (up to some arbitrary phase factors) the factors Cj2j1(m2, m1|j,m).By convention arbitrary phase factors are chosen so that all the Clebsch-Gordan coefficients are real.

Clebsch-Gordan coefficients allow also to write down a decomposition ofa product of two D-functions defined in (4.82) (see also (4.76) and (4.77)).Acting on both sides of the relation

|j1, m1〉 ⊗ |j2, m2〉 =j1+j2∑

j=|j1−j2|

|j,m1 +m2〉C∗j1j2

(m1, m2|j,m1 +m2) , (4.112)

with a rotation operator U(R) and expanding both sides one gets the relation∑

m′

1

∑

m′

2

|j1, m′1〉 ⊗ |j2, m′

2〉D(j1)m′

1,m1D

(j2)m′

2,m2

=

j1+j2∑

j=|j1−j2|

∑

m′

|j,m′〉D(j)m′,m1+m2

C∗j1j2

(m1, m2|j,m1 +m2) .

Closing now this relation from the left with the conjugation of the relation(4.112) one obtains the equality

D(j1)m′

1,m1D

(j2)m′

2,m2

=

j1+j2∑

j=|j1−j2|

Cj1j2(m1, m2|j,m1 +m2)

Cj1j2(m′1, m

′2|j,m′

1 +m′2)D

(j)m′

1+m′

2,m1+m2

. (4.113)

89

Using the relation D(j)m′m = (−1)m−m′

D(j)∗−m′,−m this can be rewritten also in

the alternative form

D(j1)m′

1,m1D

(j2)∗m′

2,m2

= (−1)m2−m′

2

j1+j2∑

j=|j1−j2|

Cj1j2(m1,−m2|j,m1 −m2)

Cj1j2(m′1,−m′

2|j,m′1 −m′

2)D(j)m′

1−m′

2,m1−m2

. (4.114)

As an illustrative example of the use of the Clebsch-Gordan coefficients letus consider the Hydrogen atom taking into account the electron spin. We firsttreat the problem in nonrelativistic quantum mechanics. Since both, L andS (where the spin operators Si = (~/2)σi) commute with the HamiltonianH = P2/2M − e2/r, the basis of states at a given energy level n is formedby the states

|n, l,m〉 ⊗ |s, σ〉 , − l ≤ m ≤ l , s =1

2, σ = ±1

2. (4.115)

They are eigenstates of L2, Lz, S2 and Sz with the eigenvalues l (l + 1)~2,m~, 3

4~2 and σ~, respectively. In contrast to H , the Dirac Hamiltonian

HD, which describes the Hydrogen atom in relativistic quantum mechanicscommutes only with the total angular momentum operators J = L + S

but not with L and S separately. Hence, in the nonrelativistic limit theeigenfunctions of HD do not reduce to the states (4.115), but rather to theirlinear combinations which are eigenfunctions of H , Jz and J2: rotationalinvariance requires exact degeneracy of groups of states with the same J2 onlyand one should expect that the first order relativistic corrections will splitenergies of those groups of eigenstates of the nonrelativistic Hamiltonian Hwhich correspond to different values of the J2 operator.21 The wave functionsof these states could be, of course, constructed just by taking the Clebsch-Gordan coefficients from tables. Instead, we will find them here by solvingdirectly the appropriate equations. In so doing we will also obtain explictlysome of the Clebsch-Gordan coefficients.

We begin by noticing that since Jz = Lz+Sz, its eigenstates correspond-ing to the eigenvalue jz~ must necessarily have the form

|n, l, s, j, jz〉 = A|n, l, jz − 12〉 ⊗ |s, 1

2〉+B|n, l, jz + 1

2〉 ⊗ |s, −1

2〉 ,

21The fact that the spectrum of the nonrelativistic Hydrogen atom exhibits large de-generacy, not required by the rotational invariance generated by the L operator (extradegeneracy of states with different values of the orbital angular momentum l) is due toan extra dynamical symmetry discussed in subsection 4.7. Moreover, even the spectrumof the full Dirac Hamiltonian HD exhibits some extra degeneracy not required by therotational invariance. Consequently, this degeneracy is preserved also by the lowest orderrelativistic corrections to the energy levels. This extra degeneracy is removed only bycorrections to the energy levels predicted by the full Quantum Electrodynamics.

90

with some constants A and B. In position representation ψσ(r) with theupper (σ = +1

2) component representing the projection onto the |s, 1

2〉 spin

state, the wave function takes the form

ψ(jjz)σ (r) =

(

ARnl(r)Yljz− 1

2

(θ, φ)

BRnl(r)Yljz+ 1

2

(θ, φ)

)

,

where Rnl are the nonrelativistic radial wave functions and Ylm are the spher-ical harmonics. The coefficients A and B should be such that ψ

(jjz)σ (r) is the

eigenfunction of J2 with the eigenvalue j(j + 1)~2. Using the decomposition

J2 = (L + S)2 = L2 + S2 + 2LzSz + L+S− + L−S+ ,

where

Sz =~

2

(

1 00 −1

)

, S+ = ~

(

0 10 0

)

, S− = ~

(

0 01 0

)

,

we get

J2 =

(

L2 + S2 + ~Lz ~L−

~L+ L2 + S2 − ~Lz

)

.

In action on ψ(jjz)σ one can replace S2 by 3

4~2. Moreover, the action of L2,

L± and Lz on the spherical harmonics Ylm is also known. The eigenequation

J2ψ(jjz)σ = j(j + 1)ψ

(jjz)σ therefore reduces to

~2

(

[l(l + 1) + 34+ jz − 1

2− j(j + 1)]AYljz− 1

2

+ pB Yljz− 1

2

pAYljz+ 1

2

+ [l(l + 1) + 34− jz − 1

2− j(j + 1)]B Yljz+ 1

2

)

= 0 ,

where p ≡√

l(l + 1)− (jz +12)(jz − 1

2) =

√

(l + 12)2 − j2z . Dividing the up-

per (lower) equation by Yljz− 1

2

(Yljz+ 1

2

) we obtain

(

[l(l + 1) + 14+ jz − j(j + 1)] pp [l(l + 1) + 1

4− jz − j(j + 1)]

)(

AB

)

= 0 ,

For a nontrivial solution to exist, the determinant of the matrix on the lefthand side should vanish:

[

l(l + 1) +1

4− j(j + 1)

]2

−(

l +1

2

)2

= 0 ,

(notice that the determinant does not depend on jz!). Thus, either j = l− 12

or j = l + 12. For example, for j = l − 1

2the coefficients A and B are related

by

A

(

l +1

2+ jz

)

+B

√

(

l +1

2

)2

− j2z = 0 .

