Lie Groups and Quantum Mechanics

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann,...

Author: D. J. Simms

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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich Series: Mathematisches Institut der Universit~t Bonn • Adviser: F. Hirzebruch

52

III

D. J. Simms Trinity College, Dublin

Lie Groups and Quantum Mechanics 1968

Springer-Verlag- Berlin-Heidelberg-New York

All rights reserved. N o part of this b o o k may be translated or reproduced in any form without written p e r m i s s i o n from Springer Verlag. © by Springer-Verlag Berlin • Heidelberg 1968 Library of C o n g r e s s Catalog Card N u m b e r 68 - 24468 Printed in Germany. Title No. 7372

Preface

These notes are based on a series of twelve lectures given at the University of Bonn in the Wintersemester 1966-67 to a mixed group of mathematicians and theoretical physicists. I am grateful to Professors H~rzebruch, Bleuler and Klingenberg for giving me the opportunity of s p e a k ~ a t

their seminar.

These notes are written primarily for the mathematicianwho has an elementary acquaintance with Lie groups and Lie algebras and who would like an account of the ideas which arise from the concept of relativistic invariance in ouantum mechanics. They may also be of interest to the theoretical physicist who wants to see familiar material presented in a f o ~ w h i c h

uses standard concepts from other

I

areas of mathematics.

The presentation owes very much to the lecture notes of Mackey [24] and [25] and of Hermann [15] . Many of the original ideas are due to Wigner and Bargmann [37] , [1] and [39] . A useful collection of reprints is contained in Dyson [11] .

I should like to thank Professor F. Hirzebruch for his help and stimulation, Dr. D. Arlt for useful criticisms, and the Mathematisches Institut Bonn for support during this time and for the typing of the manuscript.

Dublin, April 1967

D.J.

Simms

Contents

I

Relativistic invariance

i

2

Lifting projective representations

9

3

The relativistic free particle

i7

4

Lie algebras and physical observables

2o

5

Universal enveloping algebra and invariants

26

6

Induced representations

42

7

Representations of semi-direct products

z~8

8

Classification of the relativistic free particles

56

9

The Dirac equation

72

SU(3): charge and isospin

76

io

Section 1.

Relativistic Invariance.

Causality.

Let

M

be the set of all space-time events. Any choice of

observer defines a bijective map x 1, x2, x 3 event

x

M * R 4 , x~-~ (Xl,X2,X3,X4)

are space coordinates and

x4

as seen by the chosen observer.

where

is the time coordinate of the This gives

M

the structure of

a real 4-dimensionsl vector sp~co with indefinite Lorentz scalar

product

<x,y> = - xlY 1 - x2Y 2 - x3Y 3 + x4Y 4 ,

called the Minkowski structure of scalar product ~<x

<x,y>

- y, x - y>

measured

M

relative to the given observer. The

may be given the following physical interpretation:

is the time interval between events

and

y

as

by a clock which moves with uniform velocity relative to our

observer and is present at both events. Let us write x

x

is able to influence the event

means that

y

x < y

if the event

y , in the eyes of our observer. This

occurs later in time then

x

and that a physical body such

as a clock is able to be present at both events. Thus we define:

x < y

if and only if

This partial order on

M

chosen observer. An event <x,y> > 0

x4 < Y4

and

<x - y , x - y> > 0 .

expresses the idea of causalit2, as seen by our x

is time-like if

<x,x> > 0 . The relation

is an equivalence relation on the set of all time-like events,

with two equivalence classes: the future and past events. Moreover if and only if

y - x

is a future time-like event.

A change of observer determines a bijective map where

f(x)

as the event

x < y

f : M

~ M ,

is the event which appears to the new observer to be the same x

does to the old observer.

-2-

The diagram f

M

old~

commutes. Event

f(x)

~M

/ 4

ne~

will influence event

observer, if and only if

x

influences

f(y)

y

, according to the new

according to the old observe

The two observers will have the same idea of causality provided that

x < y :

for all

> f(x) < f(y)

x, y ~ M . In this case we call

f

a causal automorphism rela±

to our observer. If the idea of causality is to be preserved, we must li ourselves to observers which are related to our chosen observer by a caw automorphism.

Let

IR*

A dilatation of latio.___~nof

M

M

denote the multiplicative is a map

is a map

formation of

M

x, y E M . Here

M

of the form

x~-*~X,

A ¢ IR ~ • A

x~--~x + a , a c M . A homogeneous Lorentz tran~

is a linear map M

~M

group of non-zero real nux

A : M

-M

is given its N i n k o w s k i

with

: <x,3~

foz

structure defined by a chosez

observer. The group of translatiorsmay be identified with the additive The group ~

oE homogeneous Lorentz transformations

homomorphisms w(A) = ~ I

~ , W :~--*IR ~ , where

according to whether

A

~(A)

admits two continuou

is the determinant of

A

leaves the equivalence classes of f

and past events fixed or interchanges them~ The orthochron0us homogeneou

-3-

If a group

G

acts on a group

as a group of automorphisms,

H

H × G

h,--) h g , g ( G , then the cartesian product

with the group ope-

ration

( h l , g l ) ( h 2 , g 2) = (hlhgl , glg2 ) is called a semi-direct product and denoted by

H ~ g . We note that \

h~'~ (h,1)

and

subgroup of

HOG

g~-~ (1,g)

embed

H as a normal

subgroup and

G

as a

. Moreover

h g = ghg -I

in

H~)G

.

Any product of dilatations, Lorentz transformations

translations,

is of the form

x~-~AA-x

where

(a,A,A) ¢ M x ~

x~@B #.x + b

is

x~AB

and orthochronous

+ a

x IR• . The composition of

x~-~ A A-x + a

with

A.#.x + A A.b + a , which corresponds to a group

operation

(a,A,A).(b,B,#)

= (a + A k,b , AB , A-~ ) .

This shows that the group generated by such transformations product

)

M® relative to the natural action of ~ i

x ]Re

on

M .

is a semi-direct

-4-

Zeeman

[4o] has shown that

group of causal automorphisms of

M@(~

t x I~ )

is the complete

M .

Causal invariance in quantum mechanics.

The pure states of a quantum mechanical system are represented by the projective space Hilbert space

~ of l-dimensional subspaces of a separable complex

H . For each non-zero vector

be the l-dimensional subspace containing

The inner product

<~,~>

~

in

H , let

~ .

on the Hilbert space

H

defines a real

valued function

2

TIC!I'2 II~112 on

~ x ~ . The physical interpretation is that

<~,$~

is the transition

probabi!itY, the probability of finding the system to be in the state when it is in the state

A bijective map transition probability:

for all

$ .

T: ~ - * ~ is an automorphis~ if it preserves the

-5-

If

A

is a unitary or anti-unitary transformation of

an automorphism

~ of

~

then it defines

by:

By an anti-unitary transformation

A

we mean one such that

A( @ + ~ ) - A ~

< A ¢,~p >

for all

H

+ A~

= < @,~o >

~, ~ • H , A ( @ . The product of two anti-unitary transformations

is unitary, so that the group

U(H)

of unitary transformations is a sub-

group of index two of the group

~(H)

mations. The map

embeds

Let w:~(H) * Aut(H)

ei~

~ ei0-1

of unitary or anti-unitary transforU(1)

as a subgroup of

Aut(~) be the group of automorphisms of be the map

Theorem of Wi~ner.

U(H) .

~ , and let

w(A) = ~ . We have the fundamental result:

The sequence zr

(1) is exact.

-6-

This means that every automorphism of where if

~

is of the form

A

is a unitary or anti-unitary transformation of H . Moreover, i8 ~ = ~ then A = ei~B with e ¢ U(1) . For Wigner's proof see [38] ,

appendix to chapter 20. See also [2] .

If

f: M ...~.. M

is a bijective map arising from a change of ob-

server, it determines a bijective map

Tf: ~--~ ~ , where

Tf ~

is the

quantum mechanical state which appears to the new observer to be the same as the state

~

does to the old observer. If

g: M - @

M

is another change

of observer we shall suppose that

Tfg = TfTg .

A physical justification of this assumption could be based upon the relation between states and assemblies of events in space-time, and on the definition of

f

and

g

in terms of events in space-time.

The new observer will have the same idea of transition probability as the old observer if and only if of all transformations of

M

Tf

is an automorphism of

~ . The group

which are associated v,dth observers which have

the same idea of causality as our chosen observer is

M~(~t

x IR*)

by the

result of Zeeman. If all the observers also have the same idea of transition probability, then we have a homomorphism

T: M ( ~ ( ~

× I ~ ) --@ Aut(~t) .

In this case we say that we have causal invariance. If the weaker condition holds that all observers which are related to our chosen observer by transformations in the restricted i~_homg~eneousLorentz group

M(~ ~

have the U

-7-

same idea of transition probability then we have a homomorphism

T: M ~ O

~ ~ Aut(~) .

In this case we have rel~tivistic inv~iance. From now on we will confine ourselves to relativistic invariance.

The exact sequence (I) gives an exact sequence

where two in If

@

U(~)

is the image of

U(H)

under

~

and is a subgroup of index

Aut(~) . is a connected Lie group, then the image of any homomorphlsm

T: G --~ Aut(~)

is contained in in

G

U(~) . To see this we note that the exponential map

shows that there is a neighbourhood

V

of the identity in

which each element is a square and is therefore mapped by

Since

M~O

is a connected hie group, we have

in the case of relativistic invariance.

T

a

into

G

in U(~)

homomorphism

-8-

Let the Hilbert space and let the projective space to the surjection the state

Tf ~

@-~

~

H

be given its usual norm topology,

be given the quotient topology relative

@ . We assume that, for each fixed stste

~ ,

will depend continuously on the change of observer.

This means that the map

M®A o

H A

f~-~ Tf ~

is continuous for each

~ ~ H . This is equivalent to the

c o n t i n u i t y of

where

U(H)

~ ~

is given the weakest topology such that all maps

, are continuous,

obtained by giving

U(H)

~ ¢ ~ . The same topology on

....). ~ ,

may be

the strong operator topology, which is the

weakest topology such that all maps tinuous,

U(~)

U(~)

U(H)

~ H , A~--~A~

, are con-

~ ~ H , and then taking the quotient topology relative to the

^ . The equality of these two topologies on U (I) follows from [1] theorem 1.1. surjection

A~¢A

By a pro~ective representation

T

of a topological group

we shall mean a continuous homomorphism

T: G

given above. By a representation

G

morphism

T: G---~U(H)

defined above.

where

T U(H)

of

~U(H)

G

with the topology

we mean a continuous homo-

is given the strong operator topology

-9-

Section 2.

