L '.'•

3. MAk• .-i 7

IC/68/104

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICALPHYSICS

FORMULATION OP QUANTUM DYNAMICS

IN TERMS OF GENERALIZED SYMMETRIES

CHAPTER I - THE FRAMEWORK OF QUANTUM THEORY

IN TERMS OF GROUP REPRESENTATIONS

A.O. BARUT

196 9

MIRAMARE - TRIESTE

IC/68/104

(Limited distribution)

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

FORMULATION OF QUANTUM DYNAMICS

IN TERMS OF GENERALIZED SYMMETRIES *

CHAPTER I - THE FRAMEWORK OF QUANTUM THEORY

IN TERMS OF GROUP REPRESENTATIONS

A.O. BARUT**

MIRAMARE - TRIESTE

December 1968

* Lectures given at the Advanced School of Physics, Trieste, 1968-1969.

* * On leave of absence from the University of Colorado, Boulder, Colo. , USA.

page

APPENDIX I

WIGNER'S THEOREM 25

APPENDIX II

COVERING GROUP, PROJECTIVE AND RAYREPRESENTATIONS, GROUP EXTENSION 27

APPENDIX III

WHEN UNITARY AND WHEN ANTI-UNITARYOPERATORS? 30

APPENDIX IV

THE GROUPS O(3), SO(3), SU(2), THEIRREPRESENTATIONS AND THE QUANTUM THEORYOF ANGULAR MOMENTUM 31

- ii -

FORMULATION OF QUANTUM DYNAMICS

IN TERMS OF GENERALIZED SYMMETRIES

CHAPTER I - THE FRAMEWORK OF QUANTUM THEORY

IN TERMS OF GROUP REPRESENTATIONS

INTRODUCTION

The theory of group representations, until very recently,

has been used in quantum theory to obtain essentially kinematical

relationships. Quantum dynamics, that is, the theory of the

interactions of quantum systems, was not in any way related to

symmetry considerations, but specific "interactions" had to be

formulated in Hamiltonian or Lagrangian formalism. The

formulation of dynamics based on the concept of "system 1 + system 2

+ interaction" works satisfactorily in non-relativistic theories, but

much less satisfactorily in relativistic theories. Experience hasus

shown/that in the latter case the theories always lead to procedures

involving infinitely many steps with unknown convergence properties,

as in perturbation theory or in S-matrix theory. One can then ask

if the concept of "interaction" could not advantageously be abandoned

(even if it can be shown to exist on the basis of general principles)

and the interacting systems be treated as a whole (e.g., a "dressed"

particle, or a composite system) globally in terms of their total

quantum numbers. This idea might seem at first to be less fundamental

or exact, for we are used to considering a theory fundamental if it is

formulated in terms of few basic fields or particles and of their

interactions. One can show on the basis of special examples (e. g.

positronium) that this is not so. The global point of view has some

definite advantages not possessed by the so-called microscopic point

of view.

- iii

It is in such global theories in terms of the total quantum

numbers of the interacting systems that the concept of generalized

symmetries and the theory of group representations re-enter into

the quantum theory, this time in the formulation of dynamics. The

subject of this essay is to try to give a physical interpretation of the

dynamical groups and the use of their representations in the

formulation of quantum dynamics.

The first part of Chapter I is a review of the general

principles of quantum theory necessary for the second part, where

the notion of the symmetries of system plus interaction is developed.

The further chapters will deal in detail with specific applications.

Since von Neumann's "Mathematical Foundations of Quantum

Theory", it is customary to start with a very general framework

of quantum theory: the Hilbert space of states, all Hermitian

operators as observables, etc. One then specializes to simple systems.

In these notes we follow the opposite direction. The emphasis is

on the determination of a concrete Hilbert space, in which scalar

products can be calculated explicitly and a number obtained. We

then start from the concrete Hilbert spaces of simple systems and

enlarge them systematically to define more complex systems, always

working with specific bases of states and with the specific form of

the observables. We have further adopted an S-matrix point of

view for the observable physical processes and are not treating

the time evolution of the system.

- iv -

1. KINEMATICAL POSTULATES

1.1 States and rays

The basic framework of the quantum theoretical description of

a physical system is a Hilbert space *% whose unit rays are in one-

to-one correspondence with the "pure" states of the system. A unit

ray ¥ is the set of vectors {Mi}, [| |j = 1, X = e1", ipe $ . The

reason for the introduction of rays rather than vectors themselves

lies a) in the use of a space over the complex numbers, and b) in

the basic probability interpretation of quantum theory. The quantities

related to observable effects are the absolute values of the scalar

products, |(^(^)| ,

characterizing a ray.

products, |(^(^)| , which are independent of the parameters X, X'

1. 2 Superposition principle

This basic correspondence incorporates the superposition

principle of quantum theory. For although both in classical and

quantum theory we have to do with a continuously infinite set of

states, in quantum theory there is a set of basis states out of which

arbitrary states can be constructed by linear superposition. Thus

ti1 = ) a 0 is another vector, so that the ray { X^1 } correspondsa £_, n n ; <x

n •to another possible state of the systems, if the rays \\ip t , n = 1, 2 . . .describe physical s tates . Note that 0* - \ a 0 and X01 represent

Z ot / , n n cc

a (X 0 ) is a H different state, in general,

although 0 and (X 0 ) represent the same state. This is the famous

problem of relative phases. Such states as can be obtained from each

other by linear superposition are called "coherent" states. [ G. C. Wick,

A.S. Wightman and E. P. Wigner, Phys. Rev. 88» 101(1952)].

- 1 -

!• 3 Supers election rules

In general, there are limitations on the superposition principle.

One cannot form pure states out of the superposition of certain states;

for example, one cannot form a pure state consisting of a positively

and a negatively charged particle, or a pure state consisting of a

fermion and a boson. This does not mean that two such states cannot

interact t it only means that their formal linear combination is not

a realizable pure state. The existence of superselection rules is

connected with the measurability of the relative phase of such a

superposition and depends on further properties of the system, like

charge, baryon number, etc. The superselection rule on fermions

(i. e., separation of states of integral and half integral fermions)

follows from the rotational invariance [ see Sec. 1.13 ] . In such a

case one divides the Hilbert space % into subsets, such that the

superposition principle holds within each subset. These subsets are

called "coherent subspaces". In each subspace, ) a i) and

Z y L> n n ^a th correspond to the same state, but ) a ih and ) a1 \b ,

n*n * • /_, n^n Z_J n n

in general, correspond to distinct states.

1. 4 Probability interpretation

The physical experiments consist in preparing definite states,

in letting them interact and in observing the rate of occurrence of

other well-defined states. The transition probability between two

states is defined by |(^, if>)\ (we can also say the transition

probability between two rays ¥ and $ because this quantity is the

same for all vectors of the rays). If 0and $ are themselves linear

combinations of some basis vectors, then the phase problem is inside

these quantities. This quantity can be related, by multiplying it with

certain kinematical factors, to the observed quantities like cross-*)

sections and lifetimes .

^ For general cross-section formulas see: G. Kallen, Elementary Particle

Physics (Addison-Wesley, 1964), Ch. I;

A.O. Barut. The Theory of the Scattering Matrix, (Macmillan 1967), Ch.III.

- 2 -

1. 5 The dynamical problem

Now in order to evaluate quantities like | {ip, 0) | we must

have a definite realization of the Hilbert space %• , and must obtain a

number that can be compared with experiment, i. e., a definite

labelling of the states ip, <!> . . . and a definite expression for the

scalar product. We shall refer to this realization as the concrete

Hilbert space. This is the more important and the more difficult

part of the theory. Although all Hilbert spaces of the same dimension

are isomorphic and one can transform one realization into another,

some definite explicit realization with a physical correspondence is

necessary.

If the Hilbert space framework is the kinematical principle of

quantum theory, the explicit calculation of states ^, 0, , » , , or, of

scalar products (ijj, <f>),is the dynamical part of quantum theory.

In simple cases, the dynamical problem is solved by postulating

an equation for the states \p,<f>. . . and identifying all solutions of

the equation with all the states of the physical system, such as in the

Schrbdinger theory. For more complicated systems, or for unknown

new systems, this is not possible. Even if we know all the states of

an isolated system, measurements on the system are carried out by

additional external interactions which change the system.

Short of the complete calculation of scalar products (^,i>), some

very general principles allow one to derive a number of important

properties of these quantities. It is along these lines that the

traditional use of group representations in quantum theory has been

developed *', More recently the quantum theoretical Hilbert space

has been identified with the explicit carrier space of the representations

of general groups and algebras. In this second sense the group

representations solve the dynamical problem. We shall explain

both of these aspects.

*) E. P. Wigner, Group Theory. (Academic Press 1959).

- 3 -

1. 6 Equivalent descriptions

First of all, as in any correspondence, let us deal with the

equivalent mappings between the physical states and the rays in

Hilbert space. For the knowledge of physically equivalent descriptions

of a system reflects already, as we shall see, important properties of

the system.

If the same physical system can be described in two different

ways in the same coherent subspace of the Hilbert space, once by

rays ¥..,0, . . . , and once by rays Y , $ , . . . (for example, by two

different observers), the same physical state once described by Y-,>

in the other case by Y , then the transition probabilities must be the

same by definition of equivalence. What is then the correspondence

between the raysH and TF ? Mathematically it is more convenient to

find out the corresponding map between the vectors \p, <p, . . . in the

Hilbert space. Because only the absolute values are invariant, the

transformation in the Hilbert space can be unitary or anti-unitary.