91

Hence, for j = l − 12one can take

A = N

√

l +1

2− jz , B = −N

√

l +1

2+ jz ,

with N fixed by the normalization of the wave function

1 =

∫

d3r

(

∣

∣

∣ψ

(jjz)1

2

∣

∣

∣

2

+∣

∣

∣ψ

(jjz)

− 1

2

∣

∣

∣

2)

= |A|2 + |B|2 = |N |2(2l + 1) ,

where we have assumed that the wave functions Rnl(r) Yljz± 1

2

are properlynormalized. Thus, we finally find

ψ(j=l− 1

2, jz)

σ =

√

l+ 1

2−jz

2l+1Rnl(r)Yljz− 1

2

(θ, φ)

−√

l+ 1

2+jz

2l+1Rnl(r)Yljz+ 1

2

(θ, φ)

,

and, similarly, or just by noticing that the two functions ψ(j=l− 1

2, jz)

σ and

ψ(j=l+ 1

2, jz)

σ must be orthogonal to each other,

ψ(j=l+ 1

2, jz)

σ =

√

l+ 1

2+jz

2l+1Rnl(r)Yljz− 1

2

(θ, φ)√

l+ 1

2−jz

2l+1Rnl(r)Yljz+ 1

2

(θ, φ)

.

Comparing these explicit solutions with the general formula

|n, l, s, j, jz〉 =+l∑

m=−l

∑

σ=± 1

2

|n, l,m〉 ⊗ |s, σ〉Cl 12

(m, σ|j, jz) ,

we can read off the explicit expressions of nonzero Clebsch-Gordan coeffi-cients coupling the angular momenta l and s = 1

2:

Cl 12

(jz − 12, +1

2|l ∓ 1

2, jz) =

√

l + 12∓ jz

2l + 1,

Cl 12

(jz +12, −1

2|l ∓ 1

2, jz) = ∓

√

l + 12± jz

2l + 1.

The C-G coefficients obtained in this way could, of course, differ by a phasefactor as compared to those listed in standard tables. It is however easy tocheck that for jz = l + 1

2we have obtained Cl 1

2

(jz − 12, 12|l + 1

2, jz) = 1, as in

the the standard convention.

92

4.6 Tensor operators and the Wigner-Eckart theorem

The physical information in quantum mechanics is contained in matrix ele-ments of various operators and not in state vectors and operators separately.It is therefore possible to introduce a sort of the “Heisenberg” picture forsymmetry transformations by defining the transformed operators O′ by theequality

〈χ|O′|ψ〉 = 〈χ′|O|ψ′〉 , (4.116)

that is, so that the matrix elements of the transformed operators O′ betweenthe original states are the same as the matrix elements of the original opera-tors between the transformed states. Since |ψ′〉 = U |ψ〉 with the appropriatesymmetry operator U , one gets

O′ = U †OU ≈ O + iθa [Qa, O] , (4.117)

where in the last step we have assumed that the transformation is infinites-imal. In agreement with the intuition, in quantum mechanics of a singleninrelativistic particle, for translations the formula (4.117) gives

r′ = U †(a) rU(a) = r− a , (4.118)

whereas for rotations one finds

r′ = U †(R) rU(R) = R·r . (4.119)

Classification of operators with respect to their behaviour under rotationsproves very useful. It is precisely this property property that determines theselection rules for their various matrix elements. This leads us naturally toconsider the so-called tensor operators, i.e. sets of operators which underrotations transform one into another, and the Wigner-Eckart theorem.

From the formulae (4.117) and (4.72) it follows that for infinitesimal ro-tations

O′ = O − i

~θi[

J i, O]

, (4.120)

that is, the behaviour of the operator O under rotations is determined by itscommutators with the rotation generators J i, i.e. the operators of the totalangular momentum. One defines therefore the tensor operator T (j,m) or T

(j)m

of the type j as the set of 2j+1 operators having the following commutationrules with J:

[

Jz, T (j)m

]

= m~T (j)m ,

[

J+, T(j)m

]

= ~

√

j(j + 1)−m(m+ 1) T(j)m+1 , (4.121)

[

J−, T(j)m

]

= ~

√

j(j + 1)−m(m− 1) T(j)m−1 .

93

The above formulae can succinctly be rewritten in the form

[

J i, T (j)m

]

=

+j∑

m′=−j

T(j)m′ 〈j,m′|J i|j,m〉 , (4.122)

Using this form and the formula (4.75) the rule (4.117) of the transformationof the tensor operator under a rotation can be cast in the form

e−i~ψiJi

T (j)m e

i~ψiJi

= T (j)m − i

~

[

ψ ·J, T (j)m

]

+1

2

(

− i

~

)2[

ψ ·J,[

ψ ·J, T (j)m

]]

+ . . .

= T (j)m − i

~

∑

m′

T(j)m′

(

ψiJ i(j))

m′m

+1

2

(

− i

~

)2∑

m′′

∑

m′

T(j)m′′

(

ψiJ i(j))

m′′m′

(

ψiJ i(j))

m′m+ . . .

In this way we get the transformation rule

U †(R)T (j)m U(R) =

+j∑

m′=−j

T(j)m′ 〈j,m′|U †(R)|j,m〉

=

+j∑

m′=−j

T(j)m′ D

(j)m′m(R

−1) . (4.123)

valid for all parametrizations of the rotation R.

The operators T(j)m are the analogs of the irreducible representations of

the SU(2) rotation group on state vectors; operators of the type T(j)m behave

as if they had the angular momentum j. Indeed, it follows from the definition(4.121) that22

∑

i=x,y,z

[

J i,[

J i, T (j)m

]]

= j(j + 1)~2 T (j)m . (4.124)

The simplest example are, of course, scalar operators which commute withJ: they are the T

(0)0 tensor operators. Another example are vector operators

V i which, similarly as the momentum operator P i or the boost operator Ki

in (4.45), satisfy

[

J i, V j]

= i~ǫijkV k . (4.125)

22The formula (4.124) is the direct analog of the formula J2|j,m〉 = j(j + 1)~2|j,m〉 if

one takes into account that the action of the generators J i on tensor operators is throughthe commutator.

94

Vector operators V i are the tensor operators of the type T(1)m :

T(1)1 = − 1√

2(V x + iV y) ,

T(1)0 = V z , (4.126)

T(1)−1 =

1√2(V x − iV y) .

Let us consider now the following combination of two different tensoroperators of the type T

(j1)m1

and T(j2)m2

:

T (j)m ≡

+j2∑

m2=−j2

+j1∑

m1=−j1

T (j2)m2

T (j1)m1

Cj2j1(m2, m1|j,m) . (4.127)

Using the standard identity

[J, O2O1] = O2 [J, O1] + [J, O2]O1 , (4.128)

and the relations (4.110) it is possible to show that T(j)m defined by (4.127)

is indeed a tensor operator of the type T(j)m . Similarly, using the unitarity

relation (4.108) one can write

T (j2)m2

T (j1)m1

=

j1+j2∑

j=|j1−j2|

+j∑

m=−j

Cj2j1(m2, m1|j,m)T (j)m , (4.129)

which shows that the product of two tensor operators can be decomposedinto the sum of irreducible tensor operators.