T

If U(H)

then

Lifting Pro jectiye Representations. '

is a representation of a topological group

~-T

is a projective representation of

G

in

G :

T

G

Conversely, if

T: G

admits a liftin6 such that

~

)U(~)

>

.

is a projective representation then

if there exists a representation

T: G

T

~ U(H)

T = ~.T .

Let group

T

U(H)

G

be a connected Lie group with simply connected covering

and covering map

representation of

G

in

p

with kernel

K . Let

T

be a projective

U(~) . Using the theorem of Wigner we have the

diagram P 1

>K

')~

)G

)1

1t'

1

> u(1)

) u(H)

)

)

with both rows exact.

T-p

is a projective representation of

T,p (K) = I . Suppose

Top

admits a lifting

o(K) C U(1) . Conversely any representation that G

in

a(K) C U(1) U(~)

a: ~ a

of

~U(H) ~

in

~

and

; then U(H)

such

will define a unique projective representation

such that

T

of

Top = ~oa .

These considerations show that, if the simply connected Lie ,group

G

has the property that every projective representation of

G

a~mlts a lifting, then the determination of all the projective representations of

G

is equivalent to the determination of the representations of G

-

which map

K

into

lo-

U(1) . Bargmann in [I] Theorem 3.2, Lemma 4.9,

and § 2, proved that the possibility of lifting projective representations of

G

depends on the cohomology of the Lie algebra of

G . The remainder

of this section will be devoted to giving a topological proof of Bargmann's result. We shall prove four theorems, the fourth being the theorem of Bargmann.

Theorem I.

U(H)

and fibre

Proof.

is a principal bundle with base space

U(1) .

For the relevant definitions we refer to [17] I. 3.2.a)

[19] I!I. 4.

We first note that although

group in general since the multiplication continuous, see [28] § 33.2,

For each non-zero

V

U(H)

=

~ ¢ H

~,U(H)

the set

= IA I < A ~,~ • $ 0 I

"~ = ,(V )

therefore form an open cover of

>U(1)

for each

U(~) , and

U(H) . The sets ,-I(T~) = ~

be defined by

< A~,@

;9~(~A) = A-w~(A)

is not

is an open map.

U(H) , and such sets give an open cover of

Up: V~

)U(H)

U(1) x U ( H )

is open in

Let

and also

is not a topological

U(H) x U(H)

the multiplication

is continuous. It follows that

Then

U(~) , pro~ection

>

A c U(1) . Let

be defined by hq~(A) = (=A , Wcp(A)) •

h : V

- w

×

u(l)

.

-11

-

This is continuous with continuous inverse (=A , eiS);

~ei~[w~(A)]-IA

and is therefore a homeomorphism. This proves the bundle space property that

=-1(W'~)v is homeomorphic to the topological product

It remains to show that the structure group is If

@,

are non-zero vectors in

H

and if

~

× U(1)

@

U(1) .

=A ~ W @ ~ ] %

and

iG e

E U(1)

then h @ % I (=A,e i0) : h@

:

Thus the fibre coordinate

e

i8

I ei6~(A) -IA I

(=A , e

i0

n~(A)

-1

is multiplied by

w@(A))

.

%(A) -I

on change of coordinate neighbourhood. Moreover the function

=A ~

-~(A)-Iw@(A).

Theorem 2.

Any continuous map

connected Lie group with

is continuous. This completes the proof.

G

T: G

> U(~)

of a connected simply

can be lifted to a continuous map

r: g

> U(H)

T = ==T . G

Proof.

T

induces a

U(1)

bundle,

E

) G , with base space

to~al space

E = I(g,A) i T(g) = =(A) I C G x U(H)

G ,

-

and projection

a(g,A) = g .

12-

See C17J I. 3.3. or [33] page 98 .

The diagram

E

) U(H)

,-p co~nutes, where

=(g,A) = A . By a generalisation [3J 5., 17., and 18.,

of a result of Cartan, the second homotopy group of Since

G

G

is zero.

is also simply connected, it follows from a theorem of Hurewicz

[19J II . Corollary 9.2., and by the universal coefficient theorem [12] V. Exercise G. 3., that the singular cohomology group Since the first homotopy group of the fibre from obstruction theory [35] 35.5 and 29.8 s: G

ME

exists with

U(1)

~(G,~ is ~

)

is zero.

, it follows

that a continuous map

aos = 1 .

The continuous map

T = ¢=s

has the property required since

#oT = #==os = T=acs = T . This completes the proof.

Extensions and factor sets.

Let the pair

(G,K)

G

and

K

be Lie groups with

is a continuous map

=: G x

such that

G

...~..K

~(1,1) = I , and

z) -

for all

x, y, z ¢

G .

K

abelian. A factor set for

-

Let is the product

E~

be the topological group which as a topological space

G x K

and has group operation

(xl'kl)(x2'k2) The properties

13-

of

~

: (:'1:'2'

ensure that

imbedded in the centre of

E~

E~

~(x1'x2)k~k2) • is a topological

by the map

k,

~(1,k)

group, with . Since

E~

K is

locally Euclidean it is a Lie group by the well known result [27] 2.15 of Montgomery,

Zippin and Gleason, and we have an exact sequence of Lie

groups

I

where

~K

~E ~

~G

~1

a(x,k) = x .

Let

LG

and

symmetric bilinear map the pair

(LG,LK)

LK

be the Lie algebras of

0: LG × L ~

) LK

G

and

K . A skew-

is called a factor set for

if

0(x,[y,,1) + 0(y,[z,x]) + e(z,[x,yl) = 0 for all

x, y, z ¢ LG . The factor set

a linear map

~: LG

~ LK

is trivial if there exists

with

e(x,y) for all

0

~( [x,yl )

x, y ¢ LG . The quotient of the additive group of factor sets

by the subgroup of trivial factor sets is denoted by called the 2 nd cohomology ~roup of space

LK (with trivial action of

and [6] XIII. 8. for more details.

LG LG

H2(LG,LK)

and

with coefficients in the vector on

LK ) . See [20] III. lo.

-

14-

C o n s i d e r now the exact sequence of Lie algebras

& 0

Choose a l i n e a r map

for each If

8

• LK

B: LG

;LE W ~ *

• LE ~

~ 0 .

such that

&.# = I .

x, y ¢ LG . This is a factor set for the pair

is a trivial factor set then

~: L G

LG

~ LK

is linear,

defines a h o m o m o r p h i s m

~ = ~

and

(LG,LK)

.

where

so that

#: L G - - - ~ L E ~

connected and simply connected, such that

8(x,y) = ~([x,y])

Put

with

~@~ = I . If

G

then there is a h o m o m o r p h i s m

~o~ = I . Thus

~

is ¥: G

must be of the f o r m

~(~) -- (x,~(x)) where

A

is a continuous map

(xy,~(xy))

G

~ K . Since

_- (~,~(~))(y,~(y)) =

(~y,~(x,y)~(x)~(y))

so that

A(xy)

for all

x, y c G .

¥

: ~(x,~)~(x)~(y)

is a homomorphism,

E~

- 15 -

We have now proved the following;

Theorem ~.

Let

connected, and for

G K

(G, K)

and

K

be Lie groups t

abelian. Let

G

connected and simply

H2(LG,LK) = 0 . Then for any factor set

there is a continuous map

A: G

)K

with

(xy) : for all

x, y ~ G .

We are now ready to prove the main result on lifting projective representations.

Theorem 4 (Bar,mann). group with T: G

~U(H)

Proof. ?Toa

=

Let

G

be a connected and simply connected Lie

H2(LG,IR) = 0 . Then any pro~ective representation admits a liftin~

r: g

)U(H)

which is a representation.

By theorem 2 there exists a continuous map T

.

G

i We c a n

(and will)

)u(1) choose

~

>u(H) so that

,I o(1)

= I

.

a: G

) U(H)

with

- 16-

For each

~(=(x)o(y))

x, y ~ G

we have

: ~oo(x).~oo(y)

: T(~). T(y) : T(~y) : ,oo(xy)

.

Therefore

o(x)o(y)

where

~(x,y)

¢ U(1)

. The map

Indeed for any unit vector

[~(x,,y,)

= ~(~,y)[o(~y)

= ~(x,y)o(xy)

~

~: G × G

in

H

>U(1)

is continuous.

we have

- ~(x,y)]o(x'y,)~

- o(x'y')]o

+ o(~,)[o(y,)

- =(y)]~

+ [o(x')

- o(x)]o(y)~

so that taking norms @

The continuity of (x,y)~

~

+ll o(y,)~

- o(y)~

+if o(~,)¢

=(x)¢

U(1)

~ a(xy)~ , e(y)~ , a(x)~ . Moreover

is

~: G--->U(1)

Now put

11.

follows from the continuity of the maps

tions for a factor set for the pair of

Jt

IR

~

satiesfies the condi-

(G,U(1)). Since the Lie algebra

we can apply theorem 3 to obtain a continuous map

with

T(x) = ~ ( x ) ' ~ x )

.

T

is the required representation lifting

T .

-

Section 3.

17

-

The Relativistic Free Particle.

We have seen in section I that the relativistic invariance of a quantum mechanical system requires a projective representation

T:

of the restricted inhomogeneous Lorentz group. A permissible choice of observer defines a bijective map

M

e ~IR 4

which is a vector space

isomorphism preserving the Lorentz scalar products. The group of homogeneous Lorentz transformations of

IR4

0(3,1)

relative to its scalar

product

<x,y> = - xlY I - x2Y 2 - x3Y 3 + x4y 4 = x'Ax is the group of

4 × 4 real matrices

A

with

= <x,y~ . Thus

0(3,1) = I A i A'.A.A = A

where

A

is the diagonal matrix with entries

of translations of

IR 4

-I, -I, -I, I . The group

is naturally isomorphic to

of inhomogen~ous Lorentz transformations of

IR4

is the semi-direct

product

m @0(3,1) relative to the natural action of 0(3,1)

on

IR 4 , and the group

IR 4 .

-

18-

An inhomogeneous Lorentz transformation of will appear to the observer as a transformation of + a(a)=

M , x:

.~Ax + a

IR 4 , a(x),

) ~(Ax) +

(aAa-1)a(x) + ~(a) . The map > (~(a),aAa -1 )

(a,A),

is an isomorphism

a,

of

M@~

onto

~R4~0(3,1)

the connected component of the identity in mapped isomorphically onto

~R4~S0(3,1)

0(3,1) , then

2 x 2

S0(3,1) M ~

0

be is

.