In fact, one can prove rigorously (see Appendix I, Wigner's theorem)

that one can choose vectors 0,,<p,, . . . from rays ¥ . $ * . . . in the

first description and vectors ^9,q> , . . . from the rays ¥ , O , . . .

in the second description,such that the correspondence between

ip , . . , and qfr # t . t is either unitary or a nti-unitary. That is, one

can construct a unitary or anti-unitary correspondence. These two

possibilities come from the fact that the complex field has two {and

only two) automorphisms that preserve the absolute values: the

identity automorphism and the complex conjugation. For the Hilbert

space over a real field Wigner's theorem yields only unitary

transformations (up to a phase), because the only automorphism of the

real field is the identity automorphism. In fact, Wigner's theorem is

closely related to the fundamental theorem of protective geometry.

[J. S. Lomont and P. Mendelson, Ann. Math. _78, 548 (1963)] .

One or the other case occurs for a given situation. Whether the

transformation is unitary or anti-unitary depends on further properties

of the two equivalent descriptions of the system. It does not depend,

however, on the choice of vectors ^, <p, . . . from the rays: if the

- 4 -

transformation is, for example, unitary for a choice ,,<p , . . .

there is no other choice Xip , A'cp , . . . such that it becomes anti-

unitary and vice_versa_. [ E. P. Wigner, J. Math. Phys. _1, 409, 414

(I960)]. Furthermore, once a vector tyn is chosen, the others,

<po, X , . . . are uniquely determined from the requirement that the

correspondence is unitary (or anti-unitary).

1. 7 Symmetry transformations

The description of the symmetry properties of the system in the

standard sense belongs to the situation characterized by the above

theorem. For, if under a symmetry transformation in the "physical

space" the physical states are unchanged, we obtain automatically

two equivalent descriptions in dv, one corresponding to the

original and the other to the transformed frames and these two

descriptions must be related to each other by unitary {or anti-unitary)

transformations. Conversely, and this is more important from our

point of view, the Hilbert space of states must be isomorphic to the

carrier space oT unitary (or anti-unitary) representations of the

symmetry transformations (they may form a group-or an algebra,

etc.), Note that we wish to obtain a concrete Hilbert space to calculate

transition probabilities. Thus, if we know the symmetry transformations

of the system we can start from an arbitrary collection of irreducible

unitary (or anti-unitary) representation spaces of the symmetryrr,

transformations to build up the Hilbert space %>. This solves the

problem partly, but not completely because we do not know what

collection of irreducible representations we have.

1. 8 Uniqueness of operators

We have said that the vectors from the rays of two equivalent

descriptions can be so chosen that the mapping of vectors if/. <—> \fj^

is either unitary, or anti-unitary. There is one other important

phase problem in quantum theory and this concerns the uniqueness of

the unitary (or anti-unitary) correspondence ^.

- 5 -

1. 9 Ray representations

If there are two equivalent descriptions with rays ¥,,<!>, . . .

and T , $ , . . . , respectively, corresponding to the same states as

seen by the two different descriptions (passive view) or with rays

¥•,*$,, . . . corresponding to states la } in the first description and

to the transformed states {Ts,} in the second description (active view),

then we know that we can choose vectors \p c ¥ , . . . \Jj e Y . . .

such that

^2 = UT ^1 ' ^2 = UT Vl' ' " ' ^

That is, if ip is a vector of Y , then U^ tp is a vector of the ray ¥ .

Now if there are two operators UT and U with the property (1), they

can differ only in a constant factor of modulus 1. This result has an

implication on the group law of transformations. For the product

gives the same result as the transformation U —. Consequently,

(2)

Because U is a representation of the symmetry group, the group

law for the representations is more general than the group law itself

ST = ST. Representations of the type (2) are called "ray representations"

or "representations up to a factor". This is again the result of the

fact that we have a correspondence between physical states and rays

in Hilbert space, not vectors.

1.10 Covering group

The ray representations of groups defined by eq. (2) can be

interpreted as ordinary vector representations of the extended group,

the so-called quantum mechanical covering group. (Appendix II).

Summarizing,we have: A symmetry group Q of the physical

system induces a group Uo of invertit>le mappings of Jy on to itself,

which is unitary or anti-unitary and is a representation of the coveringItgroup of G »&nd Jy is an arbitrary collection of irreducible carrier spaces

of U s .

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1.11 Continuity

Moreover, if some sort of topology is defined in (r , telling us

which symmetry transformations are close to each other, then the

postulate of the continuity of the physical transition probabilities

implies that the mapping S «-» U must also satisfy suitable

requirements of continuity. Thus, we have to do with continuous

representations of groups.

1.12 Unitary and anti-unitary operators,

The group property of the transformations, eq. (2),and

continuity allow us to determine the unitary or the anti-unitary

character of U_.

If for every group element S of G we have

S = s2 (3)

where s is also a group transformation, we have

Ug= u(s) U^ . (4)

The square of an operator is unitary, whether the operator is unitary

or not. Thus, all !!„ is unitary. For group i connected continuously

to identity, eq. (3) is satisfied and they will be represented by unitary

operators. For the anti-unitary case, eq. (3) must break down. If

eq. (3) does not hold, further considerations are necessary to decide

the unitary or anti-unitary character of U_ (see Appendix III).

In the unitary case, one can define a normalized operator U^

that U l = U Then U3S-1

transformations.we have from (2)

such that U l = U Then U3S-1 =* w(S,S~ ) l . For two commuting

USUT=C(S.T)UTUS , O(S.T). {£f |[only if Ufp and UR also commute J

and we find C(S, T) = +1 / In general, if the commutator-1 -1 *U_U U U (which is independent of the normalizations of U,- and

1 S I S •*•

U- and which is uniquely determined from U— and U ) is a multiple

_ rt

of I, i. e., C = TST" S"1 = I, then VQ = w(T, S) I, and u(T, S) is a

characteristic of the coherent subspace only, i. e. , has a unique value

in each coherent subspace. If U and IL are members of the same one-

parametric subgroup then(j(T,S) = 1.

1*13 Superselgction rules again

In Seo.1.2 we have seen that the vectors ^ oc^ l n andn

V 7 06 (A. y ) "belong to different rays (states) although ^ and

X y belong to the same ray. If a physical symmetry transformation

of the system changes W into A W then, "because the state of the' n n ' n '

system has not changed, a superposition of the form (_^ <*n y^ isn

not possible, unless A a 1, • The relative phase A between vectors

in different coherent sectors is not observable because the physios

has not changed under the symmetry transformation. No physical

measurement can distinguish the state 2_s a n yn from the staten

ny at (X (f ), Thus, to show the existence of a superselection rulenwe need a symmetry transformation (a physical postulate) and the

existence of vectors V which go into A W under this transformation,' n n ' n

Example 1, - Rotational invariance and fermion superselection rule

Consider,for- concreteness, a state ^, •» | "|»m^ belonging to

tbe representation J>? and a state fL « In ,m^ belonging to the

representation I) ", n » integer* Consider a rotation by the Euler

angles (0,^,0). The states transform according to eq,(A.IV. 52),

i.e.,

W ^ Id,*') .

Now for m 2irt we have

Thus, we get an extra phase of (-1) in the linear combination of our

two states} hence, according to our previous discussion, there is a

superselection rule between the states with integer j and those with

half-odd integer 5"*values,

-8-

The rotation "by 2Tr in this discussion can also be replaced

"by the commutator of two rotations "by IT about axes perpendicular to

each other*-*. These two operators commute and their commutator is

independent of the normalization of these operators. One again

finds, taking DQ = (n.^) D(n2Tr) D ^ n ) "1 D^ir)""1 (n = (n^^i ) . unit

vector), that D^ - (-l)2^. Thus DQ is represented by a phase factor

whose value depends on the coherent subsp'ace (see Seel .12).

Remarkt The postulate of "dynamical groups" which says that the

collection of irreducible representations of symmetry groups is itself

a particular representation of a larger non-compact group (e.g. SL(2,C))

automatically incorporates this superselection rule, because in the

representations of these higher groups, either only integer, or only

half-integer representations of SU(2) subgroup occur, (see Ch.II).

Example 2 - SU(2) group for isospin and superselection rules

If the STJ(2) group describing the isotopic spin raultiplets of

particles were an exact symmetry group of nature as the SU(2) group

for spin, then by the result of Ex.1 there would be a superselection

rule between the integer and half-odd integer I-spin states. Now for

strong interactions which are independent of the electric charge, SU(2)^

is a good symmetry group. This means that there are no 2ure_ states

of the form | Y. / + j A \ , for example. There are,however,

pure states like j m.y + | p ^ • These superpositions violate the

superselection rule on charge (see Ex,3 below); consequently there is

no superselection rule for charge for strong interactions alone. In

the presence of electromagnetic and weak interactions, SU(2)_ is not

a symmetry Croup, but then charge superselection rule holds, a pure

state I XT / + A ^ now exists, but not a state

Y- y + I A* \ . In fact, an 311(2 rotation taking Z +

into /i (or n into p) does not leave the system unchanged but

corresponds to the weak interaction process: Z -> A TT (or

n -> p + e + V )•

Similarly, if a hypothetical "superweak interaction" violates

the rotational invariance, then we can have pure states of the form

|N^> + I TT > , and the reaction N — » TTA, where A are the quanta

of this new interaction.

G.C. Hegerfeldt, K. Kraus and E.P, Wigner, J. Math. Phys.j), 2029 (1968).

-9-

Example 3 - Superselection rules for gauge groups

Two equivalent descriptions obtained from each other by a

commutative one-parameter continuous group (not obviously related to

space-time transformations) implies the existence of an additive

quantum number q, and the eigenstates transform as

For two-states with different values of q, e»g»> +1 and ~1» we

obtain two different phases e and e , hence a superselection

rule for q. The basic physical assumption underlying all such

selection rules, such as electric charge, baryon number, lepton number,

we repeat, is the requirement that the multiplication of all states

by e produces no observable change in the system, hence equivalent

descriptions and gauge groups.