The formula most important in applications is

|j2, m2〉b ≡+j∑

m=−j

+j1∑

m1=−j1

T (j)m |j1, m1〉Cjj1(m,m1|j2, m2) . (4.130)

In the manner similar as (4.127), but by using instead of (4.128) the relation

JO|ψ〉 = [J, O] |ψ〉+O J|ψ〉 ,

the states |j2, m2〉b can be shown to be eigenstates of J2 and Jz with theeigenvalues j2(j2+1)~2 and m2~, respectively. It is important, however, thatin general the states |j2, m2〉b are not properly normalized (by definition thestates |j2, m2〉 are normalized states):

〈j2, m2|j2, m2〉b = N (j2, j, j1, T ) 6= 1 . (4.131)

95

On the other hand, because the states |j2, m2〉b with higher m2 can be ob-tained from the ones with lower m2 by acting on them successively withJ+, their normalization does not depend on m2. Therefore, as indicated in(4.131) the normalization factor N (j2, j, j1, T ) depends only on j2, j1 and onthe operator T and its type j (but not on the operator component labeledby m).23

Inverting the relation (4.130) using the unitarity relations (4.108) of theClebsch-Gordan coefficient we get

T (j)m |j1, m1〉 =

j+j1∑

j2=|j−j1|

+j2∑

m2=−j2

|j2, m2〉b Cjj1(m,m1|j2, m2) . (4.132)

Closing this relation from the left with a properly normalized state 〈j2, m2|we finally obtain

〈j2, m2|T (j)m |j1, m1〉 = Cjj1(m,m1|j2, m2)N (j2, j, j1, T ) . (4.133)

This is the content of the Wigner-Eckart theorem which says that the matrixelements of a tensor operator T

(j)m between properly normalized eigenstates

of the total angular momentum operators J2 and Jz are determined, up to asingle number N (j2, j, j1, T ), by the Clebsch-Gordan coefficients. Since thereare in principle (2j2+1)(2j+1)(2j1+1) matrix elements this is an enormoussimplification.

The number N (j2, j, j1, T ) which is sometimes written as

N (j2, j, j1, T ) ≡ (−1)j+j2−j11√

2j2 + 1〈j2||T ||j1〉 , (4.134)

where 〈j2||T ||j1〉 is called the reduced matrix element of the operator T , canbe found by computing only one (e.g. the easiest one for a given operator T )of the (2j2 + 1)(2j + 1)(2j1 + 1) matrix elements.

The Clebsch-Gordan coefficients appearing in the Wigner-Eckart formula(4.133) immediately determine the selection rules, that is, tell which of thepossible (2j2 + 1)(2j + 1)(2j1 + 1) matrix elements must vanish. In this waythe Wigner-Eckart theorem determines e.g. possible electric quadrupole (andhigher) transitions (induced and spontaneous alike) in atoms. Another itsimmediate consequence is that the system in a state characterized by the totalangular momentum j cannot have electric or magnetic multipole moments

23The factor N (j2, j, j1, T ) does depend also on quantum numbers other than thosedirectly related to the angular momentum; this additional dependence is the same for allkets with the same j (i.e. it is independent of m1).

96

of order 2l for l > 2j because the operators representing such moments aretensor operators of the type T

(l)m and

〈j,m|T (l)m′ |j,m〉 = 0 unless l ≤ 2j .

Therefore, spin 12fermions cannot (by the rotational invariance) have quadru-

pole (l = 2) moments, whereas spinless particles are forbidden to possesseven dipole moments. (Spin 1

2particles generically do have magnetic dipole

moments but electric dipole moment of a spin 12particles, allowed by the

rotational invariance, violates the CP symmetry).

For vector operators, which are most frequently encountered in elemen-tary applications the following simplified form of the Wigner-Eckart theoremproves very useful

〈n, j,m|V|n, j,m〉 = 〈n, j,m|J·V|n, j,m〉j(j + 1)~2

〈n, j,m|J|n, j,m〉 , (4.135)

where n are some quantum numbers not related to the angular momentum.The proof of (4.135) goes as follows. From (4.133) and (4.126) we have

〈n, j, jz|V (1)m |n, j, jz〉 = C1j(m, jz|j, jz)N (j, 1, j, V, n) .

Consider now the matrix element 〈n, j,m|J|n, j,m〉. Applying the Wigner-

Eckart theorem to the vector J operator itself treated as J(1)m we can write

jz~ δm0 = 〈n, j, jz|J (1)m |n, j, jz〉 = C1j(m, jz|j, jz)N (j, 1, j, J, n) . (4.136)

The first equality follows from the well known properties of the Jz and J±operators. In tables of the Clebsch-Gordan coefficients one can find thatC1j(m, jz|j, jz) = jzδm0/

√

j(j + 1). Hence,

N (j, 1, j, J, n) = ~

√

j(j + 1) .

Inverting now the second equality in (4.136) we can therefore write

C1j(m, jz|j, jz) =〈n, j, jz|Jm|n, j, jz〉

~√

j(j + 1). (4.137)

In order to find N (j, 1, j, V, n) we consider24

〈n, j, jz|V·J|n, j, jz〉 = 〈n, j, jz|V (1)0 Jz +

1√2V

(1)−1 J+ − 1√

2V

(1)1 J−|n, j, jz〉

= jz~ 〈jz|V (1)0 |jz〉+

~√2

√

j(j + 1)− jz(jz + 1) 〈jz|V (1)−1 |jz + 1〉

24Notice that the standard J+ and J− operators treated as components of the tensoroperator are not properly normalized.

97

− ~√2

√

j(j + 1)− jz(jz − 1) 〈jz|V (1)+1 |jz − 1〉

=~√2

[

jz√2C1j(0, jz|j, jz)

+√

j(j + 1)− jz(jz + 1)C1j(−1, jz + 1|j, jz)−√

j(j + 1)− jz(jz − 1)C1j(1, jz − 1|j, jz)]

N (j, 1, j, V, n) ,

where at the intermediate stage we have dropped the unnecessary labels andapplied the Wigner-Eckart theorem (4.133) to the matrix elements of V

(j)m .

Using again the properties of the Clebsch-Gordan coefficients one can showthat the expression inside the square brackets equals

√

2j(j + 1), and weobtain

N (j, 1, j, V, n) =〈n, j,m|J·V|n, j,m〉

√

2j(j + 1)~.

This, together with (4.137) completes the proof of the relation (4.135).

We illustrate the usefulness of the formula (4.135) by computing the split-ting of the Hydrogen atom energy levels in a constant uniform magnetic field(the Zeeman effect) taking into account the electron spin (i.e. its magneticmoment). To this end we have to consider the states with definite quan-tum numbers l, j and jz. The magnetic field splits the states with differentjz. The Hamiltonian Vint describing the electron spin interaction with anexternal magnetic field B reads

Vint =e

2mc(L + 2S)·B , (4.138)

(we use e > 0). The factor 2 multiplying S is the anomalous magneticmoment of the electron: it is the well known relativistic effect that the ratioof the intrinsic magnetic moment of the electron to its spin is twice25 as bigas the ratio of the electron magnetic moment due to its orbital to its orbitalangular momentum. Because of this factor of 2 the calculation of the energysplitting given by

∆Enljjz =e

2mcBz〈n, l, s, j, jz|Lz + 2Sz|n, l, s, j, jz〉 , (4.139)

is not straightforward and requires using (4.135). Since Lz + 2Sz = Jz + Sz

it is sufficient to consider the matrix element

〈l, j, jz|Sz|l, j, jz〉 =〈l, j, jz|J·S|l, j, jz〉

j(j + 1)~2〈l, j, jz|Jz|l, j, jz〉 , (4.140)

25The factor 2 comes out naturally from the Dirac equation. There are also tiny cor-rections to this result whose computation requires the use of the full Quantum Electrody-namics - see section 19.5.