S0(3,1)

The simply connected covering group of the group of

. Let

is

SL(2,¢) ,

complex matrices with determinant I . The covering

map

sT,(2,¢)

> so(3,1)

can be defined as follows. Let

TI =

[:II O

Identify each

,

T2 =

[oiI i

,

T3 =

O

x = (x 1,x2,x3,x4) ¢ IR 4

[I:] ,

O

I

with the Hermitian matrix

x I - ix21 x : x1~ I + x 2 ~ 2 + x3~ 3 + x4T 4 : (i I + x3 + ix 2

x4

x3 /

:]

-19-

so that det ~ = <x,x> . For each

rl(A)

where

A*

A e SL(2,¢)

........

:

;A#,

is the .,,,i,.m..v.e..Pse-transpose of

let

,

A . Since

det(A~A~) = det

it follows that

w(A) ¢ 0 ( 3 , 1 )

connected group

SL(2,¢)

. Since

W

is continuous it maps the

into the connected group

S0(3,1) . Finally,

is surjective [13] Part II section I § 5 , with discrete kernel

The simpl~ IR4@SL(2,¢)

, relative to the action

Let ~ o M@%

connected cover of

ym4@so(3,1)

a~: M ® , , ~ °

>A S ~)SL(2,~)

Bargmann [1]

° , M ~

of

is therefore SL(2,¢)

on

m 4 .

be the simply connected covering group of ~ o ' so that

~,: M ~ o -

M ~

~---~A~A ~

is the simply connected cover of

~(LG,IR) = 0

IR4~S0(3,1)

11,-1} .

where

M~%

. The isomorphism

induces an isomorphism .

(6.17) has shown that the cohomology group LG

is any one of the isomorphic Lie algebras of

, m 4~S0(3,1)

, IR 4 @ S L ( 2 , C )

. By the results of

section 2 we can conclude that the projective representation

T

is induced

by a representation

such that

T(I1,-ll) C U(1) . If

T

is irreducible then we call the

quantum mechanical system an elementary relativistic free p~ticle.

-

Section 4.

Let

T : G

a Hilbert space X ¢ LG

20

-

Lie Algebras and Ph2sical Observables.

~U(H)

be a representation of a Lie group

H , and let

LG

be the Lie algebra of

the l-parameter subgroup

continuous) l-parameter group

t~-* exp tX

t ~-~ To exp tX

T(X)

on

H

in

G . For each

is mapped into a (strongly of unitary operators on

By the fundamental theorem of Stone [34] Theorem B adjoint operator

G

H .

there is a unique skew-

such that

T@ exp tX = exp tT(X)

for all

t ¢ ~

. For

@ ¢ H , T(X)@

is the derivative at

t = 0

of the

map

]R

~ H , t:

The domain of the operator

T(X)

~(T.exp tX)~ .

is the set of

~

for which this derivative

exists.

There exists a dense set D T and essentially skew-adjoint for all determined by its restriction to skew-symmetric operators having

in

on which

X ¢ LG ; each

D T . Let DT

H

S(DT)

T(X)

T(X)

is defined

is therefore

be the Lie algebra of

as common invariant domain. Then

T) is a Lie algebra homomorphism. This implies for instance that if X,Y ¢ LG

then

T(X)T(Y) - ~(Y)T(X)

with unique self adjoint extension instance [31 1 , Theorem 3. I.

is esBentially self adjoint on

DT

T[X,Y] . For these results see for

-

21

-

Suppose now that the associated projective space

~

is the

system of pure states of a quantum mechanical system. Each self adjoint operator

A

on

H

is then associated with a physical observable. In

fact the spectral theorem for on

A

gives a projection valued measure

P

IR , and

is the probability of the value of that observable being found to lie in the Borel set

E C IR

the unit vector

when the system is in the state

~

determined by

@ . For an account see [26] , 2-2 .

The self adjoint operator

~.T(X) I

must therefore represent a

physical quantity. We say we have a ~hvsical interpretation of the symmetry group

G

when we have specified which physical quantity is associated with

X , for each

X ¢ LG .

Physical interpret_ation of the inhomqgeneous Lorentz grou~.

The inhomogeneous Lorentz group

M ~

is a semi-direct product.

Its Lie algebra is therefore a semi-direct sum

LM®L Z where the Lie algebra identifiable with

M

LM

of the vector group

M

itself, and the Lie algebra

linear transformations of

M

is abeli&n and naturally L~

is the algebra of

which are skew relative to the Lorentz scalar

product.

L~

= I A [ + <x,Ay~ = 0 , all

x, y ¢ M I •

o 22 -

The semi-direct sum is relative to the natural action of LM~

M . This means that

LMQL~

= LM~L~

L~

on

as a vector space, with

Lie product

[(a,A),(b,B)] = ([a,b] + Ab - Ba, [A,B]) = (Ab - Ba, [A,B]) .

See [18] for material on semi-direct sums.

A choice of observer

=,: M ® ~

M--~-~IR 4

~ m

gives a Lie group isomorphism

~®0(3,1)

which induces a Lie algebra isomorphism

=.:

LIR 4

.

is abelian and identifiable with

IR 4

as a vector space. L0(3,1)

is the Lie algebra of 4 x 4 real matrices which are skew relative to the Lorentz scalar product on

~4

:

L0(3,1) = I A i A'A + AA = 0 J

L~R4~L0(3,1) of

L0(3,1)

is the semi-direct sum relatiwe to the natural action on

LIR 4 ~

Rotations of group of

.

0(3,1) :

IR 4 .

IR 4

in the

XlX2-plane give a l-parameter sub-

-

cos t

23

-

sin t

t~-~ -sin t

cos t I

m12 ¢ LO(},I)

with

-1

0

= exp tin12 0 0

Pure Lorentz rotations in the

l-parameter subgroup of

t;~

1

¢-

I

with

0 = exp t"

0(3,1)

x3x4-plane give a

:

O"

0

I cosh t

sinh t

sinh t

cosh t

m14 ¢ L0(3,1)

. Translations of

= exp t"

0

(7

IR 4

in the

with

xI

> (t,O,O,O) = exp t~ 1

~I c LIR 4 . In this way a basis

m23, m31, m12, m14, m24, m34

for

L0(3,1)

is defined, and a basis

= exp tm14

o

l-parameter subgroup :

t :

.4

direction give a

-

for

24

-

IR 4 ; the latter corresponds to the standard basis for

under the isomorphism

L ] R 4 ~ I R 4 . We associate

IR 4

~I' ~2' ~3

with the components of linear momentum in the

x I , x2, x 3

respectively,

with the components of

~4

with~,

m23 , m31 , m12

angular momentum about the m24, m34

x S, x 2, x 3

directions

axes respectively, and

m14,

with components of relativistic angular momentum. We are

now ready to give a physical interpretation of the simply connected covering group

G = M ~

group, as follows. Let

of the restricted inhomogeneous Lorentz T: G

~ U(H)

by relativistiv invariance. Let

be the representation defined

~,: LG

isomorphism defined by choice of observer and

a,(X) = ~1

>LIR4~L0(3,1) M

a ~4

. If

be the X ¢ LG

then we interpret

~.T(X) as the self-adjoint I operator corresponding to linear momentum in the x I direction as seen by this observer. Similarly for the other components of linear momentum, the energy, and the angular momentum.

LG

is the Lie algebra

of relativistiqobservables~ ....

Effect of chan~e of observer.

Let fgf-1: M---~M g: M

~M

f: M - - ~ M

appears the same to the new observer as the transformation

did to the old observer. If we limit ourselves to relativistic

changes of observer then and

be a change of observer. The transformation

g~_~fgf-1

f

is an inhomogeneous Lorentz transformation

is an inner automorphism

I: M Q J e

I(f)

of

M~.

The map

~ Aut ( M ~ )

is a homomorphism into the group of automorphisms of

M Q~

.

-

25

-

The physical interpretation of this is that

I(f)

formation of

of observer.

M ~

induced by a change

We note that

if

A e~

and

f

is the trans-

(b,B) ¢ M ~ t h e n

I(A)(b,B) = (O,A)(b,B)(0,A) -I = (Ab,ABA -I) . This shows that the action of

I(~)

on

M

of

I(~)

On ~

is the natural action of J~ on

M , and that the action

is induced by the natural action of ~

The automorphism

i(f)

of

M~induces

on Hom(M,M)

D ~.

a Lie algebra auto-

morphlsm

Adf

where

Adf

:

LM@L~

is the derivative of

interpretation of this is that

~ LM@L~

I(f) AdfX

at the identity. The physical is the physical quantity which

appears the same to the new observer as X did to the old observer. I • Thus the self-adjoint operator ~.T(AdfX) represents the same quantity to the new observer as

$.T(X)

does to the old.

I

We thus have a Lie group homomorphism, the ad~oint representation

Ad: G

of the Lie group

G = M ~

, Aut (LG)

. This expreszes the action of the group

of changes of observer on the Lie algebra

The action of LM~M

on

LM

of relativistic observables.

is the natural action of ~

. This action preserves the Lorentz scalar product on

the elements of of

Ad(~)

LG

Ad(~)

LM

on

Hom(M,M) D L ~

L~

LM~M

on and

are therefore first order Lorentz tensors. The action is induced by the natural action of ~

. The elements of

Lorentz tensors.

G

L~

are therefore

2 nd

on order skew

-

Section 5.

26

-

Uniuversa! Enveloping Algebra~

~n the last section we have seen how the group relativistic changes of observer acts on the Lie group observables. It is of interst to know how the non-commuting operators

G

G = M ~

LG

of

of relativistic

acts on polynomials in

T(X) , and in particular to find operators

which are independent of a choice of observer. The natural setting for this is the universal enveloping algebra algebra containing product

IX,Y]

LG

ULG

which ~s an associative

as a vector subspace in such a way that the Lie

equals the commutator

XY - YX .

Tensor al&ebra

Let

L

be a real vector space a n d T L = IR @ L @ (L @ L)S(L @ L @ L)@ ...

be the ten~0r alKebra over

L . TL

may be characterised by the universal

property:

i)

TL

is an associative algebra (with identity) containing

a vector subspace, and generated by

ii)

any linear map of

a (unique) homomorphism of

L

into

In particular any linear map automorphism

A

extends to

A .

~: L-~L

extends to a unique

~ a : TL---~TL . We shall need the fact that

extends to a unique derivation

as

L .

into an associative algebra TL

L

~d: TL---~TL .

~

also

-

By a derivation algebra

A

D

27

-

of a (not necessarily associative)

we mean a linear map

D: A---~A

with

O(xy) = (Dx)y ÷ x(oy) all

x,

y

~ A

.

~a

is c~aracterised by

~a(X1@..@Xr)

: ~(Xl)@...@~(Xr)

and ~d by ~ d ( X 1 @ . . @ X r ) : ~(Xl) @ x 2 @ .. @ x r + .... + x I @ ..@Xr_1@~(Xr) . For these facts see [4], [8] and [1o] .

,

..

Universal enveloping algebra.