One can form, instead of pure states, mixed states out of vectors*)

from different coherent subspaces. But this will not interest us here <

1,14 Implications of the Buperselection rules on parity and othergroup extensions

Within a coherent subspace the parity of each state (relative

to one of them) is well determined. In fact we use the ray

representations of the full 0(3) group, or the full Lorentz group,

including reflections. In this case the parity is defined either in

the same representation space as SO(3) (or proper homogeneous

Lorentz group) or in a doubled Hilbert space (see Apps. II & IV ) .

Thus relative parities are well determined, e.g. for the levels of

H-atom, and for particle-anti-particle pair in Dirac theory. However,

for states in different coherent subspaces, the relative parity is

not determined because we oannot take a linear combination of two

such states and see how it transforms under parity. Note that the

measured relative parity between K and Tr is actually that between it

and the deuteron (*J » 1 ) ,

f Formal structure of quantum theory with superselection.rules, seet

J.M. Jauch, Helv. Phys. Act a ^ , 711 (i960) and J#M. Jauch and

B. Misra, ibid, 2£, 699 (1961).

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Very similar considerations apply to other group extensions,e»g«» ^7 charge conjugation.

The extension of the isotopic spin gr^up SU(2) by a reflection

operator implies a doubling of I a n/2 states, but not necessarily

of I s nf n a integer states. Now this extension is carried out by

C a charge conjugation, or by G s C e 2, [The use of G and G,

respectively, corresponds in the rotation group for spin to the use

of TL and parity P$ G commutes with all isospin rotations as P

commutes with all space rotations (App. IV. 7) r . G tells us

whether we have polar or axial vectors in I-spaeej e.g., f is a

polar vector. Therefore the doubling with G takes us to antiparticles.

Consequently we have among others the result that I m n/2 boson—

multiplets cannot contain antiparticles; they must lie in the other

half of the doubled space *\ In the limit of an exact STJ(2).j. the

relative G-parity (isospin-parity) between I = n/2 and I = n multiplets

is not defined} nor is it defined between states with different

charges or baryon numbers. It is defined, however, between, e.g.,

I . 1 multiplet (ir) and two I = •g-multiplets with N = O(e.g.

1.15 "Irreducibility postulate" for symmetry groups.

We said in Sec.1*7 "that the concrete Hilbert space (CHS) for

our system is a collection of irreducible unitary (or anti-unitary)

representation spaces of the symmetry transformations. Can this

information be of any use before we have determined the complete CHS?

*) This result, proved here group theoretically, has been the subject of

many recent papers where it was proved either from the assumption of

local field theories (P* Carruthers, Phys. Rev, Letters l£, 353 (1967)),

or from analyticity (crossing), (H, Lee, Phys. Rev. Letters 18, 109-8

(1967)). A proof similar to ours but using OPT was given, B. 2umino

and D. Zwanziger, Phys. Rev. I64, 1959 (1967).

-11-

The answer is yes, if we know the properties of a physical quantity

as a particular tensor operator (see definition in Appendix IV) with

respect to the symmetry group. In that case, partial information

relating to the dependence of the matrix elements on the states can

be derived from the commutation relations of tensor operators

(eq. A. IV. 45 ), In ordinary quantum mechanics the concept of

symmetry is used in the narrow sense to mean the symmetry of the

Hamiltonian. The corresponding group is the group of degeneracy

of the energy. One then introduces an additional physical postulate,

the socalled "irreducibility postulate" which says that each eigen-

space of the energy is an irreducible carrier space of the maximal

symmetry group. Only then can one relate properties of states

within each multiplet of G. We can see that the "dynamical group"

approach explains this "postulate" rather naturally (see the remark

in Sec. 1.13 after Ex. 1).

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2. QUANTUM DYNAMICS

In Sec. 1. 5 T have defined "quantum dynamics11 to be the

determination of the concrete Hilbert space (CHS) of a physical

system in order to evaluate the probability amplitudes as the scalar

product of two state vectors. In Sec.l. 7 it was shown that this CHS

must be a collection of carrier spaces of the unitary (or anti-unitary)

irreducible representations of the symmetry groups of the system.

To proceed further it is important to distinguish between an

isolated system and a system plus interaction. An isolated system

has a CHS which is a single irreducible representation of the

geometrical symmetry groups (such as the space-time symmetry

groups). External interactions cause transitions to other states;

they cause the system to reveal its internal structure. We must then

consider the system together with interactions (i.e., with other

systems) as a larger unit which now possesses in terms of the new

relative co-ordinates a larger symmetry than the original system.

According to this plan I shall begin with the simple quantum

systems and simple interactions and then define a new generalized

concept of symmetry, the dynamical symmetry, and then give a group

theoretical formulation of quantum dynamics. Eventually the boundary

between "kinematics" and "dynamics" may completely disappear.

2.1 Definition of simple quantum systems

Using the equivalence of descriptions provided by the symmetry

transformations we can already define special quantum systems by

the irreducible representations of the symmetry groups. Such simple

systems are approximations to the more realistic physical sjstems,

under certain conditions, when the external interactions restrict the

complete freedom of the system (or when those simple interactions

are involved which only cause transitions to a limited number of

states out of all the possible states).

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Consider the rotational invariance and the concrete Hilbert

space (CHS) resulting from the irreducible representations of the

rotation group.

Definition: A j - rotator is a system whose states belong to the

irreducible D -representation of the rotation group. The Hilbert

space is (2j + l)-dimensional and a basis is labelled by jj;m)> ,

j = fixed [Appendix IV] .

A j - rota tor cannot move (rest frame) and does not have any

other degrees of freedom . The system is completely characterized

by the statement that any two descriptions related to each other by

a rotation a re equivalent (Sec. 1.6). Indeed, if <p = \ qj | m)>- ,

Z Z_j r n

ip ] m y a re arbitrary states, the description of the system

by the states |(p, ip, ... I and by the states j DJqj, D ip, . . . 1 are

equivalent:

(Dj<p,D3<&) = (qj.D

m

The following considerations apply equally to an I-isorotator, a

system with isospin I whose other quantum numbers do not enter

into the processes considered.

The system can be easily generalized to include discrete symmetries.

For example, if we include reflection symmetry as well, that is,

consider the full group O(3), then the parity quantum number is

introduced. The parity of the state Jj;m)> can be assigned as

(-1) , for j = integer and the Hilbert space need not be doubled; for

3 = half-odd integer, we have to double the Hilbert space in order to

define parity eigenstates (Appendix IV.7. 3).

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• • v , >*• • •

External interactions allow us to observe the change of the

state in such a system. When these are small enough not to form

new systems (i.e., interactions which do not "break" the rotator ),

they can be represented by operators acting on the carrier space of

D . The simplest interaction is one which just rotates the rotator,

in which case this active point of view is equivalent to a passive

point of view of a rotation of the co-ordinate frame. The effect of

an external interaction will be represented by the "interaction

vertex"

Interaction J

and the matrix element

M = < m 1 ) . j | m > - (2.1)

is the transition probability amplitude that the measurement J (i. e.

interaction) will cause a transition from state m to the state m1 .

This is the only measurable quantity; the wave function itself need

not be introduced. If, for example, the measurement is a

rotation and m is the value of the angular momentum along the z-axis

of the system before the measurement, then

< m ' | e ^ " ? ] m > = D ^ M ) (2.2)

is the probability amplitude that after the rotation the angular

momentum has the projection m' . Clearly, the sum of

probabilities over all final states is \ JD J | = 1 . Accordingr i_~i m m

m1

to the fomulas (A. IV. 52), we can introduce rotated states

(with respect to a fixed direction) ] m j 0 > = e * | m > , then (2. 2)

is the overlap ( m 1 , 0jm> = <m ' |m , -

When the interaction "breaks" the rotator, the Hilbert space must be enlarged to include the

degrees of freedom of the constituents.

- 1 5 -

More generally, the interaction vertex may be described by a

tensor operator (A. IV. 11) so that

M - (2.3)

for example, a "vector vertex" may be of the form

M - g

where g is a coupling constant and the matrix elements <( m" j j j m

have been evaluated in {A, IV. 54).

Note that we have not introduced the notion of the "time-

evolution of the state", nor the Hamiltonian. The states of the

system are asymptotic states labelled by jm^ only (j is fixed).

An experiment is an interaction vertex and we evaluate the

probability of transition to the final state. This formulation is the

same for relativistic and non-relativistic systems. The example

of the j-rotator is indeed very simple and restrictive. More

realistic physical systems undergo transitions with a change of

mass (or energy), of momentum and of other properties under the

interactions. In order to account for these properties we have to

enlarge the Hilbert space and the symmetry groups. We shall

consider systems with increasing complexity as shown in the

following Table:

Rotation group

Galilei group

Poincare' group

(0(4)Higher groups JO(3,1)

l0(4,l)

Higher groups $ndPoincare group

System endowed with one spin only

Non-relativistic system endowed with spin, energy andmomentum

Relativistic system endowed with spin and four-momentum

System at rest with many spin states and other degrees offreedom

Relativistic systems with many mass and spin states

-16-

2. 2 Elementary systems

We now go over immediately to the largest symmetry groups

associated with the geometrical transformations of space-time: the

Galilei group or the Poincare* group. All the transformations of

these groups have a physical, geometrical interpretation:

a) space and time translations (displacements of the co-

ordinate frame);

b) rotations and reflections;

c) transformations giving a system a velocity ("boost"

transformations).