98

where we have used (4.135) and omitted inessential quantum numbers. Wecan now use the identity

J·S = −1

2(J− S)2 +

1

2J2 +

1

2S2 =

1

2

(

J2 + S2 − L2)

. (4.141)

Since the states |l, j, jz〉 are eigenstates of all these operators we get

〈l, j, jz|Sz|l, j, jz〉 =j(j + 1) + 3

4− l(l + 1)

2j(j + 1)〈l, j, jz|Jz|l, j, jz〉 . (4.142)

Thus we finally obtain

∆Enljjz =e

2mcBz

[

1 +j(j + 1) + 3

4− l(l + 1)

2j(j + 1)

]

〈l, j, jz|Jz|l, j, jz〉

=e

2mcBz g 〈l, j, jz|Jz|l, j, jz〉 =

e

2mcBz g jz~ , (4.143)

with

g = 1 +j(j + 1) + 3

4− l(l + 1)

2j(j + 1), (4.144)

the so-called Lande factor.

4.7 Higher symmetries

In some special cases the spectrum of the Hamiltonian exhibits degeneracieshigher than could be expected from the rotational invariance alone. Con-sider for example the two-dimensional isotropic harmonic oscillator with theHamiltonian

H =1

2M

(

P 2x + P 2

y

)

+1

2Mω2

(

x2 + y2)

. (4.145)

Its eigenvalues are given by Enxny= ~ω(nx + ny + 1) and depend only on

nx + ny and not on nx and ny separately. Hence, the N -th energy level has(N + 1)-fold degeneracy. Of course, L = xPy − yPx commutes with theHamiltonian, but since the group of two-dimensional rotations it generates isAbelian, all the representations of its Lie algebra are one-dimensional. Thus,on the basis of the rotational symmetry one would not expect any degeneracy.

Similarly, the eigenvalues of the Hamiltonian of the three-dimensionalisotropic harmonic oscillator are given by Enxnynz

= ~ω(nx + ny + nz +32)

and the degeneracy of the N -th energy level is 12(N +1)(N +2)-fold. Again,

these multiplicities do not correspond to (2l + 1)-fold degeneracies expectedon the basis of the invariance with respect to three-dimensional rotations.

99

The degeneracy of the spectra of the two- and three- (and also higher)dimensional isotropic harmonic oscillators can be understood by noticing thatin each case one can construct a number of Hermitian operators Qa, whichcommute with the Hamiltonian and form a closed algebra

[

Qa, Qb]

= i~f abc Qc , [Qa, H ] = 0 , (4.146)

with some structure constants f abc . The operators L are part of the algebra of

Qa’s. Since these algebras are closed, the operators Qa generate some higherdimensional symmetry groups. In the case of two dimensions the algebraof Qa’s coincides with the one generated by L in three dimensions. Hence,the two-dimensional isotropic harmonic oscillator has the SU(2) symmetrygroup which appears because of the special form of the potential. Similarly,the algebra of Qa’s in the case of the three-dimensional isotropic harmonicoscillator coincides with the algebra of the generators of the SU(3) group.

These symmetries are easy to uncover if the Hamiltonian of the n-dimen-sional isotropic harmonic oscillator is written in terms of the creation andannihilation operators:

H = ~ωn

∑

i=1

(

a†iai +1

2

)

. (4.147)

This form of H as well as the commutation relation [ai, a†j ] = δij are easily

seen to be invariant under the transformations

ai →(

e−iθaTa)

ijaj , a†i → a†j

(

eiθaTa)

ji, (4.148)

where the n2−1 Hermitian n×n matrices T a form the so-called fundamental(i.e. the nontrivial one of the lowest dimension) representation of the SU(n)group.26 For SU(2) the matrices T a are the Pauli matrices divided by 2:T a = 1

2σa; for SU(3) T a = 1

2λa, where λa are the eight 3 × 3 Gell-Mann

matrices, etc.27

Invariance of the Hamiltonian under the transformations (4.148) imme-diately suggests that the hidden symmetry group is just SU(n). Indeed, it is

26In fact the Hamiltonian (4.147) is invariant under the larger, U(n) symmetry groupgenerated by the n2 − 1 matrices T a and one additional matrix T which is simply propor-tional to the unit n×n matrix (it generates transformations changing the phases of all theoperators ai in the same way). However, as can be checked, the Hilbert space operatorgenerating these additional transformations coincides (up to a multiplicative constant fac-tor) with the Hamiltonian itself. Hence, this additional symmetry is the time translationsymmetry and is not reflected in the degeneracy of the energy levels.

27For the generators in the fundamental representation we adopt the normalizationtr(T aT b) = 1

2δab.

100

easy to see that if there exist unitary symmetry operators U(θ) ≈ 1− i~θaQa

which commute with H and such that

U †(θ) ai U(θ) =(

e−iθaTa)

ijaj , U †(θ) a†i U(θ) = a†j

(

eiθaTa)

ji, (4.149)

then the Hamiltonian is automatically invariant

U †(θ)H U(θ) = H , (4.150)

or, in other words, [H, U(θ)] = 0. For infinitesimal parameters θa we shouldtherefore have

i

~[Qa, ak] = −iT akjaj ,

i

~

[

Qa, a†k

]

= ia†jTajk , (4.151)

and it is easy to see that these commutation rules are satisfied by the oper-ators Qa of the form

Qa = ~ a†k Takj aj . (4.152)

It is also straightforward to check that together with the Hamiltonian (4.147)the operators Qa (4.152) satisfy the commutation rules (4.146) with the samestructure constants of the SU(n) group as do the finite dimensional c-numbermatrices T a:

T aikTbkj − T bikT

akj = if ab

c T cij . (4.153)

The symmetry group of the n-dimensional harmonic oscillator is there-fore SU(n) and the degeneracies of the successive energy levels correspondto the number of basis states of higher and higher dimensional representa-tions of this group. For n = 2 these representations are labeled by the valuesk(k + 1) with k = 0, 1

2, 1, . . . of the Casimir operator Q2 =

∑3a=1Q

aQa andthe multiplicity of the N -th energy level is easily seen to correspond to therepresentation with k = N/2. For the three-dimensional oscillator, n = 3, therepresentations are in principle labeled by the two Racah operators (one ofthem, the Casimir operator, is of the form Q2 =

∑8a=1Q

aQa and the secondone is built as some trilinear combination of Qa’s), but usually the represen-tations of the SU(3) Lie algebra are distinguished just by their dimensionor by giving a pair of integers (u, v), where u, v = 0, 1, 2, . . ., in which casethe dimension of the representation is given by 1

2(u + 1)(v + 1)(u + v + 2).