Let UL

L

be a real Lie algebrs. The universal enveloping al6ebra

is the quotient of the tensor algebra

TL

by the ideal

J

generated

by the elements

X@Y-

X,Y c L .

i)

UL

UL

Y~X-

KX,YS

may be characterised by the universal property:

is an associative algebra (with identity) containing

vector subspace, generated by

L , and with

Cx,YS = xY - YX

each

X,Y c L .

L

as a

-

ii) with UL

any linear map

A

of

L

AlE,Y] = A(X)~(Y) - ~(Y)A(X)

28

-

into an associative algebra

A

extends to a (unique) homomorphism

'A .

In effect

UL

is the algebra of all non-commutative poly-

nomials in the elements of product

IX,Y]

in

L

equals the commutator

XY - YX

in

UL . Any

automorphism

A: L

the ideal

invariant and hence induces a (unique) automorphism

J

A a: U L - - ~ U L leaving

J

•L

L , subject to the condition that the Lie

induces a (unique) automo~@hism of

. Any derivation

~: L - - - ~ L

TL

leaving

induces a derivation of

invariant and thus a unique derivation

kd: UL

~b~

TL

.

For more details see [2o] V. and [30] LA 3 •

Symmetric algebra~

Let

L

be a real ~ector~space. The s~mmetric algebra

is the quotient of the tensor algebra

TL

by the ideal

I

generated

by the elements of the form

x@y- y@x x, y c L .

i)

SL

containing

SL

may be characterised by the universal property:

is an associative commutative algebra (with identity) L

as a vector subspace and generated by

L .

SL

of

L

-

ii) with

any linear map

A(x)A(y) = A(y)A(x)

homomorphism

If

A

SL

A: L ....>. L

derivation

;A

of

L

all

29

-

into an associative algebra x,y ¢ L , extends to a (unique)

.

is any linear map then the automorphism

Ad

of

TL

leave the ideal

I

x 1,...,x n

SL~

is a basis for

the extension to

SL

L

then

,

Xl,...,x n . With this identification,

of the linear map

f(x 1,...,xn):

and the

SL .

]R[Xl,...,Xn]

the algebra of polynomials over

Aa

invariant and therefore

induce an automorphism and a derivation of

If

A

L---~R

, x i ~ - - ~ a i , is

"~ f(a 1,...,a n )

See [lo] V. 18., 19. for details.

Automorphisms and derivations.

Let

A

be a, not necessarily associative, finite dimensional

real algebra. Then the Lie group algebra

D(A)

that if

D

Aut(A)

of automorphisms of

the algebra of derivations of

is any derivation of

A

~ i+j=k

then from

.k' f!j!

•

has Lie

A . To see this we note D(xy) = (Dx)y + x(Dy)

we have by induction

Dk(xy) =

A

" DXx DJy .

-30-

Therefore

tk k

i

= (etDx)(etDy)

so that

tJDJy

~-~ tiDix

exp tD ¢ Aut(A)

all

,

t ¢ IR .

Conversely, if etD(xy) = (etDx)(etDy)

all

t ~ ]R

then t2 D 2 xy + tD(xy) + ~ (xy) + ........

=

all

t2 t2 (x + tDx + ~ . D2x + ...)(y + tDy + ~ D2y + ... )

t ~ ]R , so that

D(xy) . (Dx)y + x(Dy) and

D

is a derivation.

-

31

-

Action of change of observer.

Let

G = M Q~

be the connected group of restricted

relativistic changes of observer. Let

LG

S(DT)

be the Lie algebra homomorphism defined by the requirement of relativistic invariance, see sections 3 and 4. By the universal property

T

extends to

a unique homomorphism of associative algebras

• ULG ~

of

ULG

Op(DT)

into the associative algebra

For each

u c ULG , T(u)

generated by

is ian operator with domain

of products of operators induces an automorphism

0p(DT)

T(~), X ~ LG . A change Ad g : LG ~ L G

DT

g ¢ G

S(DT) -

and is a sum of observer

of relativistic observables,

which extends to a unique a u t o m o ~ h i s m of

ULG

which we also denote by

The physical interpretation of this is that the operator the same role in the eyes of the new observer as

T(Adg(U)) plays

T(u) did for the old.

The map

gives the action of

An element Adg(U) = u

for all

G

Ad

:

on

ULG .

u c ULG

G

~Aut

(ULG)

is celled an i_nvariant of

g ¢ G . In this case the operator

G T(u)

if has a

physical significance independent of a choice of relativistic observer.

Ad

g

-

32

-

Characterisation of Invariants.

Let Let

TLG

LG

be the Lie algebra of a connected Lie group

be the tensor algebra over

symmetric tensors, SLG

LG , STLG

G .

the vector subspace of

the symmetric algebra,

ULG

the universal

enveloping algebra. We have a diagram

STLG

TLG

C

>

SLG

ULG

A where

~, @

maps

are the canonical surjections. By [9] III § 5 Proposition 6

STLG

bijectively to SLG , with inverse

A = @W . By a dimension argument the linear map

~ A

say. We write is bijective.

See [9] V. § 6 proposition 2 .

The automorphism of the tensor algebra SLG

also denoted by

of LG induces a unique automorphism g TLG , which induces automorphisms of ULG and

Ad

g

Ad

.

The Lie group homomorphism

Ad : G

~Aut

(LG)

induces a Lie

algebra homomorphism

ad

called the

: L~---~D(LG)

adjoint representation of

is in fact the map

LG .

,

The derivation

a~:

adx(Y ) = [X,Y] ; see [7] IV § XI for example.

LG

~LG

-

LG

-

The derivation

a~

tensor algebra

TLG , which induces derivations of

also denoted by

of

33

induces a unique derivation of the ULG

and

SLG

ad x

It follows that the diagrams

A SLG - - ~ -

SLG •

ULG

SLG

~ ULG

SLG

A

> ULG

~ ULG A

commute, for each

g c G

fore correspond under

A

and

X c LG . The invariants of

to the elements of

I s c SLG I Adg(S) = s , all

Theore m .

ULG

UrLG

e~uals the

C .... C UrLG C .....

X E LG

,

the derivation

finite dimensional subspace r

G

is the vector subspace spanned by products of the form

X i c LG . For each

U LG

g c G I •

has a natural filtration by finite dimensional subspaces

X ~ Xi~ ........ Xir

to

Ad G :

ULG.

UILG

where

there-

invariant under

The set of invariants of a £onnected Lie group

centre of

Proof.

SLG

g

U LG r

+

2'

of

ULG

leaves the

invariant, so that the restriction

of the series

.

ad x

3'

-

converges.

34

-

It follows that the series converges

defines an endamorphism derivation.

exp(a~)

The restriction of

by [7] IV § IX extension to

proposition

of

ULG

exp(adx)

on

ULG, and

since

ad x

LG

equals

to

is a Adexp X

I , and therefore by uniqueness

of the

ULG ,

exp(adx) = Adexp X

on

ULG . We further note that since

derivation extending the derivation

the map

ULG--~ULG

a X(U) for all

a~

LG~LG

on

ULG

, Y~--~[X,Y]

is the unique ,

it must equal

, u : ~ Xu - uX . Thus

: X u - uX

u c ULG .

Since

G

is connected i~ is generated by the set of elements

exp X , X ¢ LG . Therefore

u ¢ ULG

is invariant

all

iff

Adex p tx(U) = u ,

iff

(exp@adtx)

iff

u + t(adx)u__ + ~:~'(adx)2U + ....

iff

a~X(U ) = o ,

iff

Xu - uX = o ,

iff

u ¢ centre ULG .

This completes the proof.

X ~ LG

(u ) = u ,

"

all

, t ¢ ]R

,

-- u

X ¢ LG

T!

-

35

-

Invariants of the inhomo~enegusLorentz group.

With the notation of section 4, (~I" ~2' ~3' ~4 ) orthono=mal basis of the Minkowski space of ~

on

LM

LM~M

is an

, and the action

in the adjoint representation is the same as the natural

action of ~f on

M . The Lorentz scalar product gives isomorphisms

~}2M ~ Hom(M,M)~ (~}2M)* which commute with that action of ~

, where

*

denotes the vector

space dual. These isomorphisms are characterised by

~@~ ~----,- (

z ~-.

< y, z>

~)

and

A ~

( x@y~.<

~, Ay >)

respectively. We identify these spaces, (this being the customary identification of contravariant, mixed, and covariant tensors) and note that the skew-symmetric tensors in ~ 2 M Indeed

correspond to

L ~ f C Hom(M,M) .

~i~j

- ~ . ~ .~i ~ corresponds to mij . The scalar product <-~..> is an element of ( ~ M ) invariant under ~f and corresponds to:

i,j in ~ 2 M

.

It follows that the corresponding element of 2 -

is an invariant of

~1

2 -

~2

2 -

~3

2 +

~4

UL~

-

Let

wi =

Wi =

2.3'

3'

=

:

in

j,k,l Sijkl

j.k,l

, where

¢1234 = 1 . Then :

j,k,1

~j ^ ~k ^ ~I

i. j. k. 1

~3M--~M

1, 2, 3, 4 • The Lorentz

is any even permutation of ~I' =2' ~3' ~4

defines an isomorphism

, characterised by

w

for all

~3M

¢ijkl

scalar product and the basis 6

-

is completely skew-symmetric and

Sijkl

where

2.3!

36

^

x =

w E A3M

< 8(w),

and

x

> ~I

^ #2

^ ~3

x g M . Moreover

@

^ #4

commutes with all Lorentz

transformations of determinant I . Under this isomorphism, wl, w2, w 3, w 4 correspond to

- if1' - #2' - ~3

- w~w~

~n

#.®/~.

element of

UL~

-

i~ inva~i~n~

•

j.k.l

respectively, Therefore

- w3~w 3 +

un~e~

~.

w~w~

I~ ~o~ow~

~t

t~

:

is an invariant o f ~

2.3!

w2@w 2

' #4

¢ijkl

2

2

2

- w I

- w2

- w3

, where here

•

~jmkl

in

w.

I

ULZ

+

2 w4

denotes the element

•

oor~on~n~

-

37

-

The unquantised relativistic relation between the linear momentum

(PI' P2' P3 ) ' the energy

E , and the rest mass

particle is: - Pl2 - P22 - P32 + E 2 = m 2 " invariant

2 2 2 2 - ~I - ~2 - =3 + ~4

More specifically, if Hilbert space

H

system, and if

T

m

of a

We therefore associate the

with the square of the mass.

is the representation of

M ~

in the

associated with the states of a quantum mechanical ) Pj = .--1T(~j" z

are the self-adjoint operators

representing linear momentum and energy, then the self-adjoint positive 2 2 2 square root (if any) of the operator - PI - P2 - P3 + represents the mass of the system.