An isolated system must allow equivalent descriptions under

the Poincare group. Consequently we can define elementary systems

whose concrete Hilbert space (CHS) is the car r ie r space of a single

irreducible representation of the full Poincare' group $>. An

elementary system is characterized by the invariants of $, mass (m )

and spin (j( j"+l)) or helicity. The eigenstates of the displacements

are labelled by |p ;or|> ; the rest frame states Ip^p*55 O;cr)> ,

= O) a re rotational invariant and a velocity imparting is given by

if • M

Here M are the generators of pure Lorentz transformations for the

rest states and § = pen — = psh — for massive particles.

m m

An elementary system may reveal under external probing a

more complex internal structure. We can give an operational

definition of an elementary particle . To do this we first say that

two elementary systems are connected if physical interactions can' connect the Hilbert spaces 7v(m , j .) and f ( m j ) of the two_ ^

A.O. Barut, Dynamical groups and a criterion of elementarity, in

Lectures in Theoretical Physics, Vol. K B , p. 273 (Gordon and Breach,

NY, 1967).

-17-

systems. For example, the connection of Is and 2p states of the

H-atom by photo-absorption or the connection of the neutron and

proton states by /3-decay. We have then:

Definition: An elementary particle (EP) is an elementary system

whose states in no way can be physically connected to the states of

other systems. Its Hilbert space is isolated, i.e. , the states of

one elementary particle |l)> do not form a linear space with those

of other systems ] 2 )> , that is,the superposition |l)> + [2 > is

physically meaningless. The only effect that outside interactions

can have on an EP is to change the state within the irreducible

representation, i. e., to change its momentum. It follows that an EP

can have only those internal quantum numbers for which there are

absolute superselection rules (see (A. I. 3) and (A.. 1.13)).

This operational definition of an elementary particle reflects

the dependence of the concept on the nature of interactions, as

it should be. Clearly, in the kinetic theory of gases, for example,

the molecules are elementary particles for, under the processes

considered,the internal structure of the molecule is not excited

and there is no connection to other parts of the Hilbert space.

Similarly, nuclei are elementary particles in atomic phenomena, and

so on.

As we did in the previous section, a "measurement" on an

elementary particle will be described by an interaction vertex

M = <FjOr'jJ]p;cr> - <>'|e J e *

Here m and j are fixed on both sides (hence the label § inside the

state vectors is not necessary in this case). Again J can be a scalar

or a general tensor operator.

-18-

i-t

2. 3 The generalized concept of symmetry

There are some transformations which are important in

determining the concrete Hilbert space of the system, but which

obviously do not correspond to the geometrical transformations

discussed in the previous section. As prototypes/the O(4) group for

the bound states and the O(3,1) group for the scattering states of the

H-atom for a given energy, the SU(3) group for the multiplets of

hadrons and the O(4,1) group for all levels of the H-atom. At first

there seems to be no physical interpretation of the corresponding

transformations. Furthermore, they violate, for isolated systems,

the conservation laws and the supers election rules: we cannot

perform a superposition of a proton and a neutron state, for example.

However, these groups clearly account for the internal degrees of

freedom and for the complete concrete Hilbert space of the

system, in the same way as the Poincare' group accounts for the

external quantum numbers (momentum and spin) of the system. How

are we going then to interpret these larger groups physically?

The clue to this problem lies in the interactions. We see from our

discussion leading to this point that the new transformations referred

to above are not accessible to isolated systems and they make no

sense. But thej are physically meaningful for the larger system

consisting of (system + interaction). We assert now that they

are "symmetry" transformatioreof this larger system.

First of all the clash with the superselection rules and

conservation laws disappears once we identify the new transformations

with the external interactions changing physically the state of the

system. For example, a neutron is indeed transformed into a

proton under the weak forces . For our original system alone

these transformations constitute physical changes, not symmetries;

:f. the discussion in A. O. Barut, Phys. Rev. 156, 1538 (1967).

-19-

they lead to an inequivalent description. But for the combined system

they can be considered as symmetries, in the sense that one obtains

from one possible state of the system another possible state of the

system, corresponding,in general, even to a different total mass and

spin. It follows from our definition of elementary particles in the

previous section that the systems admitting these more general

transformations cannot be elementary. This is also intuitively clear

from the existence of new quantum numbers corresponding to internal

degrees of freedom.

2. 4 Physical interpretation of dynamical transformations

Once the composite structure of the system is taken into account,

it will be possible to give a physical interpretation of the new

transformations. They correspond to active changes in the internal

constitution, changes in the distribution of matter by outside agents.

The characteristic of geometrical transformations was that the active

point of view (i.e., rotating a system) was equivalent to the passive

point of view (i. e., rotation of the co-ordinate frame). This is now no

longer true; there is no simple passive transformation of the observer

which would have the same effect as the external interaction on the

system (see however. Sec. 2. 5).

As an example we consider the O(4, 2) group and the H-atom.

Now the geometrical symmetry transformations connect states of the

atom with the same internal shape, but at different points in space,

at rest or moving with some velocity and with different orientations of

the plane of the orbit. In addition we can now make dynamical

transformations by changing the relative distance between the electron

and proton (e.g. dilatations), by giving an extra tilt to the orbit or

by giving extra energy to the electron (photo-absorption) so that

it moves faster. The important point is that although a continuous,

infinity of such transformations in the internal structure could be made,

all possible states may be obtained from a denumerable basis set .

The same situation occurs, by the way,also for rotations: although

there are infinitely many orientations of the plane of the orbit, there

- 2 0 -

are only 2(2j + 1) distinct basis states out of which all other states

can be constructed. In fact the dynamical group idea suggests

that, in the rest frame of the system, the dynamical transformations

on the system form a relatively simple group.

2. 5 An analogy. General theory of relativity

Although a non-quantum theory, the general relativity offers,

at least conceptually, a complete analogy to our dynamical

transformations. Here we have,in addition to the Poincare group,

physical changes corresponding to gravitational interactions of the

system . Under these interactions the system makes transitions to

new states not accessible to free systems, just like the dynamical

transformations in the H-atom, or the SU(3) transformations changing

neutron into proton. The further important feature in general

relativity is that the active point of view (gravitation) can again be

made equivalent to a passive point of view by enlarging the

transformations of the Poincare'group by all co-ordinate transformations.

Thus changes under gravitation become "symmetry" transformations

on the same footing as changes under rotations or displacements.

The concept of symmetry is now generalized. Equivalent descriptions

of the system now also include descriptions which differ from each

other by the presence or absence of gravitation. In other words

gravitational forces may be eliminated in favour of the dynamical

geometry .

*>

That the symmetnesof general relativity are of dynamical nature is

also the conclusion of Cartan, Fock and Wigner, see

K.M. F. Houtappel, H. van Dam and E. P. Wigner, Rev. Mod. Phys.

37, 595 (1965) for a very detailed discussion of geometrical symmetries

and for other references. See also A.O. Barut and A. B6*hm, Phys.

Rev. 139. B1107 (1965).

-21-

2. 6 Complete and approximate geometrization of dynamics

If the interaction can be completely eliminated as in general

relativity the system and its gravitational interactions are described

completely, in quantum theory, by a single unitary irreducible

representation of the generalized symmetry group (dynamical group).

The transition probability under gravitation is just the overlap

between the two relevant states )O and ]2)>

M = <1|J]2> ;

in particular we expect J = 1.

If, however, we have not eliminated the complete interaction in

the Hamiltonian

H - H0 + H l + H 2 '

but a part H + H only and succeeded in obtaining the states under

H + H from a dynamical group, then the interaction operator in

M '

will be quite simple in contrastto the situation when only H~ would

have been replaced by symmetry transformations. This is indeed

the case. In the example of the electromagnetic interactions of the

H-atom, for example, the use of the larger dynamical group O(4, 2)

brings forth that the interaction operator is to a large extent a

constant operator, V , with a small part depending linearly on the

momentum.

- 2 2 -

2. 7 The concept of "interaction" versus global point of view inrelativistic theories

The dominant point of view in microscopic theories is based

on the separation

system + system + interaction

This view presupposes that the systems under consideration still

preserve somewhat their individuality even under the interaction.

Whether such a separation is possible for relativistic systems is

not clear. There are some well known difficulties connected with

the proper times of the systems even in classical relativistic

mechanics. Besides these fundamental difficulties, a theory based

on such a separation, in the relativistic case, always leads to an

infinite system of equations; we know this from our experience in

perturbation theory or in S-matrix theory. In axiomatic field

theory J, or in Heisenberg's non-linear theory the above

separation is not made, but one operates with interpolating or with a few

self-coupled basic fields. The dynamical group approach, on the

other hand, deals with the observable quantum numbers only; it is a

global description of interacting systems in which the concept of

interaction (e. g. potential) has been eliminated in favour of total

observable quantum numbers and their range. There is thus the

possibility that the behaviour of the system can be describedthe

relativistically in finitely many (few) steps. The example of [relativistic

H-atom and positronium in which the recoil effects are evaluated

globally is very illustrative % Thus the approach of dynamical

groups is not only an algebraic reformulation of the present theories,

but may also help to solve some of the difficulties of the relativistic

quantum theories. •!•_

H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo CimentoU 425(1955),-A.S. Whightman, Phys. Rev. 101, 860 (1956).

* *'W. Heisenberg, Introduction to the Unified Field Theory of Elementary Particles (London, Interscience,1966). i.

* * * ) A . O . Barutand A. Baiquni, ICTP, Trieste, preprint IC/69/3.

- 23 -

2.8. Survey of further chapters

The ideas expressed in this chapter will be extended to larger

and larger groups. According to the Table in 2.1, we shall considerand

the concrete Hilbert spaces of the Poincare' groupithose of the larger

groups containing the internal quantum numbers, together with the

corresponding physical systems in which dynamics has been

expressed as "geometry".

-24-

APPENDIX I

WIGNER'S THEOREM ^

Let the rays tjr , $.. , . . . represent the states of the system in

one description and "(jr , $ 0 , . . . represent the same states in the second

description.

Let ^ , (p1 , . . . and 0 , <p , . . . be two sets of orthonormal

bases chosen from the first and second set of rays. We are given a

transformation which preserves the absolute values. The problem is

to construct a corresponding transformation U on the vectors by the

suitable choice of the phases. It could be a priori that U0.. = c^09 ,

Uqpn = c <p , . . . and that no relations between the c's can be es-1 <p 2

tablished, in which case U is not even a linear operator. We want,

in fact, to show 1hut U can be so defined that it is a unitary or an anti-

unitary operator.