Not all representations are realized in the spectrum of the three-dimensionaloscillator: for example, the representations denoted in particle physics as 1,3, 6 and 10 are realized at the 0-th, first, second and third energy levels, re-spectively, but the adjoint representation 8 does not appear. In fact, realizedare only those representations which in the (u, v) classification are denotedas (u, 0). It is left for the reader to investigate the reason for this.

101

Another celebrated example of a symmetry higher than expected explainsthe structure of the spectrum the nonrelativistic Hydrogen-like atom. Solvingthe Schrodinger equation with the potential V (r) = −e2/r (for simplicity weconsider the Hydrogen atom with Z = 1) one finds that the n = 1 groundlevel is nondegenerate and consists of a single 1S state.28 At the second n = 2energy level there are four states: one 2S and three 2P states. For n = 3there are: one 3S state, three 3P states and five 3D states, and so on. Oneimmediately realizes that the multiplicity of states at the n-the energy levelis n2 instead of being equal (2l + 1) for some integer l = 0, 1, 2, . . ., as couldbe expected from the rotational invariance of the potential V (r). Clearly,some higher symmetry must be responsible for this degeneracy. However,as we now show, in contrast with the harmonic oscillator, this symmetryis only a dynamical one, that is, the Hilbert space symmetry generatorsforming a closed algebra can be defined only for subspaces corresponding toindividual energy levels. As a result, the symmetry considerations will allowto explain not only the degeneracy of the energy levels but also to find purelyalgebraically the eigenvalues of the Hamiltonian.

As it is well known, from classical mechanics, the motion of a particle ofmassm in the central potential V (r) = −e2/r is characterized not only by thefour usual constants of motion (the energy E and the three components of theorbital angular momentum L), but also by three other constant quantities -the three components of the so-called Lenz vector

M = r× L− e2r

r, (4.154)

(the dot represents the time derivative). Indeed, differentiating M with re-spect to time and using the Newton’s equation

d

dtmr = −e2 r

r3, (4.155)

one easily finds that M is constant. While the constancy of L expressesgeometrically the fact that the motion occurs in a fixed plane, the constancyof M is related to the fact that the axes of the ellipses do not precess withtime.

The analogy with classical mechanics suggests that in the quantum theorythe three components of M will, together with the three components of L,play the role of the generators of some higher symmetry, which explains thedegeneracy of the spectrum. There is however an ambiguity in the definitionof M as the quantum mechanical operator because L and P do not commute

28In the considerations we ignore the doubling of each energy eigenstate due to the spinof the electron.

102

with each other. It turns out that the correct definition is provided by thesymmetric ordering

M =1

2m(P× L− L×P)− e2

r

r

=1

mP× L+

i~

mP− e2

r

r. (4.156)

With this definition it is straightforward (but in fact terribly tedious!) tocheck that the following commutation rules are satisfied:

[H, L] = 0 ,[

Li, Lj]

= i~ ǫijkLk ,

[H, M] = 0 ,[

Li, M j]

= i~ ǫijkMk . (4.157)

The last relation expresses simply the fact that M is a vector operator.

To have a closed algebra of generators - a property necessary to identifythe results of the action of QaQb − QbQa on states with another transfor-mation belonging to the group of symmetries (see section 4.2) - it shouldalso be possible to express the commutator of components of M as a linearcombination of Li and M i. However, computing the commutator explicitlyone finds (after a long calculation) instead:

[

M i, M j]

= − 2

mi~ ǫijkHMk , (4.158)

(with the Hamiltonian H on the right hand side). In other words, one findsthat the algebra of the operators L and M does not close.

The solution is provided by the observation, that if we restrict the con-sideration to a given discrete (with E < 0) energy level, we can form the newoperators

N =(

− m

2E

)1/2

M , (4.159)

which are also vector operators commuting with H and satisfy the rule

[

N i, N j]

= i~ ǫijkNk , (4.160)

that is, on the subspace of states corresponding to fixed E < 0 they form,together with L, a closed algebra of generators. Therefore, the six generatorsL and N can be promoted to the generators of the transformations formingthe SO(4) symmetry group. Of course, the operators N are defined only fora given energy level (different operators N form closed algebras at differentenergy levels), but apart from this, their implications for the degeneracies ofenergy levels are the same as of “normal” symmetry generators.

103

Before finding the representations of the Lie algebra generated by L andN it is useful to note two relations. Firstly, computing (laboriously!) thesquare of the operator M one finds that

M2 =2

mH

(

L2 + ~2)

+ e4 . (4.161)

On the subspace of states corresponding to fixed E < 0 this is equivalent to

N2 + L2 = −~2 − me4

2E. (4.162)

Secondly, for the specific realization of the SO(4) Lie algebra provided bythe operators L and N we have M · L = L ·M = 0, or, equivalently,

N · L = L ·N = 0 . (4.163)

It is important to stress that this relation does not follow from the commu-tation rules of the SO(4) Lie algebra; it is specific only for its realization byoperators acting in the Hilbert space of the nonrelativistic Hydrogen atom.

We can now find the representations of the Lie algebra generated by L

and N. To this end, it is easier to define the new generators29

I ≡ 1

2(L+N) , K ≡ 1

2(L−N) , (4.164)

which satisfy simpler commutation rules:[

I i, Ij]

= i~ǫijkIk ,[

Ki, Kj]

= i~ǫijkKk ,[

I i, Kj]

= 0 . (4.165)

Thus the Lie algebra of L and N is equivalent to the direct sum su(2) ⊕su(2) of the two su(2) Lie algebras generated by the I and K operatorsindependently. The symmetry group is therefore

SU(2)× SU(2) ∼ SO(4) . (4.166)

With this result it is easy to see that the Lie algebra of L and N is ofthe rank two with I2 and K2 playing the roles of the two Racah operators.Irreducible representations of the su(2)⊕ su(2) Lie algebra spanned by theHilbert space vectors belonging to a given energy level are in general labeledby pairs of numbers30 (i, k) where i, k = 0, 1

2, 1, 3

2, . . ., which correspond to the

eigenvalues of I2 and K2, respectively and have dimensions (2i+ 1)(2k + 1).

29A warning: the operator K defined here has nothing to do with the boost generatordefined in (4.40)!

30That is, they can be written as |n, iz, kz〉 with the matrices of the first (second) SU(2)group in the product SU(2)×SU(2) acting only on the first (second) group index iz (kz).The index iz (kz) is simply the Iz (Kz) eigenvalue (modulo ~) of the vector |n, iz, kz〉.