Determination of invariants

We now indicate how invariants of a semi-simple Lie group

G

may be constructed. Take any finite dimensional, not necessarily unitary, representation

T

of

G

with representation space

V . For each

the commutative diagram T

G

~ GL(V)

1

1T(g)(')~'(g-1)

T

G

-~ GL(v)

gives a commutative diagram

LG

Ad

>

Horn(V)

dett

T(g)(.~

g LG

f

Horn(V)

~

IR

g ¢ G

-

where

dettA = det(~-tl)

for each

det(T(X)

say, where

°

t g ~R. We will have

~

- tl) =

is a polynomial

Qi

38

Qi(x)ti

function on

LG

and

LG

Ad

g

LG

is cQmmutative.

This shows that

Qi

is a polynomial

function on

LG

invariant under

AdG . We now establish a relation between such poly-

nomial functions

and the invariants

Let isomorphism

(LG)" P~-~

N

from the algebra

u: SLG---~]R

linear map A~

A: LG

functions

SLG

on

is the homomorphism • LG

P~-~P

(LG)': for each

extending

P c SLG

if the polynomial

function

of

over

LG

u ¢ (LG)"

u: L G - - * ~

~:

(LG) •

~IR

Ad

define

SLG . The transpose

~ith these endomorphisms.

is invariant under

to

. Each

of the algebra of polynomial functions

commutes

rations show that

LG . There is a natural

of polynomials

induces an endomorphism

induces an endomorphism

(LG) ~ . The map

G .

be the vector space dual of

the algebra of polynomial

where

of

on

These conside-

, g ¢ G , if and only g is invariant under (Adg)" .

-

The Killing form on a linear map morphism of

a: L G LG

-

: < X,Y > : trace

• (LG) ¢ , ~(X)Y = < X,Y>

.

If

(adxo a ~ ) A

defines

is any auto-

then the diagram

A

l

1,(A_1),

LC

is commutative

LG

39

~

~

(LG)*

since

~(AX)Y : ~AX,Y> : < X , 7 1 Y >

: ~(X)A-Iy : [(A-~)*~](X)y

.

We have here used the fact that the Killing form is invariant under all automorphisms.

We have now shown that a polynomial

invariant under

(Adg -I )"

invariant under

Ad

Since is bijective. under (AdG) ~

G

defines a polynomial function

P

~u

on (~G) ~ on

LG

, g • G .

is semi-simple

the Killing form is non-singular

It follows that every polynomial

function on

LG

and

a

invariant

Ad

gives rise to a polynomial function on (LG) ~ invariant under g and hence to an element of SLG invariant under AdG , and finally

to an invariant of

N0teo Let over

g

function

G .

P = P(XI,..,Xn)

LG , expressed as a polynomial

¢ SLG = IR[XI,..,Xn] in the basis elements

be a polynomial X I,..,X n

-~o-

The corresponding p o l y n o m i a l Q(x)

function

= ~(,(x))

Q = ~a

on

L@

is given by

- a(x)(v)

v(a(x)xl,...,~,(x)x n)

:

= P(,x,xl~ ,... ,<x,x#)

E~le.

G = S0(3) , the rotation group of

~3

LG = so(3) , skew-symmetric matrices. The basis XI =

o

I

o

-I

o

o

O

O

O

I

I o P

X2 =

O

o

o

o

o

=

o

I

-I

o

is orthonormal with respect to the Killing form. The identity representation gives det t LG ~

so(3)

.......

I -t aIX I + a2X 2 + a3X3~-~det

-a 1

-a 2

-t -a 3

= - t3

2 _

2

2). t

(a I + a 2 + a 3

•

-

41

-

Thu s

Q(alXl + a2X2 + a3X3) = a12 + a~ + a~ is a polynomial function on

LG

ponding polynomial

is given by

P • SLG

2

2

invariant under

AdG . The corres-

2

% * % * =3 = QCx)= P(<x,x1,, <x,x2>, <x,x# ) = P < - 2a I , - 2a 2 , - 2a 3 > so that

P =~

(X

+ X2 + X

e SLG . The corresponding invariant of

is

x(P)

=y

(x

.

+ x~) c tmG .

This invariant, however, can be obtained much more ~irectly by the method used earlier in constructing invariants of the inhamogeneous Lorentz group .

-

Section 6.

Let Let

G

-

Induced Re~resentatiQns.

be a locally compact group, and

a : K----~ U(V)

space

42

K

be a unitary representation of

a closed subgroup. K

on a Hilbert

V . We will now describe a process whereby a unitary representation

of the whole group

G

is defined.

On the topological product

G x V

define an equivalence relation

(gk,v)~ (g, o (k)v) for all

k ~ K . Let

G×KV=

I [g,v] I gc G, v

be the quotient topological space, where of

[g,v]

Vl is the equivalence class

(g,v) . Let

~: G X K V

be the map

-~ G/K

~[g,v] = gK .

For any

p ~ G/K

the inverse image

~

-1

(p)

has a natural

Hilbert space structure. Indeed if

p = gK

is of the form

uniquely determined. The bljection -1 gives ~ (p) the structure of a

[g,v] ~

v

[g,v] of

with

-1(p)

v ¢ V

onto

V

then each element of

Hilbert space. This does not depend on the choice of since if

p = gK = gIK

then the diagram

g

with

-1(p)

p = gK

-

[gjv] ;

43

-

~ V V

-I

o(g!g)

(p) V [gl ,v] ~

v

I

commutes. Since

a(g1!g )

is unitary our assertion follows.

Hilbert bundles.

The considerations above suggest the following definition. A triple spaces,

~ = (X,~,Y) ~

is a Hilbert bundle if

a continuous surjection of

a Hilbert space structure for each

X

X

on

p ~ Y .

and

Y

Y , and X

are topological -1(p)

is given

is then called the total

space

of the bundle ~ , ~ the projection, and Y the base space. The -I Hilbert space ~ (p) is the fibre over p . A section @ of the bundle is a map

@: Y

~X

on the base space each

such that Y

~@ = ~

with its value

. Thus a section is a function -I ~(p) in the fibre ~ (p) for

p .

Let

~ = (X,~,Y)

and

A Hilbert bundle isomorphism a: X ~ X

i) ii)

I

and

# : Y----~YI

~I = (XI'~I'Y1)

~~ I

is a pair

be Hilbert bundles. (a,B)

of homeomorphisms

such that

~I = = ~ =

maps the Hilbert space

for each

p ¢ Y .

-l(p]

isometrically onto ~11(~p)

-44-

Let

G

be a topological group.

given continuous actions of ag: x ~ - ~ g (~g, Bg)

x of

and ~

for each

such that

If

a

a Hilbert space

a

X

~1

~ ~

and

on

Y

if there are

such that the pair

is a Hilbert bundle automorphism

g c G . If

~ and

is a Hilbert G-bundle

on

Bg: Y ; ~ g y

then a G-isomorphism (a,~)

G

~

~1

is also a Hilbert G-bundle

is a Hilbert bundle isomorphism

commute with the action of

G .

is a unitary representation of a subgroup

K

of

G

V , then the triple

~o = (G ×~: v , ,, , ~/K) introduced above is a Hilbert G-bundle relative to the actions

g[gl,vl]

: [ggl,vl]

and

g(glK) = gglX of

G

on

G xK V

and

of

K

in

W

o

and

G/K and

respectively. If T

T

are equivalent representations, with the

equivalence given by an isometry

A:V

then

~a

and

~T

is a representation

~W,

are G-isomorphlc under the map

rg,~]

.~

~-

and the identity map on the base space

rg,Av] G/K .

in

-

45

-

~ea~ure theoretical notions.

Let

X

be a topological space. The class of Borel sets of

is the smallest family of subsets of

X

which includes the open sets

and which is closed under i) complements ii) countable unions. If a topological space then a Borel function ~-I(E)

is Borel in

is a function

~

X

whenever

E

X

~: X - - * Y

is Borel in

on the class of Borel sets of

Y

is a map for which

Y . A measure in X

is

X

with values in

[0,oo]

such that OO

OO

i)

E i) =

7S..(E i) i=I

for each countable sequence

ii)

Two measures

~(E) < ~

~,~

on

IEil

of mutually disjoint Borel sets.

for each compact Borel set

X

for the same Borel sets in

E .

are equivalent if they assume the value zero X . The measure class of

~

is its equivalence

class under this equivalence ~elation. For any two measures

#,v

in the

same measure cla~s there exists a unique non-negative Borel function on X , dv denoted by ~ and called the Radon-Nikod~m derivative of ~ with respect to

~ , such that

/x

:

X

f(p) -~dv (p)d,u(p)

for each integrable complex Borel function

Now let

K

M

and each

on

X .

be a closed subgroup of a locally compact group

There is a unique non-zero measure class in

f

g ¢ G

the measure

: ,,(g-lE)

M ~g:

in

G/K

~.

such that for each

-

is also in

M .

M

46

-

is called the invariant measure class on

G/K .

See [5] § 2, no 5 •

Induced r e presentations.

Let

~ = (X, =, Y)

topological group, and let For each

~ ¢ M

~ = I @i@

be a Hilbert G-bundle, where M

G

is a

be an invariant measure class on

Y .

let

a Borel section of

~ , and

f y <@(p), @(p)> d~(p) < ~ I

where < @(p), @(p) > denotes the inner product in the Hilbert space -I (p) . With the inner product

and with sections identified if they differ only on a set of zero,

~

is a Hilbert space.

Define an action of

g

on ~

(g @)(p) =

for each

w-measure

by

(p)g(~(g-.Ip))

p ¢ Y . This is the usual rule for shifting functions on

apart from the "~eighting factor

~ I u 4 ~

Y

. It is a unitary action since

-

=

d/~

=/y

d/~(p)

<@(g-.Ip), %o(g-.lp)> d//(g-,Ip)

It can be shown that this action of continuous) representation of

~:

~

~ "~

-

<~,~>.

=

T (~) . If

~7

G

on ~

G . We denote this representation by

is any other measure in the measure class of defines an isometry

valence between

T (~)

and

~

~ ~u

M . If

Y

M

g

then

which gives an equi-

Tv (~) . The equivalence class of

is therefore independent of the choice of class

gives a (strongly

Tp(~)

in the invariant measure

has a unique invariant measure class then we denote

the representation of

G

simply by

T(~) . We note that

G-isomorphic

bundles give equivalent representations.

When

~a = (G xK V, ~, G/K)

a locally compact group in a Hilbert space

G

is the Hilbert G-bundle defined by

and a representation

a

of a closed subgroup

V , and the unique invariant measure class is taken

on the base space

G/K , then the representation

T(~)

representation of

G

and

T

and

~T

representations of

induced b.y a . If K

then the bundles

a ~

and yield equivalent induced representations

T(~a)

is called the

are equivalent are and

G-isomorphic T(~T) .