We single out the unit vector 01 and choose U^i = 0O . This& 1 1 2

is the only arbi trary choice and shall show that all other phases are

uniquely determined. Thus U is determined up to an overall phase

factor.

Next consider the vector 01 + <p1 . A representative vector

of the corresponding ray in the second description is aip + b<p .

We have then

b'cp2

where we must have c = I /a by the previous choice, and put

cb = b1 = b/a . We now define Uep. by 11(0. + qpj - 0 , or simply

*> E. P. Wigner, "Group Theory",(Academic P re s s , N. Y., 1959) Appx. 20.

For more details see V. Barg'mann, J. Math. Phys. J5, 862 (1964);

U. Uhlhorn, Arch. Fysik, 23, 307 (1963).

-25-

by b'<P9 • Hence

<p,) = Uf + U<p.

Similarly for a general ^ = a ^ + a ^ + , . . we choose a represent-

ative f2 " 5 ^ 2 + 3 ^ ^ + . . . . write Uf 1 = cf2 with ca^ = a^ , so

that

U ( a / l

Now we form

and

a , + acp

These two numbers must be equal. This plus the fact that la ' I = I a |• (p ' ' cp'

allows us to calculate a1 in terms of a and a, . One obtains twocp cp 0

solutions:I 1 n

<Pand i1 = a*

cp cp

Clearly for the first solution U is linear and unitary. For the

second solution we find Uf. = a ,/a* [a* IM, + a*Ucp.. + . . . 1. An over-1 if/ if/ if/ 1 ( p i

all phase factor is unimportant and by a new normalization of U -

which by the way does not change tty = 0 - we obtain an anti-unitary

operator.

- 26 -

MWitf ;)"U* I

APPENDIX II

COVERING GROUP, PROJECTIVE AND RAY REPRESENTATIONS,GROUP EXTENSION

Ray representations:

It was shown in Appendix I that corresponding to every element

x of the group of transformations between equivalent descriptions, a

unitary (or anti-unitary) operator Dfx) can be constructed which is

determined up to an over-all factor w(x) . We have then written the

group law as (Sec. 1. 9)

D(x)D(y) = u(x.y)D(x,y) . (A.II.l)

Although the phases u(x) are completely arbitrary, u(x, y) is not.

First of all, two phase systems u(x, y) and u'(x,y) may be defined to

be equivalent if

c(x) arbitrary function, because then the corresponding D(x) and

D (x) differ only by a phase D (x) = c(x)D(x), hence have the same

physical content. Therefore, functions w(x, y) not related by (A. II, 2)

determine inequivalent ray representations. Furthermore the

associativity law of the group multiplication puts another restriction on

the phase system ufx, y) :

ufx, y)u(xy, z) = u(y, z)u(x,yz) . (A.II.3)

Note that u'(x,y) defined in (AJI.2)satisfies GUL3), if w(x, y) does.

It is easy to see by taking very simple examples that (A. II. 3) even

up to equivalence(pLLI.2) does not uniquely determine the phases ufx, y)

so that we have, in general, a number of new inequivalent ray represent-

ations for a given group G , besides its vector representations.

Protective representations:

Let Dfx) be a representation up to a factor of G , x e G ,

acting on vectors v c 3( , Let v. be the components of v in some

- 2 7 -

co-ordinate frame. A ray can be represented by the quantities

v. = v./v , for clearly all vectors in the ray £Xv} induce the same

v where v is any component singled out. By choosing a special

vector v with v1 = 1 we see that the transformations induced on v

by D are

V' = vI 1

oil

^i = N [ Z DiK(x)^K + D i l ] ' i " 2, 3. . . . (A. II. 4)

These transformations are non-linear and are called protective

transformations. If H is infinite dimensional eq.^..II.4) makes sense

only if the sum converges. It is easy to check that representations

D(x) and c(x)D(x) , as well as inequivalent ray representations

u(xy) i 0 , induce the same projective representation £1.11.4). The phase

ambiguity has completely disappeared in this formulation, but it is only

hidden, because the inverse problem of finding all inequivalent project-

ive representations is equivalent to finding all inequivalent ray represent-

ations.

Group extension and the covering group:

The remaining phase ambiguity precludes the application of

mathematical theory of vector representations when w(x,y) / 1 . In

this case we can try to construct a larger group E whose vector re-

presentations give all the inequivalent ray representations ( .JI.1) of G .

This can be done simply as follows: Let K be the abelian group

generated by multiplying the inequivalent phases w(x,y) satisfying (A.II.3),

Consider the pairs (u, x) , u e K , x e G . In particular, K ={(w, 1)}

and G = [(1, x)^ . The pairs (w, x) form a group with the multi-

plication law of a semidirect product

- 28 -

V 1 V X2 ) = (U1W(X1X2)W2' X1X2) " (A* I L 5 )

In the present case we can think of (u,x) as ux . The group

6 = {(u, x)3 is a central extension of G by K and we see that the

vector representations of O contain all ray representations of G ,

Thus the extended group 6 may be considered as the proper quantum

mechanical group. The theory and applications of group extensions

will be discussed later. Here we point out only that the quantum

mechanical group of a continuous group G is the so-called covering

group G .' If the group space S of G is simply connected, then G

is the same as G ; if G is not simply connected we construct a

simply connected covering space S to S , then G is that group whose

group space is S , G and G have the same number of parameters

and the same Lie algebra. In general there are several Lie groups

with the same Lie algebra, among them there is only one which is

simply connected and this is the covering group; the others a re factor

groups of G by one of its discrete invariant subgroups D . A simple

example discussed in Sec. 2 of this Chapter is G = SO(3) , G = SU(2) ,

SU(2)/Z = SO(3). Other examples and applications will be discussed

as we go along.

In particular we have the following useful result. Finite-

dimensional representations of simply connected continuous groups are

equivalent to vector representations.

To prove this, first, quite generally, take the determinant of

eq.(A.H 1) tdet D{x) det D(y) = un(x, y) det D(x,y) , where n is the

dimension of the representations. The new representation

D'(x) = D(x)/[det D(x)]1/11 formally satisfies D'(X) D'{y) = D'(X y):

There a re different values of [det D] ' n and we can pass to an equi-

valent phase system such that D*(x) D'(y) = w'(xy) D'(xy) with u ' n = 1 .

Now if the group space is simply connected [u ] can be uniquely

defined and is the same for all x by continuity. Hence we arr ive at

an equivalent vector representation.

Similarly the ray representations of the one-parameter subgroups

of Lie groups are always equivalent to vector representations.

-29-

APPENDIX III

WHEN UNITARY AND WHEN ANTI-UNITARY OPERATORS?

We have seen that the anti-unitary case cannot occur if the sym-

metry transformations are continuously connected to the identity {see Sec.

1.12). One can also see that the invariance of a state which is a super-

position of two stationary states with different energies at time t also•4

eliminates the anti-unitary case. For then the operator correspondingto the second solution denoted by A would give

E E E E

J = e

whereas the correct solution of the Schrodinger equation is

E l E2

e A ^ + e Acp . On the other hand, for the "reversal of

the direction of motion" (time reversal), the stationary states 0(t) and

En- l t

En- i —" t

e (p and \ en

n nThis last state muHt be (by time reversal invariance) at the same time

.E . .E .e

n n

.E . .E .1 -h O - 1 ft

iL = ) e i> •

n /_, nThus, here only the second solution is correct.

*)' E .P . Wigner, Group Theory (Acad. Press, NY, 1959) Appx. 20.

- 3 0 -

APPENDIX IV

THE GROUPS O(3), SO(3), SU(2), THEIR REPRESENTATIONS

AND THE QUANTUM THEORY OF ANGULAR MOMENTUM

1. Parametrization and group space

The rotations may be parametrized by the unit vector n , the

axis of rotation,and the angle $ of rotation round n . Other para-

metrizations will be given below. The group space S can then be

represented as the 3-ball of radius it with the diametrically opposite

points being identified; S is connected, but not simply connected.

For the rotation group O(3), S is double connected, for O(2) it is in-

finitely many times connected. The simply connected covering space

S of S for O(3) is the space of two 3-balls with a point P on one

sphere identified with a point P' on the second sphere; for O{2), s'

consists of infinitely many segments with the endpoints identified.

For O(3), S is also equivalent to the surface of the 4-dimensional sphere:2 2 2 2

x1 + x + x + x = constant.1 2* o 4

The group whose parameter space is the simply connected cover-

ing space S of S is called the covering group G of G . (See also

Appendix IX} For SO(3): A point Q(n, 6) in ball 1 and the corresponding

point Q(n, 6) in ball 2 represent the same element of G , but different

elements of G . I shall show that the covering group of SO(3) is SU(2).

2. The defining three-dimensional representation

For the rotation of a vector r round n by an angle d6 we have

dr = d0 fi x r = d0 !.*> ?= d0(?«fi) r* , (A. IV. 1)

where in a Cartesian co-ordinate basis

=T-n . (A. IV. 2)

-31-

Hence [I., I.] = 6... I, .

Integrating eq. (A. IV. 1) we obtain

r = R(n0)r , R(f*0) = e n , (A. IV. 3)

or, because I*, = -IA ,n n

Hence

R(&0) - I + L, sin0 + I? {l-cos0) .n n

r = ?Q + sin© fi x r + (1-Cos0) n x (S x ? ) . (A. IV. 4)

In components, writing

we getRk^ = cos0 6k^ + (1 -cos0) n V - sin0 £k^ j nj . (A. IV. 5)

One can verify that RR = I or R R = 5 ; using the identities

t t j l t H

Thus R is orthogonal (unitary and real), det R = +1 , i. e., R € SO(3),

and is identical with the unitary irreducible representation D (n*0).