104

However, because of the operator relations (4.163), in the Hydrogen atomHilbert space we necessarily have

I2 =1

4

(

L2 +N2 +N · L+ L ·N)

=1

4

(

L2 +N2)

,

K2 =1

4

(

L2 +N2 −N · L− L ·N)

=1

4

(

L2 +N2)

, (4.167)

i.e. I2 = K2. Hence, in this particular case realized can only be thosesu(2) ⊕ su(2) algebra representations, for which k = i, and which, for thisreason, have dimensions (2k + 1)2 with k = 0, 1

2, 1, 3

2, . . .. This explains

why the multiplicity of states at each energy level is a square of an integern = 2k + 1.

One can go further and notice that

1

2

(

L2 +N2)

= I2 +K2 , (4.168)

so that the operators appearing on the left-hand side of (4.162) can take onlyvalues 2 · 2 · k(k + 1)~2. Hence,

−~2 − me4

2E= 4k(k + 1)~2 , (4.169)

which leads to the well known result

E = − me4

2~2(2k + 1)2, with k = 0,

1

2, 1,

3

2, . . . (4.170)

This reproduces the spectrum of the nonrelativistic bound states of the Hy-drogen atom and explains the rule that the multiplicity of the n-th energylevel equals n2.

4.8 Discrete symmetries: space and time reflection

The set of classical Galileo transformations (4.33) can be supplemented withtwo discrete symmetry transformations: P and T - the space and time reflec-tion, respectively (often called simply parity and time reversal) - which acton coordinates in the following way

r′ = −r , t′ = t , (space reflection)

r′ = r , t′ = −t , (time reversal). (4.171)

From this it is easy to see that on coordinates P−1 = P , T−1 = T and that

P · (R, V, a, τ) = (R, −V, − a, τ) · P ,T · (R, V, a, τ) = (R, −V, a, − τ) · T . (4.172)

105

In the Hilbert space these transformations are represented by the operatorsU(P ) = P and U(T ) = T , which, according to the Wigner theorem, mustbe either unitary and linear or antiunitary and antilinear. The commutationrules of P and T with the generators of the Galileo group can be deduced byusing the same procedure that led us to (4.45). Repeating the steps we find

P iJ i P† = iJ i , T iJ i T † = iJ i ,

P iKi P† = −iKi , T iKi T † = −iKi ,

P iP iP† = −iP i , T iP i T † = iP i ,

P iH P† = iH , T iH T † = −iH .

We have left the factors of i, for we still have to decide whether the operatorsP and T are linear or antilinear. This can be decided by looking at thelast pair of relations: if the operations of space and time reflections are tobe symmetries of the given physical system, they should transform it into(taking the active view) another realizable system with the same energylevels. In particular, the energy spectrum of the transformed system shouldnot be unbounded from below (typically the spectrum of H is unboundedfrom above - there can be arbitrarily high excitations; recall also that P†HPand T †HT give on the states of the original system the same values asdoes H on the states of the parity transformed and time reversed systems,respectively). Hence we should require that

PH P† = H , T H T † = H , (4.173)

from which it follows that P must be unitary and linear whereas T has to beantiunitary and antilinear (multiplying these relations from the right by Pand T one gets the commutation relations in the standard form [H, P] = 0,[H, T ] = 0). Thus,

P J i P† = J i , T J i T † = −J i ,PKi P† = −Ki , T Ki T † = Ki , (4.174)

P P iP† = −P i , T P i T † = −P i .

We now come to specific realizations of P and T in Hilbert spaces.

We consider first the space reflection in the conventional nonrelativisticquantum mechanics of a single particle. As we will see studying the rela-tivistic case, each particle31 can be characterized by a number η called its

31That is, each irreducible representation of the Poincare group identified with some typeof particles. Of course, parity is known to be broken in Nature by the weak interactions,so strictly speaking the parity operator P acting on full interacting state vectors of theStandard Theory cannot be constructed. However, considering the action of P on freeparticle states is still useful because in physics - as opposed to mathematics - we are notmerely interested in the statement that the space reflection is not a symmetry, but ratherin how it is broken by interactions.

106

intrinsic parity which is the eigenvalue of the parity operator P on the vectorstate of this particle. Since classically P 2 is the identity, in the Hilbert spacewe can have P2 = eiϕ. Therefore η2, and hence also η, is a complex numberof modulus 1. We will see later, that for bosons (particles having integerspin) the possible eigenvalues of P are ±1, whereas for fermions (particleshaving half-integer spin) η can assume values ±1 and ±i. (In nonrelativistictheory this is sometimes justified with the argument that since P 2 is like therotation by 2π, for bosons one must have P2 = +1, while for fermions P2

can also equal −1; this argument is however not very convincing).

The intrinsic parity η is unique for a given particle and plays, hence,completely no role in quantum mechanic of a single particle. In principle thestate vector |ψ〉 (the wave function ψ(r) of the space reflected system shouldsatisfy the relation

ψ′(r) ≡ 〈r|ψ′〉 = ηψ(P−1r) = ηψ(−r) = η 〈−r|ψ〉 , (4.175)

and if the space reflection is a symmetry of the physical system then theoperator P such that

|ψ′〉 = P|ψ〉 , (4.176)

commutes with the Hamiltonian.

In quantum mechanics of a single particle the action of P on the positionoperator r can be deduced by noticing that since PJP† = J, where in generalJ = r×P+ S, cf. (4.73), and P PP† = −P, one must have

P rP† = −r , (4.177)

(and also PSP† = S). This is in line with the physically motivated expecta-tion that if 〈ψ|r|ψ〉 = r, then 〈ψ′|r|ψ′〉 = 〈ψ|P†rP|ψ〉 = −r.

If the Hamiltonian H of a system of many particles (not necessarily allidentical) or, more frequently, its unperturbed part H0, commutes with P,the N -particle eigenstates of H or of H0 denoted by |ψN〉, forming a basis ofthe system Hilbert space (see Section 5) can be chosen so that they satisfy

P|ψN 〉 = ηψηψ1. . . ηψN

|ψN〉 , (4.178)

with ηψ = ±1 the parity of the state |ψN 〉 and ηψithe intrinsic parities of the

particles in this state. In such a case parity can be used to establish certainselection rules for matrix elements of those operators O which have definitereflection properties

P†OP = ηOO . (4.179)

107

Indeed, in this case we have:

ηO〈χN |O|ψN〉 = 〈χN |P†OP|ψN〉= ηχηψη

∗χ1. . . η∗χN

ηψ1. . . ηψN

〈χN |O|ψN〉 , (4.180)

and the matrix element must vanish if

η∗Oηχηψη∗χ1. . . η∗χN

ηψ1. . . ηψN

6= 1 . (4.181)

In particular, in quantum mechanics of a single particle (N = 1) the operatorr has ηr = −1 and the electric dipole transitions can occur only betweenstates having opposite parities.