For more material on induced representations see [22], [23], and

[25]

.

-

Section 7.

Let

G

be a separable locally compact group with an N . Let

H

the multiplication

N × H

~G

map

(nlhl)(n2h 2)

product

nh~-~-(n,h)

N QH

-

Representations of Semi-direct Product s.

abelian normal subgroup

shows that

48

=

be another subgroup such that is bijective. Then the equation

n1(hln2h; I) hlh 2

is an isomorphism of

X

of

N

onto the semi-direct

n: r h n h -1

relative to the action

A character

G

of

H

on

N .

is a continuous homomorphism

x: of

N

into the complex numbers of absolute value

1

. The set of

A

characters form an abelian group

N

with group operation given by

(XIZ2)(n) = Xl(n) x2(n ) . With respect to a suitable topology ('the compact-open) ~ dual of

is a separable locally compact group called the

N . See [16] § 23 for details. Any bijective map

induces a map

~--*N

,

X"----)'a~Z

(=x)(n) In this way t ~ 2~--~-g X

of

action G

on

n~gng

, defined by

: X(='ln) -1

of

. G

on

N

induces an action

N , (gx)(n) = x(g-lng) . For each

C-x= I g x

~: N ~ N

I g~ G I

X E N

we write

- 49

for the orbit

of

X

-

G

under this action of

on

A

A

N ; so that

N

is a disjoint union of orbits. We denote by

G :I gi g c G

and g x : x l

X

the isotropy group of

X

under the action of

G . Since

N

is

A

abelian

N

acts trivially on

a subgroup of

G . Let X

N

and hence on

L = HA X

N , so that

N

is

G , then X

=N

is a semi-direct product.

L

is called the little ~Foup

of

X ;

X it is the isotropy group of

X

under

H . There is a natural bijection

Gx X gG~---*g2

, which can be shown to preserve Borel sets, and which

commutes with the action of

G . This bijection will even be a homeo-

morphism under quite general conditions; in future often identify

G/G

and

see rS] Appendice I . We will

GX .

X

Bundles over an orbit. ,,,H H

Let

~

be a unitary representation of the little group

with representation

space. V . Then

Xc:

(n,~) w--~ X (n)a (~)

is a unitary representation of

G X

= N~L

in X

V .

L

X

-

Let

~X

space

) be the Hilbert G-bundle, with base X G/GX = GX , defined by the representation Xa • Thus for each o

over the orbit

GX •

of the little group

The bundle

then

-

= (G XGxV , : , G/G

representation

orbit

5o

GX

(a X

depends

with

we have a G-bundle

~X

up to G-isomorphism, only on the

and not on the choice of

XI = gIX

LX

X • Indeed, if

gl ¢ G . The little group of

GX = GX I ,

X1

is

-1

LX1 : glLXgl Let

aI

be the representation of

oi(l)

The bundles

~°

and

G xG v

. LXI

on

: o

V

given by

) •

are then G-isomorphic under the maps

Gi

.'~G ×G

X

V

,

[g,v1:

~ [g1~g~ I , v]

2i

and GX

~ GX 1

Theorem of Nmckey.

Let

,

P .

G = N~H

i)

T(~ X)

~

under

G

•

be a semi-direct product satisfying

the conditions above, and suppose that meets each orbit in

> gl p

~

contains a Borel subset with

in just one point. Then

is an irreducible representation of

and each irreducible representation

a

of

L

. X

G

for each

X c

-

ii)

and

a

-

each irreducible representation of

one of the form i

51

T(~)

with the orbit

GX

G

is equivalent to

uniquely determined

determined up to equivalence.

We shall call a semiTdirect product regular

if it satisfies

the conditions of Mackey's theorem. See [22] Theorem 14.2

for a proof

of this theorem, or [24] Theorem 3.12 for a more general result.

Action of

N t~C

The ind~ed representation form when restricted to the subgroup section of the bundle say, so that

~

, and let

~(p) = [g,v]

(n~)(p) = n(@(p))

for some

since

N

T(~)

takes a particularly simple

N . Let

~: GX

n ¢ N . For

~ G x G V be a X p c GX we have P = gx

v ¢ V . Then

acts trivially on

GX C N

= n[g,v] = [gg-lng,v]

= rg, x ( g - l n g ) v ] = [g, (gx)(n)v] = p(n)rg,v]

_- n(p) ~(p)

where

n ~ N

n(p) = p(n) .

is considered as a function on

~

defined by the rule

-

Thus the operation of

n

52

on the section

cation of the section by the function space

-

@

is simply multipli-

p~--~-n(p)

on the base

GX .

Action of

H . A

Since

N

acts trivially on

;GX,

H/L X

N , the map

hLx,'

-hx ~._ be a fixed

is bijective and we shall identify these sets. Let section of the bundle

H

~ H / L v . This means that

w

is a map

#%

GX

) H

such that

~(P)X = P

for each ~

-I

p ¢ GX . Each element in the fibre

(p)

of the b u n d l e

has a unique expression in the form

v] with

v ¢ V . It follows that any section

@

of this bundle has a

unique expression in the form

@(p) =

where

@~

is a function on

GX

with values in

V .

-

53

-

For simplicity we assume now that the action of sections of

~

is relative to a measure

invariant under

G :

For eabh

we then have

h • H

~

on

GX

G

on the

which is

(h~)(p) = h(~(h-lp)) = h[w(h-lp),¢w(h-lp) ]

= [w(p) w(p)-lhw(h-lp),~w(h'Ip)]

= [w(p),a(w(p)-lh~(h-lp))@w(h-lp)]

Thus the induced action of

H

on

~w

.

is:

(h@w)(p) = a(u(p)-lhw(h-lp))¢w(h-lp)

Let ~

be the Hilbert space of square integrable Borel

sections of the bundle

~a X . The linear map

gives a Hilbert space structure to the vector space

which consists of certain functions on the orbit

GX

with values in

V .

-

The inner product in ~ w

54

-

is given by

<@w,~w > = <@,~> : J GX <@(p),~(p)> d~(p)

which is the usual inner product of functions on

GX

in a ffilbert space

will not in general

V . We note however that

be a Borel function unless

Let ~w

T

w

@w

is a Borel section.

be the representation of

H

on the Hilbert space

defined above:

(T(h)@w)(p) = o(w(p)-lhw(h-lp))~w(h-lp)

The group

L

acts on the bundle

X H

W

rH

h~---~lhl

. Let X X of the operator T(X)

equivariant section of t:

,

---Gx

1 ~ L

meter group

,

H~GX

.

by

-I

II II Gx

each

p:

;-Ip

belong to the Lie algebra of on

~w

H---~G X

L

X

w

is an

relative to the action of the l-para-

~ exp tX . Indeed, in this case we have

t E ~R , so that

. The action

is particularly simple when

wCCex p tx)p) = (exp tX)w(p)(exp-tX)

all

with values

-

55

-

(T(exp tX)~w)(p) = a(exp tX)@w((exp-tX)p) . Thus

(~(X)~.(p) : ~d iT(exp tX)~}(p) It:o

: ~(X)(~w(p)) + ~ This means that on ~ .

w

}(x)

d

~((exp-tX)p)It=0

is the sum of operators

TI(X)

•

and

T2(X)

, where

}~(x)% = ;(x)~,~ and

T2(X)@~ = ~t @w ((exp-tx)p)It=0

"

We may note that

is a unitary operator on ~ particular that

TI(X )

for each

i ¢ L . This implies in X is the generator of a 1-parameter group of

unitary transformations and therefore skew adjoint by Stones theorem. T(X)

is also skew adjoint by Stones theorem, so that

T2(X)

is

as

well. I

•

We note further that the operator ~T 1 (X) same spectrum as the operator ~ ~(X) JL

on V .

on

=~" has the

- 56

Section 8.

-

Classification of the Relativistic Free Particles.

In section 3 we have defined a quantum mechanical system to be an elementary relativistic free particle if it is associated with anirreducible representation of the covering group

M ~

of the restricted inhomogeneous Lorentz group. A choice of relativistic observer

M

~

~IR 4

induces an isomorphism

: M ® Z o--

m4®sL(2,c)

so that, in order to classify the possible elementary relativistic free particles, we must determine the irreducible representations of

]R4~SL(2,C)

. To do this we will apply the theorem of Mackey

from the previous section.

Orbits and little groups.

Each

p ~ IR 4

defines a character

Xp : where

Xp(X) = e i

The map ~4

P~-~Xp

and

m ~

~ u(1)

denotes the Lorentz scholar product.

is an isomorphism of

]R 4

onto its character group

as a topological group, see [16] (23.27). If

a homogeneous Lorentz transformation then

A: ]R4----~]R 4

is

- 57 -

XAp(X) = ei<Ap,~ = ei = xp(A -I~) = Axp(x) so that

XA p = AXp . The isomorphism

action of

IR4~

~4

thus preserves the

SL(2,C) . We will therefore identify

by the map

p:

IR 4

with its dual

rXp . Thus

p(x) : x(p) = e i

In applying the theorem of Mackey to determine the irreducible representations of

]R4@SL(2,~)

we have

G = IR4~SL(2,C)

,

#%

H = SL(2,C)

and

N = IR 4 = ~ . The orbits in

N

under

G

are the

A

orbits

in

N

under

H

since

determine the orbits in

The action of SL(2,¢) ~

IR4

N

acts

under

SL(2,C)

on

S0(3,1) . The orbits of

the orbits under

trivially.

We must t h e r e f o r e

SL(2,~) .

]R $ IR A

is via the covering map under

SL(2,C)

S0(3,I) . These orbits are Mc +

= I p [ = c > 0 , P4 > 0 1

M c_ = [ p l = c > 0 , P4 < 0 1

M-c=

I pL

= -c < 0

M +° =

Ipl

: 0, p$ > o I

M ° : IpJ = O, P4 < 0 I 101 .

are therefore

- 58 -

To see this we note that

i)

any point in

~4

[ P I Pl = P2 = 0 , P3 ) 0 I ii)

can be mapped into the half-plane by a space rotation.

The orbits under the group of pure Lorentz transformations

I ~W

cosh u

sinh u

sinh u

cosh u

u E ]~ , in this half-plane, are the hyperbolas and straight lines and point: 2

2

-p3+p~=

c

> o,

p~ > o

2 2 -P3 + P4 = c > 0 , P4 < 0 2 2 -P3 + P4 = -c < 0

ps=p~> O -P3 = P4 < 0

Io]

•

-

iii)

iv)

the function

59

-

p : >

is constant on each orbit

each orbit is connected since

From properties i)

and ii)

SL(2,C)

is connected.

we conclude that each of the sets in the

family

IM+I , and

M°+ , M_° , [Ol

IM-Cl

IM_°I ,

(c > o)

are c o n t a i n e d i n one o r b i t .