The group law is

R(n\0) = R^ejR^ej ,

fl 1 9 1 9

cos- = cos-y cos~^" " " i ' " 2 Sin~2~ Sin~2~

(A. IV. 6)

ii sin~ = ft sin— cos— + n sin-r- cos— + n,x fi sin— sin—£t \. ct Ct ct ct it £t Ct £t

This result can be put in a more elegant form. Let

-32-

, ?= f i s in | , 0 ^ 0 4pr (A.IV. 7)

2 -*2 JH _>2 V /^^-^

with a + a =1 , a = + l l-a ,xe., the group space isl/the 4-dimensional

unit sphere/ The group of motions of the group space S is O(4). The

law of imposit ion (A. IV. 6) now reads

3 =«<2

If we now associate a real quaternion to the three group parameters by

o <y oi - j = k = - 1 , i j = - j i = k (cyc l ic ) ; aQ, a^ j

(A. IV. 8)

then the group law becomes

Thus we have obtained a representation of SO(3) in terms of quaternions:

q = 1 is the identity rotation, q = a - ia - jot - ka is the inverse

rotation; qq = *qq = 1 .

*") For complex a , a. we can get a representation of the Lorentzp

group.

-33-

3, Passage to the 2-dimensional double-valued representation

The quaternions can be represented by 2 x 2 matrices via the

correspondence

io J-H>-HT k^>-i<T '

q-> aQ - i ^ s + U . (A. IV. 9)

We have then U+U = UU+ = (a - ia-c)(a + ia-3) = a2 + B2 = 1 .

Thus we have a representation of the rotations in terms of the matrices

of SU(2)O In terms ol the original parameters {A. IV. 7):

+U = an - !<?•? = cosf - io*-n*sinf = e" 1 ^" °l2

This can also be written as

n !<?? cosf ionsinf e . (A. IV. 10)

±U = (^ h\ , | a | 2 + | b | 2 = l (A. IV. l la )

where the new parameters of the rotation group a re

9 . . 6 , . . v . 9a = cos- - i n s in- , b - (- i n - n ) s in- . (A. IV. l ib)

ii O Z X £, £i

Another set of parameters, i. e.,the real Euler angles, characterizing

the rotations is given by

a = e i (* + ^ 2 cosf , b=e*~v)!2si4 . (A. IV. 12a)

2 + -1<r. = 1 , t r c = 0 , tr. = or. = or. , <r x a = 2i a or

X 1 1 1 1 ~ ~ —

CT.CT. = 5 . . + i £. or,l j 13 ljk k

( a * c r ) ( b * < 7 ) = ( a « b ) + i ( a x b ) ' t r

Z ( a , ) . (or, ) . = 2 6 , 6 U - 6 ^ 6 ,k a b k c d a d b e a b c d

k

t r ( o \ o \ ) = 2 6 . . ; t r { < r . c r . o \ ) = 2 i 6 . . ' t r ( o - . c r . c r . c r ) = 2 ( 6 , . 6 t - 6 . , 6. + 8 .i ] 13 1 ki jkZ ] k i m lk ^m it km 3m

-34-

Then

>/20

0 e" iy/2

= e e e " = U(A*,£,IO or D(K,£ ,V) . (A.IV.12b)

Thus

U^Oijg.i;) = U(-*if-5,-i/) = U (-V,-5 , -M) . (A. IV.13)

The basis for the two-dimensional representation is |—; m >jm = +— j

so that

<m' | U ,n, ?, v) | m> = e <ro' | e * | m>

= e i 0 i m - + v m ) d l / 2 ( } ^ (A. IV. 14)m m

1/2In fact the values of d ' (?) a re given in eq. (A. IV. 12b).m'm

4. The homomorphism between the two- and three-dimensional

representations

The explicit connection between the two- and three-dimensional

representations is expressed by the formula

R (&) = R..(U) = ^ t r (U(9)<r. U+(P) a.) . (A. IV. 15)i] — IJ 2 l j

This formula can be proved directly by inserting eq. (A. IV. 10) into the

right-hand side and taking the trace; the left-hand side is given in

(A. IV. 5).

It is also ii'structive to prove it as follows: Consider hermitian

and traceless matrix X = r <r. . The only matrices which transform

hermitian and traceless matrices into hermitian and traceless matrices

- 3 5 -

I 4.again, are multiples of unitary matrices. Hence X = UXU is alsohermitian and traceless and consequently must be of the form

1 'i ' 2 12

X = r a. . Because det X = det X , we find r = r , i. e. , the

induced transformation on the space of r is an orthogonal transform-

ation. Writing r = R« r ^ one obtains from JC = UXU+

r" CTk = U r 1 CT. U = R £ r V (A. IV. 16)

hence eq. (A. IV. 15). Q. E.D.

Using the formula (A.IV.lEj) one can explicitly verify

(a) R(UJ R(UJ = R(U n )

I 1 2 1

(b) R(U)* = R(U) , R(U)T = R(U)"1

io(c) e v U—>R(U)(d) det R = +1 . *)

From (a) - (d) we see that we have a homomorphism between U(2)

and SO(3) ( reflections cannot be obtained by this homomorphism), i. e . ,

a single-valued map from U(2) onto SO(3) that preserves the multi-

plication . The elements of U(2) mapped into the identity element of

SO(3) are of the form U = e • (kernel of the automorphism). We can

write U = e1^ V , det V = +1 , V € SU(2). We have then the homo-

morphism + V—>R(V) . The kernel of this homomorphism is the

discrete abelian subgroup CL = {I, -i} of SU(2). Thus there are the

following isomorphisms (one-to-one homomorphic maps):

.*/ For any 3 x 3 matrix R

6... det R = € R R . Ri]k imn tx mj nX

Insert this into (A.IV.15) and use the identity

imn m n . r / r 1 s. ~\f u M i a * o - a* 6 Aab cd w ad be cb ad

to obtain £.„ det R = € . Hence det R = +1

-36-

(A. IV, 17)

u ( 2 ) ^ £ ( i m S M ^so(3)HU(l)

The group space S of SU(2) consists of two balls corresponding to V

and -V with one point of each being identified. (In eq. (A.IV. 10) the

point 9 = -K of U and 6 = -x of -U correspond to the same rotation.)

Thus SU(2) is the covering group of SO(3); the unique simply connected

group with the same Lie algebra as SO(3).*'

The inverse formula to (A. IV. 15) can be derived as follows.

Fi rs t from {A. IV. 15), taking the trace of both sides, one gets

Tr R * | trier. U o\ U+]

= t r U t r U+ -1 = (tr U)2 -1 .

Then from eq. (A. IV. 16) one finds

a. R.. a. - 2 U t r U+ - I .

Combining these two results , we have

I + cr. R.. a.

U = + i 1 ^ ,3 . (A. IV. 18)2 4 t r R + I

Now we come back to the quantum mechanical ray representations.

For the three-dimensional representation we had found the group law

i. e . , it is a vector representation. But for the two-dimensional r e -

*' All unitary groups a re simply connected (C. Chevalley, "Theory of

Lie groups" (Princeton Univ. P ress , 1946).

- 3 7 -

presentation (A. IV. 11) we can have

2) U(^) = lUfg*^) , (A. IV. 19)

because U and -U represent the same rotation. Thus the phase

6) (12) of the ray representation has only two values: +1 . The + sign

occurs if both 6 , 6 are in the same ball in the group space, the -

sign if they are in the opposite balls. In accordance with the discussion

of Appendix II, we can also write the elements of SU(2) as pairs

{u, x) , x e SO(3) , w = +1 , with the multiplication law

(6>ljXl) (w2Jx2) =(w1^2>x1x2) (A.IV.20)

which is, in this case, that of a direct product. Thus all ray represent-then

ationsof SO(3) are vector representations of SU(2) which is/the quantum

mechanical rotation group.

5. Relation to stereographic projections

The correspondence R*- > + U has another interesting geometric2 2

interpretation. Consider a sphere of radius r : r = rn and project

it from north pole to a plane with co-ordinates z = x + iy tangent to

the sphere at the south pole. We have the following correspondence:

r l r2 r 0 ' r 3 r 0 " r 3 °

(A. IV. 21)

1 ' " 2Let r1 = R r , then z! = x1 + iy1 ~ 2r_ = f(z) . One obtainsr0 3

precisely

z , = az + b ^ (A. IV. 22)-b z + a1**

i. e., bilinear fractional transformations which form SU(2) [ subgroup

of z1 = a Z , . , ad - be f 0 which is isomorphic to SL(2,C)] .cz + a

- 38 -

. _ - . J. vU— . . - . , ' 1 -

6, Spinor calculus

Vectors in the two-dimensional representation space of SU(2) are

called spinors. .This is the fundamental representation of SU(2). We

shall see that all other representations may be obtained from this one

by the methods of tensor calculus. The spinors transform as

U b | b I , 2 . (A. IV. 23)

+-1 * T-lTogether with U = U also U = U is a representation; but these

two representations are equivalent (in the two-dimensional case) because

(inner automorphism)

U* = C"1 U C , C = (° "*) = -icr2 = -C T = -C"1 ; det C = +1 .

(A. IV. 24)

U are not equivalent. Hence,

transforming according to

In general, for SU(n), U* and U are not equivalent. Hence, the spinors

5

are related to the spinors f by

5a = C a b 5 b : «a = C - X a b | b . (A. IV. 26)

Consequently

e,a n , = ?a ^ = i n v a r i a n t _ - * *) (A. IV. 27)a a

In particular, | £, = 0 „

In the case of the SU(2) a lower dotted index is the same as an upper

undotted one. For SL(2, C) it is different .