Consider now the time reversal. If [H, T ] = 0, i.e. if the time reversal isthe symmetry of the system, for an arbitrary state |ψ(t)〉 one can write

T |ψ(t)〉 = T e−iH~t|ψ(0)〉 = e−i

H~(−t) |ψ′(0)〉 = |ψ′(−t)〉 , (4.182)

where we have used the antilinearity of T and defined

|ψ′(0)〉 ≡ T |ψ(0)〉 . (4.183)

Thus, T |ψ(t)〉 = |ψ′(−t)〉. With this definition it is easy to see that if |ψ(t)〉satisfies the Schrodinger equation, so does also |ψ′(t)〉:

i~d

dt|ψ′(t)〉 ≡ i~

d

dtT |ψ(−t)〉 = T i~ d

d(−t) |ψ(−t)〉

= T H|ψ(−t)〉 = T HT †T |ψ(−t)〉 = H|ψ′(t)〉 , (4.184)

(we have used again the antilinearity of T ).

In concrete representations of quantum mechanics one represents the op-erator T as the product T = UK, in which U is a suitable unitary operatorand K is the complex conjugation. The operator U is chosen so that thecommutation rules (4.174) of T are satisfied.

For complex H (e.g. if the potential energy V is a complex function of theposition, which is the case e.g. when inelastic scattering is modeled withinthe framework of quantum mechanics of a single particle) the time reversalsymmetry would require

UH∗U † = H , (4.185)

which usually is impossible - the time reversal is broken because the absorp-tion effects (described by the imaginary part of V ) single out (like friction orviscosity) one time direction.

108

In quantum mechanics of a single particle J = r×P+S (c.f. (4.73)) andfrom (4.174) one deduces that

T r T † = r , (4.186)

and T ST † = −S. The wave function of a spinless particle has only one com-ponent and it is easy to see that in usual the position representation the rules(4.174) are satisfied by U = 1. Instead, in the momentum representation, inwhich P = p and r = i~∂/∂p, one must have UpU † = −p in order to satisfythe relations (4.174) with P and J = L.

For spinning particles the commutation rule (4.174) with J = r ×P+ S

imposes additional conditions on U . In the position representation assumingthat the matrices Si are such that Sx and Sz are real while Sy is purelyimaginary (this can always be arranged for), we must have

USx,zU † = −Sx,z , USyU † = Sy . (4.187)

It is easy to check that in this representation

T = exp

(

−π i~Sy

)

K , (4.188)

satisfies the requirements. In particular, for spin 12particles, for which Si =

(~/2)σi, one has T = −iσyK.

Consider now a system of N identical particles. In the ordinary posi-tion representation, in which the state is represented by the wave functionψs1,...,sN (r1, . . . , rN), where sk is the spin label of the k-th particle, the timereversal operator takes the form

T = e−πi~Sy1 · . . . · e−π i

~Sy

NK , (4.189)

where Syk acts on spin variables of the k-th particle only (so that [Sik, Sjl ] = 0

for k 6= l). If all the particles have integer spin, T 2 = 1 independently oftheir number N . However,

T 2 = (−1)N =

+1 for N even−1 for N odd

. (4.190)

if there are N half-integer spin particles. Consider now the energy eigenstates|n〉. If the time reversal is a symmetry, i.e. if [H, T ] = 0, the state T |n〉 hasthe same energy as |n〉. Suppose now that T |n〉 and |n〉 represent the samestate, i.e. T |n〉 = λ|n〉 with λ a phase factor, that is, there is no degeneracyof the n-th energy level. Applying T twice we get

T 2|n〉 = T λ|n〉 = λ∗T |n〉 = |λ|2 |n〉 = |n〉 , (4.191)

109

(antilinearity!) which is only possible if T 2 = 1. Hence, for an odd numberN of fermions, the states |n〉 and T |n〉 must be distinct and, therefore, theremust be at least a two-fold degeneracy - called the Kramers degeneracy -of the energy levels of the Hamiltonian. The states |n〉 and T |n〉 are thenorthogonal to each other: for arbitrary two states Ψ and Φ (we again resortto the mathematical notation) we can write (cf. 4.11)

(Φ|Ψ) = (Φ|T †T Ψ) = (T Ψ|T Φ) . (4.192)

Setting now Ψ = T |n〉 ≡ T Ψn and Φ = |n〉 ≡ Ψn we get

(Ψn|T Ψn) = (T 2Ψn|T Ψn) = ±(Ψn|T Ψn) , (4.193)

so that for the minus sign (T 2 = −1) we must have (Ψn|T Ψn) = 0. Usuallythe state T |n〉 together with |n〉 belong to a larger multiplet of some con-tinuous symmetry,32 so that the Kramers degeneracy does not introduce anyadditional doubling of states with a given energy. The two-fold degeneracyexclusively due to the time reversal can be observed in some special situa-tions, though. Consider for example a crystal which has a low symmetry.Each atom of the crystal lattice can be then viewed as feeling the electro-static field of the neighbouring atoms. This field is not rotationally invariantand, superficially, one would not expect any degeneracy of the atomic energylevels. However, the electrostatic field felt by the electron at the positionr of a given atom described by the Hamiltonian term

∑

i φ(r − ri) (ri arethe positions of the other atoms) does not break the time reversal symmetry(recall that T r T † = r so that T φ(r − ri)T † = φ(r − ri)). Therefore, if thenumber of atom’s electrons is odd, there must still be a two-fold degeneracyof each atomic energy level. It is lifted only if the crystal is placed in anexternal magnetic field, because the term H ′ ∝ B · (L + 2S) does break thetime reversal: T B · (L+2S)T † = B · T (L+ 2S)T † = −B · (L+ 2S) and thefull Hamiltonian H0 + Vint does not commute with T .

Another important conclusion is that fermions cannot have electric dipolemoments without violating the time reversal invariance. Consider a free spins (half-integer) fermion at rest,33 so that only its spin degrees of freedommatter. It has (2s + 1) degenerate spin states. If the particle possesses anelectric dipole moment d, an external electric field E lifts this degeneracy

32For example, in the Hydrogen atom the Kramers doubling of energy states is relatedto the doubling introduced by the spin degree of freedom: indeed, since T †

JT = −J onehas T |n, l,m, sz〉 = |n, l,−m,−sz〉, but the fact that En,l,m,sz = En,l,−m,−sz follows alsofrom the rotational invariance. (If the spin degree of freedom is neglected, electrons aretreated as bosons and the fact that the ground state is nondegenerate is not in conflictwith the time reversal invariance.)

33Clearly, this reasoning applies to massive fermions only.

110

completely.34 If the time reversal were a good symmetry, a two-fold degen-eracy should remain. Since it is not the case, symmetry with respect tothe time-reversal must be broken. In the context of relativistic quantumfield theory this means that also the CP (the combination of parity and thecharge conjugation - the operation of replacing particles by their antiparti-cles) must be broken, because the CPT operation is always a good symme-try of relativistically invariant quantum field theory models (with HermitianHamiltonians).

34The Wigner-Eckart theorem (4.135) can be used to calculate shifts of the energy levelsdue to the interaction Vint = E·d. In particular, application of this theorem makes it clearthat the degeneracy is lifted completely.

111

Appendix A Rotation matrices

In this appendix we present a method of finding the matrices d(j)m′m(β) defined

in (4.83) and collect various properties of the d(j)m′m(β) and D

(j)m′m(α, β, γ)

matrices.