From i i i )

and i v )

we conclude that no two of these sets is contained in the same orbit.

We note that

~4~

SL(2,C)

is a regular semi-direct product

since the Borel set

I(O,O,0,p4)lp4

E ~

U {(O,O,P3,0) lP3 > 0 1 U

I(o,o,1,1)}ui(o,o,1,-1)1

meets each orbit in just one point.

For each orbit

GX

in

each irreducible representation irreducible representation representative point

X

can be taken as follows.

T(~)~

IR 4 o

with representative

X , and

of the little group of

IR4~SL(2,C)

L we have an X . The choice of

in each orbit, and the resulting little group,

-

Orbit

6o

-

Little group

Representative

Gx

X

Mc

(o,o,o,¢c)

SU(2)

Mc

(o,o,o,.,,rc)

SU(2)

(o,¢c,o,o)

SL(2,m)

(0,0, I , I )

A

~o

(0,0,1 ,-I )

a

101

(0,0,0,0)

ST,(2,r)

+

m

u-c M° +

L X

where

A=II

iO

e O

z -i8 1 e

t8¢~,ze¢

1

These little groups are determined directly from the definition of the action of

i)

SL(2,¢)

on

IR 4 . We have for instance

x = (0,0,0, ~rc) corresponds to the matrix

so that the little group is the subgroup of

~ =

(~c o) o ~rc

SL(2,C) :

f A, A.{'°o ~c ° ).A. =(~°o ¢c ° 1 1 :IAI~*:II = su(2)

.

! ~0

i

o

II

~Z O

O 01

0 o ,-~

°

I

***1

,~a

®

II

~

0 v

N

O

0

o

0

o

tl

° o

I

o o !

.~

0

•~"¢'!

0

II

o

,r-

T

0

<~

II

o

~-

I!

<~ << I!

T

0

I

II tD

II

il

,e-

II

O !

.T-

"7

O

,,--.. o

4~ o

~

d w

O

II

O

MZ O

O

O o

10

.el

4.~

@ tl

O V

1'4

,el

O

0

0

0

O

II

O

O

t~l

c~

,~t

to

,D

V

0

t! @

II

TI

<1

-

62

-

Momentumr e ~ r g y and mass.

Let

T = T(~)

G : ~R4@SL(2,£) of

x c IR4

. In

be an i r r e d u c i b l e

of

section 7 we determined the action

on a section

space the orbit

representation

~

of the Hilbert bundle

~

T(x)

with base

@X • Indeed

(T(x)~)(p) : x(p)~(p) : ei
T(x) : e i
where by this is understood the operator given by multiplication of sections by the function p ~_~ei defined on the base space GX of the bundle.

Let mentum in the

~ xI

¢ L]R $

be the element corresponding to linear mo-

direction, then

T(e~p t, I) = T((t,0,0,0)) : e-itpl so that

T(~I) : -iPl and @

~("1)

: -Pl

"

-

Thus the self-adjoint operator in the • x I

63

-

Pi ' which represents linear momentum

direction, is simply multiplication by the function

-Pi "

Similarly for the other components of linear momentum. The operator I' P4 = ~T(~4) representing the energy is multiplication by the function

P4

The operator 2

2

2

2

- P1 - P2 - P3 + P4

is therefore multiplication by orbit

which is a constant on the

GX . The system is therefore an eigenspace of the mass operator,

and the mass is the constant value of

V

on the orbit.

We can now write down the following table giving the spectrum of the energy operator, and the mas , of the particle as'~>ociated with the representation and not on

T(~ X) . We note that it depends only on the orbit

o .

Orbit

Ener&y s pe_ctrum

Mass

Me

[v%,oo)

v'c

Mc

(---o~,-V"c]

~'C

M- °

(..-,o,oo)

¢-0

M+°

(0,~)

o

M°

<-~,o)

o

Iol

Iol

o

+

GX

-

64

-

The case of imaginary mass is excluded since the mass operator would not be self-adjoint.

The only particle with zero energy spectrum is

the vacuum state. If we exclude negative energy states then we are left with the orbits

M +c , c > 0 , and

M° +

which we will refer to in future

as the cases of non-zero mass and zero mass respectively.

An6ularmomentum

and spin.

The self adjoint operator corresponding to angular momentum 1. about the x 3 axis is ~ ( m 1 2 ) . In the case of non-zero mams the little group

LX

is

SU(2)

; in the case of zero mass the little group is

In both cases the 1-parameter subgroup of

SL(2,C)

A .

:

it t:

corresponding to the space rotations of a subgroup of

IR 4

about the I. L X . We can therefore determine ~T(m12 )

x3

by the procedure

given at the end of section 7 under the heading: action of assumption that the orbit

GX

axis, is

H . The

has an invariant measure is satisfied in

this case, since d.(p) =

dPldP2dP3

P4 is

such

a measure,

for

X = (O,O,O,~rc)

of

X = (0,0,1,1)

The procedure requires the choice of a section which is equivariant under rotations about the

x3

.

~:@X ~ S L ( 2 , ¢ )

axis. The section property is

-

(p)x for all

:

65

-

p

p E GX • The physical interpretation of this in the non-zero

mass case, is that

w(p)

is a Lorentz transformation to a new frame

of reference so that a classical particle with p = (pl,P2,P3,P4)

has new

4-velocity

4-velocity

X = (O,O,O,~c) . Thus

w(p)

is a transformation to the rest frame of the particle. In the case of zero mass, ~(p) photon with

is a change to a frame of reference in which a classical

4-velocity

p

therefore moving along the

has new x3

4-velocity

X = (0,0,1,1)

and is

axis.

A specific equivariant section

w

is suggested in each case by

the following physical considerations.

i)

Non-zero mass.

A convenient section

w: GX

• SL(2,¢)

to be the unique pure Lorentz transformation such ~(p)

is defined by taking

~(p)

~(P)X = P • Physically

is a change to an observer moving with uniform velocity. To express

this analytically we note that each matrix

h C SL(2,C)

has a unique polar

decomposition

h=~'u

where

~

is positive definite h e r m i t i a n a n d

u c SU(2) . This corresponds

to the unique expression of a Lorentz transformation as a product of a pure Lorentz transformation and a space rotation; see [36] page 168. Thus ~(h SU(2)) = ~ . It follows that for each

v ¢ SU(2)

the diagram

-

sL(2,c)

C-X

~ sL(2,c)

..........

"n"

66

-

-I

"--

VITV

[ >

h SU(2) .'

~c-x

vh

II

II ,, SU(2) is commutative. This shows that

ii)

w

s~(2)

is

~v

-1

su(2)

SU(2)-equivariant.

Zero mass.

In this case we can define

~: GX

, SL(2,C)

so that

~(p)

is the composition of the pure Lorentz transformation mapping (0,0,1,1) : ~ (0,O,P4,P4)

and the space rotation mapping

(O,O,P4,P4) ~--~(pl,P2,P3,p~)__ . This section is equivariant under rotations about the

x4

axis. Physically it corresponds to a change of observer given

by a change of velocity followed by a rotation.

We can now apply the results of pages 52-55 • The section X to a Hilbert reduces the Hilbert space ~ of sections of the bundle ~a space

~

of wave functions defined on the orbit

representation space

V

of

~ . The Hilbert space

and the dimension of

V

with values in the is the fibre of the

bundle

~X

states

of the relativistic particle. The representation

a representation

T

on ~

V

GX

is called the number of polarisation

. The self-adjoint operator

representing angular momentum about the S 12 + 012 W

x3

T . T(~ X)

defines

~ T (m12)

on

axis can be written as a sum

~w

-

of self-adjoint operators, where operator

~(m12 )

on

12 1 d ( 0 ~)(p) : r ~

1 d z ~

:

1,

V,

67

-

S 12 has the same spectrum as the

and where

~(exp(- trot2) p) I t:0

P2sin t , Plsin t + P2COS t , P3" P4)It=O

~ ( P l c°s t

. ~

-

p2"."q

a

) %(p)

The physical quantity represented by the operator the spin angular momentum about the

x~

S 12 is called

axis relative to the section

The quantity represented by

012

on

~

1 : _z (pi.~.a op2 - P 2 " - -aPa I )

is called the orbital angular momentum about the

relative to the section

axis

~ .

We note that, in the non-zero mass case, we have spin and orbital angular momentum defined about all space axes, relative to the same section

~ , since

~

is

SU(2)-equivariant. I

•

The largest absolute value of the eigenvalues of ~ a(m12) is called the spin of the particle (about the

x3

axis) .

W

°

-

68

-

Determination of spin.

We will determine the spin of a particle associated with the representation little group

T(~)

. Here

a

is an irreducible representation of the

LX

and the spin about the 1 . value of the eigenvalues of T o(m12) .

x3

axis is the largest absolute

i) Non-zero mass. The little group

SU(2)

is compact, so that its irreducible

representations are finite dimensional, [16] (22.13). described as follows. For each ~rA

be its

r

th

A c SU(2)

: ~rc2

~r

They can be

let

C2

tensor power. Let

ar(A): Vr+ I be the restriction of @ r A symmetric tensors. Since

) Vr+ 1

to the SU(2)

(r+l)-dimensional subspace of

is compact, [161 (22.23)

V+I

can be

given a Hilbert space structure so that

r: SU(2)-----~U(Vr+l) is a(unitary) representation. The representations

, r = 0,1,2,... r are all irreducible, and every irreducible representation of SU(2) is

equivalent to some

a

r

.

a

- 69 -

The element

m12

to rotations about the

x3

in the Lie algebra Of axis

corresponding

$U(2)

is

0

m12

Therefore

I Q T at(m12)

=I -~i11 o

is a diagonal matrix with entries

r .~. , - ~r -

Thus the spin about ~ the

~ , ~r. - 2

x3

,

....

,

-~-

r

r axis (and hence about any axis) is ~-

We see that the particle associated with mass

m > 0

and spin ~r

rep~ntation

.

where

T(~

.

) has

r X = (0,0,0,m2 ) . This proves that the

associated with a particle of non-zero mass is uniquely

determined by its mass

m • 0

s = 0, ~I,

and spin

1 , l2 ' .....

The fibre of the Hilbert bundle associated with a particle of spin are

s =~

2s + I

is

Vr+ I

which is

(2s + 1)-dimensional. Therefore there

polarisation states for a particle of spin

s

and

mass.

ii)

Zero mass.