**) Note: (U+)b = U b ; U a Ud = (U+U)d - 6d .a a c a c c

-39-

Quite generally, for any SU(n), higher order spinors are obtained

from the direct products of f . For example, the bases in the spinorEL , ,

spaces with two indices are given by | n, , ? *7, (£ *7 ) and | 17 .

The product space is reducible: the symmetric spinors T transform

among themselves, so do the antisymmetric spinors. (This much is

true for general 1'near transformations e GL(n, C).) For the mixedV* W m Q

the trace i Ta

irreducible subspace:

V* W m Q

spinors T the trace i__ T is invariant and forms a one-dimensionala a a

13 = T?' uffTb' = 7 T 3 • (A.IV.28)^ a a b1 a' ^ a

abThus tensors T.. ' ' ° with symmetric and antisymmetric indices are

1 J • . »

tensors of lower dimensions, so is the contracted tensor

T . * ' " = S, * . A tensor antisymmetric in two lower (upper) indicesaj.. . J . • .

transforms like a tensor with these two indices replaced by an upper

(lower) index.

7. The full orthogonal group O(3)

It consists of O(3) and one reflection operator £ through a plane

with the normal n : O(3) = { SO(3) , coset E SO(3) { 7

7 1 Properties < f Z (a "symmetry")n_. —

Def: E r = r - 2 j~~£ i? . (A. IV. 29)n (n- n)

This definition holds for arbi trary pseudo-orthogonal spaces; (n, n) = 1

for orthogonal groups. We have

(1°) E2 = I , E"1 = E , ET = En n n n n

(2 ) E^ n - -n

(3°) If m- it = 0 , then E^ m = in

(5°) R E R"1 = E_ , R e SO{3) .n Kn

-40-

This is the most important property. It shows that reflection

through any arbitrary plane can be obtained from the reflection through

rf and rotations. Hence, det E - det E = -1 .

(6°) t r E n = ti

<?O) E£ l \ ? = * • 2 C ? ' I I 1 ) *1 " 2 ( ? ' n 2 ) S + 4(?-n2

) ( i ?2*"l ) *1

[(n-n) = +1]det (E E ) = +1 .

The product of two reflections is always a rotation. We find

n x nLi — rvln, v) , n — : , v — £<p , n., • n_ ~ coscp

n, n sm<p ^ 1 2 •

Hence we have the re la t ion (E E ) ' T = I .S i S 2 -

(8°) [E ,E ] ? - 4 ( ^ . 0 [(? i t) Tt- - ( r - n J i t l

= 0 if n" • n*o = 0 , or if n* = if .

(9°) Every x eSO(3) is the product of at most three reflections."*)

(10°) Parity P = E1 E E commutes with all rotations:

1 It O

TJ T3T3 _ T> T T*1 T T? — V T* V = f T T : P

P r = -r .

''For a space with the metric g.. r r . : "every transformation leavingi j y j

g.. r r invariant is the product of at most n reflections. Proofi]

see: A.O. Barut, "Electrodynamics and;classical theory of fields andparticles" (MacMillan, 1964), Ch. I.

-41-

7.2 Structure of O(3)

(a) SO(3) is a normal subgroup of O(3). For

E R E~* = R El1 E = R1 = FRn n Kn n

2and F is an automorphism of SO(3). F = I.

(b) O(3) is the minimal extension of SO(3) by an element

O(3) : {SO{3) , E SO(3)].

(c) O(3) is the semi-direct product of SO(3) with the "group

generated by reflections": O(3) - SO(3) 0 ( 2 ] .

If we write the elements of O(3) as pairs (E, R) we have

the multiplication table

.(A. IV. 30)

both sides give acting on "r (compare 7.1, property (7 ))

The multiplication table of cosets is, with Re 80(3),

R

ER

R

R

ER

ER

ER

R

(d) If we use the parity P , O(3) becomes a direct product

SO(3) S P . For any E : ER = PR1 , for some R1 .

PRP"1 = R

-42-

7. 3 Is there a two-dimensional representation of O(3)?

We have seen that all elements of U(2), including those with

det = -1 , correspond to rotations, i .e . , elements of SO(3). Parity

commutes with all elements of U(2). Because U(2) is an irreducible

system, only multiples of identity commute with all U(2). Hence

there is no two-disnensional representation of O(3). In this case we

can double the representation space: The four-dimensional matrices

'u(0) 0 \ /o 1D(e) = 1 1 D(P) =

0 U(0) / \ 1 0

form a representation of O(3).

This result is a special case of a more general one in the re -

presentations of extended group (see Ch. III). If P O P = -6

(as is true for parity) and if D (-6) is inequivalent to D (£) fcis is

true for S = j = half odd integer) then the extended group (G, PG) has

the representation

> D(P) =

0 DS{-0) /

For SO(3), the odd-dimensional representations allow a parity operator

without doubling, as in the three-dimensional case:

P =

8. Lie algebra in the two- and three-dimensional representations

The two-dimensional representation of SO(3) given by {A. IV. 10)

in the parametrization (n, 6) is generated by the fixed matrices

J. ~ +7T^*- with the commutation relationsl 2 l

[J. ,JJ = + i £ i j k J

k • (A. IV. 32a)

Similarly the three-dimensional representation (A. IV. 3), i. e.,

-43-

R(n0) = e is generated by 3 x 3 matrices with the same commutation

relations.

Introducing the dual operators

L.. = 6...L. = -L..

we can write the commutation relations also in the form

or

lLij'V"Lji, • (A. IV. 32b)

The operator L.. is the infinitesimal generator of a rotation in the

i« j-plane.

The commutation relations (A. IV. 32) are valid for every represent-

ation of the group. The elements L.. are said to generate a Lie algebra.

The Lie algebra is most conveniently defined in terms of one-parameter

subgroups.

9. Passage to tensor (and spinor) calculus

The index i on the basis elements of Lie algebra runs from

i = 1, 2 , . . . r ; for f>O(3) , r = 3 . If we make a transformation by an

arbitrary non-singular matrix (a change of basis)

S-Lj (A. IV. 33)

we obtain

where

The two Lie algebras generated by fL.l and {L.* are isomorphic.1 1 k

There is even a change of basis for which the structure constants C..k 1*'

are invariant (i. e., C. are isotropic tensors):

-44-

S; = 67 + a C j . (S )T = 6, - a C . . (A. IV. 35)

Then

13 13 01 1*3 aj i j ' ij ak' '

= C , by Jacobi-identity:

or

cK.t cl. + c ., c3 . + c ,, ck. = o

3if ia 131 aj ak1 ji

C k . C1 + cyclic in (....) = 0 (A. IV. 36)1

which fol lows f r o m

[L . , [L . ,L 1 ] + cycl. = 0 . (A. IV. 37)1 3 K

For SO(3), the transformation (A. IV. 35) is

i i k iS = 6! + 6 n C .

i i ki

and coincides exactly with the infinitesimal rotation (A. IV. 5). Thus,

under rotations (in the Lie algebra space) the structure constants are

invariant.

Corresponding to L. = S, L, we can define a matrix transformation

JT f 1 . T ' . <,j r

Then

JW.,1^.} A'1 = c ! Jk <fL, 'if"1 . (A.IV.38)1 3 13 k

In particular, all transfer mations A which leave the structure con-

stants invariant form a group which is isomorphic to the original group.

Similarly, all S. which leave the structure constants invariant form an

r x r matrix group. In fact we shall show that for every group element

x there corresponds an S.

- 4 5 -

S.(x) : r-dimensional adjoint representation of G

) : any representation of the group

10. The adjoint representation

A one-to-one continuous map of the group elements

x —• x1 = a x a , a = fixed group element,

(A. IV. 39)

is an inner automorphism (F ) [x,x' a re in the same conjugate class] .

If we consider a curve in the group space x(t) , x(0) = 1 and form the

tangent to the curve at t - 0 , we obtain for the Lie algebra

€ —• ?' = a ? a"1 . (A. IV. 40a)

For an arbitrary representation of the group

D(x) —>D(x') = D(a) D{x) D(a)"1 (A. IV. 41a)

which gives for the representation of the Lie algebra

L L' = D(a) LD(a)" 1 . (A. IV. 41b)

This is of the form (A. IV. 38) with D(a) —«• A . Eq. (A. IV. 40), i. e . ,

§• = a ? a , provides an r-dimensional representation of the group in

the space spanned by §. (or L.) , it is called the adjoint representation;

Ad(a) ? = a | a"1 . (A. IV. 40b)

Indeed, Ad(I) = I \ Adfa"1) = Ad"1 (a) ; Ad(ab) § - Ad(a) Ad(b) | .

What is the Lie algebra of the adjoint representation? We con-

sider a curve aft) t

Ad(a(t))f = a( t ) f a(t)"1 .

Then

d Ad[aW I f - i« - €»l - [*l. 5J • Ad(i|) ? (A. IV. 42a)dt I t = 0

Ad(r7) is a representation of the Lie algebra,

-46-

In terms of a matrix representation and with a definite base L.

we haveAd(L.) L. = [L., L.] = Ck. L . (A. IV. 42b)

1 3 ! 3. !J K

i. e . , Ad(L.) is represented by the "left-multiplication" in the Lie1 k

product sense. In the basis L. the matrix C,... represents L. .3 (1)3 1

In wother form, from L - D(a) L D(a) we get infinitesimally

= (I + Ad(L )) La

and this is precisely the S that we had introduced in eq. (A. IV. 33).

When applied to the two- and three-dimensional representations of

SO(3)*U(2) , eq. (A. IV. 41b) implies the relations

- 1 •*)

D J. D = R J ' (holds also for parity)(A. IV. 43)

U tr. U = R, . cr,l ki k

Thus these relations are true for arbitrary representations D and U .

(The second equation is then a special case of the first.)