The general form of the d(j)m′m(β) matrices

We begin with j = 12and replace (denote) the basis spin states |1

2, 12〉 and

|12,−1

2〉 by the variables ξ and η, respectively. Action on these states of the

generators J i

Jz ξ =~

2ξ , Jz η = −~

2η ,

J+ ξ = 0 , J+ η = ~ ξ ,

J− ξ = ~ η , J− η = 0 ,

can formally be represented by the action on ξ and η of the differentialoperators

Jz =~

2

(

ξ∂

∂ξ− η

∂

∂η

)

, J+ = ~ ξ∂

∂η, J− = ~ η

∂

∂ξ. (A.1)

The operator J2 = (Jz)2 + 12(J+J− + J−J+) takes in this representation the

form

J2 =M(M + ~) with M ≡ ~

2

(

ξ∂

∂ξ+ η

∂

∂η

)

. (A.2)

It is then easy to find that

Jz ξaηb =~

2(a− b) ξaηb ,

J2 ξaηb =~2

4(a + b)(a+ b+ 2) ξaηb , (A.3)

so that for a = j +m and b = j −m (with −j ≤ m ≤ j) the monomials oforder j

u(j,m) ≡ Nj,m ξj+m ηj−m , (A.4)

are the eigenfunctions of Jz and J2 with the right eigenvalues ~m and ~2j(j+

1), respectively, to represent the states |j,m〉. For this identification to bepossible the normalization factorsNj,m must be chosen so that (cf. eq. (4.78))

J±u(j,m) = ~√

(j ∓m)(j ±m+ 1)u(j,m± 1). This requires

Nj,m±1 = Nj,m

√

j ∓m

j ±m+ 1,

861

and determines Nj,m (up to a multiplicative constant independent of j andm) to be

Nj,m =1

√

(j +m)!(j −m)!. (A.5)

The formula (A.4) expresses simply the fact that (as can be shown) thestates |j,m〉 for a given j can be constructed as completely symmetrizedtensor products of 2j states |1

2,±1

2〉:

|j,m〉 ∝∑

P

(

| 12, mP (1)〉 ⊗ . . .⊗ | 1

2, mP (2j)〉

)

, (A.6)

where the sum is over all permutations P (i) of 2j numbers mi = ±12such

that m1 + . . .+m2j = m.

The above results allow to obtain the matrices d(j)m′m(β) explicitly by em-

ploying the formula (cf. eqs (4.76) and (4.83))

e−i~βJy |j,m〉 =

+j∑

m′=−j

|j,m′〉 d(j)m′m(β) . (A.7)

Acting with the exponentiated generator Jy represented by the differentialoperator

Jy = − i

2(J+ − J−) =

i

2~

(

η∂

∂ξ− ξ

∂

∂η

)

,

on the state |j,m〉 represented by the function u(j,m) one obtains the sum

of monomials in ξ and η of order j. The function d(j)m′m(β) is then given by

the coefficient of u(j,m′). For j = 12we get

e−i~βJy

ξ =

∞∑

n=0

1

n!

(

β

2

)n(

η∂

∂ξ− ξ

∂

∂η

)n

ξ = ξ cosβ

2+ η sin

β

2,

and similarly

e−i~βJy

η = −ξ sin β2+ η cos

β

2,

and ideed, the coefficients of ξ and η coincide with the appropriate elementsof the matrix (4.84). Generalizing one obtains

e−i~βJy

u(j,m) = Nj,m

(

ξ cosβ

2+ η sin

β

2

)j+m(

−ξ sin β2+ η cos

β

2

)j−m

= Nj,m

j+m∑

k=0

j−m∑

l=0

(−1)j−m−l

(

j +mk

)(

j −ml

)

(A.8)

×(

cosβ

2

)j+m−k+l(

sinβ

2

)j−m+k−l

ξ2j−k−l ηk+l,

862

with Nj,m given by (A.5). This simply expresses the fact that once we knowhow the individual states |1

2,±1

2〉 in the formula (A.6) transform under the

action of a rotation, we also know how does their (symmetrized) producttransform. Substituting in (A.8) 2j − k − l = j + m′, k + l = j − m′ andextracting the coeficient of Nj,m′ξj+m

′

ηj−m′

we arrive at

d(j)m′m(β) =

Nj,m

Nj,m′

∑

k=0

(−1)k−m+m′

(

j +mj +m− k

)(

j −mm′ −m+ k

)

×(

cosβ

2

)2j+m−m′−2k (

sinβ

2

)m′−m+2k

. (A.9)

Useful properties of the rotation matrices

Orthogonality of the D(j) matrices:

∫ 2π

0

dα

∫ π

0

dβ

∫ 2π

0

dγ sin β D(j2)∗m′

2m2

(α, β, γ)D(j1)m′

1m1

(α, β, γ)

=8π2

2j + 1δj1j2δm1m2

δm′

1m′

2. (A.10)

The completness relation reads

1

8π2

∑

j

+j∑

m=−j

+j∑

m′=−j

(2j + 1)D(j)∗m′m(α, β, γ)D

(j)m′m(α, β, γ)

= δ(α− α)δ(cos β − cos β)δ(γ − γ) . (A.11)

The D(j)-matrices satisfy the following decomposition rule

D(j1)m′

1,m1D

(j2)m′

2,m2

=

j1+j2∑

j=|j1−j2|

C∗j1j2(m1, m2|j,m1 +m2)

Cj1j2(m′1, m

′2|j,m′

1 +m′2)D

(j)m′

1+m′

2,m1+m2

. (A.12)

For integer j = l elements of the D(j) matrices are related to the sphericalharmonics

D(l)∗m0 (ϕ, θ, 0) =

√

4π

2l + 1Y lm(θ, ϕ) . (A.13)

The orthogonality relation satisfied by the d(j) matrices reads∫ π

0

dβ sin β d(j′)∗m′m(β) d

(j)m′m(β) =

2

2j + 1δj′j . (A.14)

863

The corresponding inverse relation is

∑

j

(j +1

2) d

(j)∗m′m(β) d

(j)m′m(β) = δ(cos β − cos β) . (A.15)

They have the following properties

d(j)m′m(0) = δm′m ,

d(j)m′m(±π) = (−1)j∓mδ−m′m ,

d(j)m′m(−β) = d

(j)mm′(β) , (A.16)

d(j)m′m(β) = (−1)m

′−md(j)mm′(β) ,

d(j)m′m(β) = (−1)m

′−md(j)−m′−m(β) .

For integer j = l the elements of the d(l) matrices are related to the Legendrepolynomials:

d(l)00(β) = Pl(cos β) ,

d(l)10(β) = −d(l)01(β) = − sin β

√

l (l + 1)P ′l (cos β) , (A.17)

d(l)11(β) = d

(l)−1−1(β) = Pl(cos β) +

1− cos β

l (l + 1)P ′l (cos β) ,

where prime denotes derivative of Pl(x) with respect to x.

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