From J one can form tensors J. J, , etc. and their linear com-l l k

binations T.. , etc. These tensors have the transformation property

D T., D"1 = R.,. R, , . T , , etc. (A. IV. 44)

]k, . . ]'] k'j j'k1

(in particular for a scalar operator DAD = A).

11. Tensor operators

In these equations we have always the adjoint representation R..

on the right-hand side, because the indices j , k . . . run from 1 to 3 .

The generalization of these indices lead to the concept of tensor

operators; T " ' * These transform like, e. g.,m 1 m o . . .

D T J ' D~ = D , T , (A. IV. 45)m m'm m'

In fact the commutation rules for J can be obtained from this

equation. -47-

J f (u) d/u(u) = J f(gu) cl/i(u) I u e G/K , g e G . (A. IV. 50)

Thus we shall obtain a unitary representation of G in this Hilbert2

space of L -functions, which in the quantum mechanical application are

the physical wave functions.

For G = SO(3) ; K = SO{2) , G/K is isomorphic to the surface of

the unit 3-sphere". Hence we consider the functions f(0,<p) and the

scalar product

(f,g)= r f *gdn , dfi= sin0 d0d<p .

The unitary representation of G = SO(3) is then given by

D(R) f(0cp) = ftR'N^q))) . (A. IV. 51)

The representation defined by this equation (regular representation) is

in general reducible. The irreducible functions are precisely the

YL(0,q>) , eq. (A. IV. 49). The reduction of the regular representation

inj;o its irreducible parts is the generalization of Fourier expansion

(see Sec. 13 of this Appendix),

We emphasise that the variables (0,<p) are just auxiliary quantities

introduced in order to express the Lie algebra elements as differential

operators ; they are not in any way essential in expressing the symmetry

conditions. In quantum theory, also, the observable quantities do not

depend on (0,9) ; in the scalar product / 0 (0qp) 0 (0<p) d& one

integrates over these angles. Indeed it is possible to formulate quantum

theory without the use of the customary wave functions. {See Sec. II

text.)

12.3 Transformation properties of the eigenfunctions.

From our fundamental relation D(R) J D(R)~ = j ' and letting

j j,m> and I j , m>' be the eigenvectors of J and J* both with eigen-

values m , we obtain

-50-

j . m1 >

a n d , (A.IV.52a)m1

Hence

The following notations are all equivalent;

Parity: Under parity 0' = T - 0, ^' •= )r + cp, hence

From eq. (A. IV. 52') follows (take 0 =cp = 0, R, {

Thus

I rf^«P) CA.IV.52b)

D l o ( W f l ' ) = ((UTT)) ^m<«-") • (A.IV. 52c)

12.4 Matrix elements of the generators.

We recall the standard quantum mechanical treatment :

J±= <Ji±iJ2> y r

[ J ± , J 3 J = T J ± ; £J + J_1 = Jg • (A. IV. 53)

Let ni y be the eigenvector of J_ with the largest eigenvalue.

-51-

The J_J m > = (m -1) J m a x ^ . Hence ( J i m ) ) is an3 - I m a x ' max - ' ' - I m a x '

J m )> = h m -1 / e igenvec tor of Zn with the eigenvaluev - I m a x x - I max J 3 s

v vr-——— -""(m -1): repeat ing this p roces s we find (J ) |m y=a\m - r ^

max ' r & - I m a x ' I max '2 2 2

Now J = J J, + J_ + Jn, hence J m )> = m (m -1)3 " 3 m a x ' max max3 3Jm )>,. The eigenvalues of J must be r e a l and non-negative:

| J[ V |j > 0. F o r finite-dimensional representa t ion ,

the re exis ts an n such that J tm -n]> = 0.— I max

J 2 | m - n > = (J J + j f - J ) j m - n > = ( ( m - n ) 2 - (m -n1 max ' + - 3 z I max ' v max max

= m (m +1)max max

or,nm = —max 2

mmax

Now from <( jmj [ J , J_] | jm.)> = m, we can determine all the matrix

< j m | J 3 | j m ' > = m'

+elements

< j , m ± 1 | j ± | jm> = ^ i / ( j ± m + 1) (j T m)' } (A. IV. 54)

in Condon-Shortlcu phase oonvention.

12. 5 Representation in terms of boson creation and annihilationoperators

The expressions

**)satisfy the commutation relations of the Lie algebra of 0(3) . Here

*) 2For the same reason m (m +l)-r is also real aid non-negative.

max' max 6

In differential form X ^ ^ ^

-52-

[a. , a* ] -6... We consider the following basis in the Hilbert space:

. m> E Nj a / + m a* m | 0> . (A. IV. 56a)

2I t i s c l e a r t h a t J | $, m)> = m j # m > a n d J J ^ m ) > = 0(0+1) | 0 , m )>>

by direct computation. Also J, | 0> $y - 0, J | 0, - ^> ~ 0. From

(J ) = J one obtains the matrix normalization

* = [ (0 + m) 1 (* - m)l ] "^2 . (A. IV. 56b)m

Hence the matrix elements (A. IV. 54) of J , J .

Thus, arbitrary unitary representations have been obtained from

the two-dimensional fundamental representation.

12. 6 Representations of the group elements (SU(2)).

Let the group elements be parametrized by the two-dimensional

fundamental representation (eq. 11)

_ fa $\ -1 . ftf-P \u - y a) >u - U - J '

Consider the basis functions (A. IV. 56 and A. IV. 57)

Now transform ava by U , the induced transformation on | 0,1 it

is the desired representation of U:

D(U) l*,m> = A " ' l J"~ " " " ^ " ~

Use the binomial theorem to expand the right-hand side,re-arrange terms

to obtain, for m > m1

-53-

Im1

• m-ra1

N' / pp \N'

m1

Thus

r/A+^V fi .^^!jV2 W^Ti (m-m')l

j m>m' j |

(A. IV. 57)

In particular, for a = cos-!-, /3 = sinf-

m-m1

X F (-0-m1, -p+m, 1+m-m' ; -tan •*•) . '

(A. IV. 58)

or, by a transformation of the hypergeometric functions,

-54-

m'm [_ ($-m)l (<p+m')i J (m-ni1)! 2

-m-m' * m-m1

2 fx F(-<£-m', 1+^-m1 ; m-m' + 1 ; sin f) ;

Similarly, for mr > m, one obtains

)I (0-m'j: J

m'-m( c o s 2} ( s m

x F (-0+m1, tf+m1 + 1 ; m'-m + 1 j \fi\ .) ; m'> m .

(A. IV. 59).

13. Functions and integrals over the group space - harmonic analysis.

The representations D(x) are special examples of functions over

the group space [ i . e . , set of functions D (x)] . If x is parametrized

by Ou,f ,v), for example, then we have the functions D (l*%v). Formn

irreducible representations D , these sets of functions satisfy some

important orthogonality properties (also valid for any finite or compact

group).

Quite generally, for functions over the group space we can define

f{x) dju(x). The measure is

I T f if f >, 0.otftdn; F (f+g)du = Ifdix+ Tgdu ; f

- 5 5 -

invariant if / f(yx)dju(x) = / f(x)dju(x). For compact groups there

is always a unique invariant measure (up to a constant) such that

/ 6n(x) <oo . If we write dju(x) = fx(6.)d&^ . . . d0 in terms of group

parameters 0. we can determine ju (0) from the invariance condition

nf(6)n (0) d 0 = ni(6)n (0) dn0 (A. IV. 60)

where 0' are the parameters of x, £= those of yx, 8 =

0" = fixed. The two differentials in (A. IV. 60) are related by a

Jacobian, d0 =|fgT, Jj d©', hence

Let us now introduce new co-ordinates V> . Then

and

9h.

» ' . - 80-.

90.I

" i

j

90.l

9 < P k

90 ' .J

301'J

- 1 .

Thus

M*> &

The last two equalities follow because the equation is true for arbitrary

d',9t(p . Hence u(6) = N ||r7j| | and N can be so chosen that

fG^

Let us apply this for the rotation group. All rotations by a

fixed angle 0' around any axis form a conjugate class. In group space

this is a sphere of radius 0'. The invariant measure has the same value

of this sphere, thus depends only on a single variable 0' (u{6) is a

class function). The combination of two rotations (01, 0, 0) •, (0* Q1' 9')± A o

= {By 0g, 0 J gives infinitesimally the equations- 5 6 -

$1 = 8' + 0^

62 M ' V 2 si

„2 sine-

1_ r, \

2 fi2 J

Hence, for fixed $',

M30" 2(1 - cos0')

From

We find N = l/ftr . Finally we have

1 (1 - COS0). 2 J.

(A. IV. 61)

Having defined an invariant integral over the group space^we have

the following orthogonality relations for the matrix elements of thesirreducible representations D of compact groups:

0 if D and D are inequivalent

-L 6. 6. i f D S ^ D S 'n lm jnl m

(A. IV. 62)

-57-

where n is the dimension of the representation. For our rotation

matrices, in particular,

/i ' *i

2J mm' nn'

(A.I V. 63)The functions D..(x) form therefore an orthonormal basis (up to

normalization factor) for the square integrable functions f (x) over

the group. Thus we can expand

whe re

Then

Ln

G

j m, n=lm n

Dj (x)mn

• "i f * « "x) D L ( X )

mnm, n

(A. IV. 64)

14. Characteristics of the representations of the rotation group

For an irreducible representation D"1 , the character is

X1 = Tr (A. IV. 65)

The character is a class function. Hence for the rotation group it is

only a function of 9, not of direction n. Thus we can in the evaluation

of it pick ft to be the z-axis. Then in the basis | jm)>

- 5 8 -

(d) = J e"1111" = ° l l l l j;2'~ . (A. IV. 66)m = -j

Hence x\-B) = X2(e), X3(JT) = cos^j .

The orthogonality relations of the characters follow then

explicitly from (61) and (66)

I" d/j(e) xhe) x j %) = P

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