The little group

relative to the action

A

is isomorphic to the semi-direct product

(e,z)~-~eiSz

of

IR/~v

on

¢ .

non-zero

-

70

-

The isomorphism is given by the map

i~

ze

[: -il

l_

~

(z,2d)

e-i~

C

and its dual

¢

can be identified with IR 2 • so that IR /4~ acts 2 by ro~ations. The orbits in ~ under rotations are the origin and circles with centre the origin. product

IR2~IR/4 ~

A

is therefore a regular semi-direct

and we apply the theorem of Mackey to determine

its irreducible representations.

For each orbit we can choose a re-

presentative and determine the little group as follows.

Orbit

Representative

I(o,o)l

(o,o)

I(x,Y) J x 2 + y2 = c21

(c,O)

Little group

IO,2~J

No known particles have been associated with representations arising 2 2 from the orbits I(x,y)i x + y = c21 . We shall therefore confine ourselves to the orbit

I(0,0)~ . Each irreducible representation of

is l-dimensional and is of the form

~,

r = O, + ~

I

3 , + I , +~,

~e

ri e

.... , The induced representation of

rr: (x,y,8) ~

e

rio

]R

]R/4~

-

so that the representation

T

of

r

71

A

-

is

2ri~ e-ie

i_ t

Thus

2 e

rit ex? tm 12

I......

0

~

e

e

so that 1. ) T Tr(m12 = r . We see that the particle (about the

x3

T(~)

with mass zero has spin

axis) . Moreover, for a "r given spin

two mass zero particles, corresponding to

Ts

and

s

[rl ,

there are exactly

T_s .

The Hilbert

bundle has a 1-dimensional fibre in each case, so that there are two polarisation states for each spin s

and mass zero. The sign of

r

is called the

helicit~ of the particle.

We conclude with three examples of elementary relativistic free 1 particles. The electron is a particle with non-zero mass and spin ~ ; I the neutrino has mass zero and spin ~ ; the photon has mass zero an~ spin I

-

Section 9.

72

-

The Dirac Equation.

In this section we shall give a brief treatment of the Dirac wave equation, and show that it is associated with a particle of I spin ~- and mass m > 0 .

Minkowski-Clifford algebra.

For each

x e IR 4

let

x = -xlT I - X2T 2 - X3T 3 + X4T4

~ = x i T i + x2r 2 + x3r 3 + x4~ 4 , where the

Tj

are the matrices

defined in section 3 • Let

~'(x):I ° ~) o

Then

~: l%$----~Hom(C4)

is linear in~ective and

[~(x)]2 = <x,=~ Thus

¥(~R 4)

.

generates a Clifford algebra over

the Lorentz scalar product.

The map

T:

~/A

0

)

" \ 0

(A')-~

A:

IR @

with respect to

-

is an isomorphism of

SL(2,¢)

73

-

onto the spin group

of the Clifford algebra, which is a subgroup of Let

SL(2,C) .....~. ~ S0(3,1)

Spin (3,1)

GL(4,C) .

be the covering map, then we have a

commutative diagram

su(2)

"

,~ s o O )

sL(2,c) -° = so(3,~) Spin(3,J)---~ SO(Ym )

.

This means simply that

for each

x c IR 4

and

A c SL(2,C).

Dirac bun~. e: Let

X = (O,0,0,m)

with

m > 0 . Let

be the Hilbert

6X

bundle associated with the restriction of the representation

T(A) =

all

A c su(2) . ~ u s

Dirac bundle.

~~

I

A

0

0

A

1

i, a b u n ~ e

T to

E U(4)

~

fibre

C~ ~

~Ie,I

the

SU(2) :

-

Let

~X

be the bundle with fibre

representation

SU(2)

~X aI

C2

defined by the inclusion ¢2

in

C4

by the map

> ( x 1,x2,x 1,x 2)

is a sub-bundle of

The map

-

~ ~ U(2) . If we embed

(x1,x 2)~ then

7~

X ~T

.

G ×G C4-"~ GX x ¢4 given by X [A,v] ~---~ (Ax,T (A)v)

for each

A ¢ SL(2,C)

product bundle over

gives a bundle isomorphism of GX

with fibre

Define an action of (p,v) : for each

x ~ ]R 4

G

onto the

C4 . The Hilbert space structure

in the fibres is not preserved however since is not a unitary representation. By

X ~w

G

SL(2,C)

T ~ Spin(3,1)

we mean the group

IR4QSL(2,C)

on the product bundle by

.~ (p,x(p)v)

and

(p,v) ~---->(Ap,r(A)v) for each

A ¢ SL(2,C) . The bundle isomorphism is then a G-isomorphism. Each section

P:

~ (p,~(p))

on the orbit

~

of the bundle

X ~T

of the product bundle, where GX

~(p) = [w(p),~(p)]

with values in where

;(p) :

~

corresponds to a section ~

is a function defined

C4 . More specifically, if

is a section

GX----*SL(2,C),

then

.

-

The function

all

¢

75

-

is a solution of the Dirac

wave equation:

P ¢ @X • This is equiv~ent to

Y(p)T(w(p))@(p) : mT(m(p))@(p) i.e.

i.e. ¥(w(p)-~p)@(p) = m@(p) i.e.

i.e. m

0

o

,,

m m

• ~@) m

o

0

m

:

• ~(p)

m m

i.e. ~(p)

Thus

@

satisfies the Dirac equation if and only if

of the sub-bundle has mass particle.

E c2 .

m

~X

¢

is a section

. The particle associated with the bundle

~X a~

and s p i n ~ , so that the Dirac equation describes such a

-

Section 1o.

su(3) :

?6

-

Charge and Isospin.

We have seen that a representation of the Lie group IR4~SL(2,C)

on the Hilbert space associated with a quantum mechanical

system leads to a definition of the physical concepts of linear momentum, energy, angular momentum, mass, and spin, and leads to a classification of elementary relativistic systems.

More recently the Lie group

SU(3)

has been used in an effort

to explain the quantities electric charge and isospin. We will sketch some of the ideas involved.

Physical interpretation of

The Lie algebra of su(3)

of

3 × 3

SU(3) .

SU(3)

is the 8-dimensional algebra

skew hermitian matrices of trace zero. As a physical

interpretation, the matrix

Q=

11

2.

is associated with electric charge and the matrices

- 77 -

I

11 =

0

0

O

0

0

0

0

0

~-i

0

- 17 o o

12 =

0

0

0 1.

o

I3=

0

0

0

0

o

0

are associated with the three components of is ospin.

If it is assumed that, for a given quantum mechanical system with Hilbert space and

SU(3)

on

H , we have representations of

H

which commute, then we will have a representation

of the direct product

G × SU(3)

one in the sense that

T

T = ~ @ w

where

space

Ho , w

space

W , and

G = IR4~SL(2,¢)

e

on

H . If the system is an elementary

is irreducible, then

T

will be of the form

is an irreducible representation of

is an irreducible representation of H = H

O

@ W . Since

T

SU(3)

SU(3)

is compact

W

G

on a Hilbert on a Hilbert

is finite

dimensional.

Let by

U(W)

su(3)

be the Lie algebra representation defined

v . This extends to a unique homomorphism of complex Lie algebras e

sl (3,¢)

where

s1(3,¢)

is the complex Lie algebra of

with trace zero. Let ~ spanned by

Q

7 > Hom(W)

3 × 3 complex matrices

be the abelian (Cartan) sub-algebra of

and

13 :

= ~

C =

°

c2

ci ~ C

-(oi+% )

1

s1(3,¢)

- 78 -

Since

Q

and

13

commute, the skew-adjoint operators

~(13)

have common eigenvectors

T(~) " 0

and

e 1, ... , e n which form a basis for all

X c ~ ; let

e. are eigenvectors of ~(X) J be the eigenvalue of ~(X) on ej .

uj(X)

The linear forms ~ights

W . The

~1,..,Wn

on the complex vector space •

of the representation

weight veqtors

of

~ . The eigenvectors

for

are called

el,..,e n

are called

~ .

We have an isomorphism

H=H

@W~H

• .... @ H

O

O

(n

O

factors)

defined by

@ e.

,3

~(o,...,o,~,o,..,o)

(Moreover this decomposition of representation

T

gives a decomposition of the

when restricted to

TI G = ~ @ Since

H

.

.... @~

G :

.

is an irreducible representation of

G = IE4~SL(2,C)

it

is associated with a relativistic elementary particle of a definite mass

m

and spin

s

say. Thus the system will, under observation,

manifest itself as any one of Furthermore, for

X = Q

or

I ~1(x)

n

particles each of mass

13 , .Iz-~(X)

,

' i~n

m

and

has a discrete spectrum

spin

s °

-

and

H = H

Thus the of mass isospin

@ .. @ H

Oth j

79

-

is the corresponding eigenspace decomposition.

o

subspace

H

represents an elementary relativistic particle o 1 3rd m , spin s , electric charge ~ -~j(Q) and component of 1 [ ~j(I3) .

Determination of weights.

It remains to determine the weights the possible irreducible representations of

~. associated with J SU(3) , and to identify

the corresponding particles.

The weights of the adjoint representation of called the root_~s of

0

where

a and

8

, 0

sl(3,C)

, =

,8

are

and are

,

-a, -8 , = + ~ ,

are the linear forms on

c~--~ - c I - 2c 2

sl(3,C)

and

~

-=

-8

:

c~---~2c I + c 2

respectively .

Let ~l

be the vector space of linear forms on

~

let

V

be the ratiomal vector space contained in ~P and spanned by

and

~ . The bilinear (Killing) form on

sl(3,C)

(dual of ~

)

and a

defined by

<x,y> = trace (adx,ady)

is a symmetric scalar product whose restriction to I~ is non-singular. This induces a symmetric scalar product of this to

V

<°,->

on ~' . The restriction

is positive definite and rational valued, so that

V

is

-

80

-

a rational Euclidean space when equipped with this scalar product. 1

The vectors

~ 13

and

1

~ M =~

1

~3(Q-I3)

are orthogonal in

and equal in length. Relative to the dual basis in ~Jany element in ~/ will have coordinates

l i ~(13) ' In particular the root ,

~

~(M))

has coordinates (~, --~) ~2

and

~

has

) . By our choice of basis vectors in ~ which are

orthogonal and equal in length we ensure that the map

I

~ ,-------. (

~'(T3) ' T ~(M)

V ~IR

2 :

)

preserves angles.

We shall use the following facts about ~',eights, which are all special cases of general theorems on the weights of semi-simple algebras. See [2oS IV Theorem I, VII, [30] LA 7, and [29] •

I

All weights of all finite dimensional representations of

belong to the rational Euclidean ~pace

2

If

representation space

~

is any root and

~

V .

is any weight of a finite dimensional

~ , then the reflection of

V ) in the line perpendicular to

I r ¢ ~ I ~ + r ¥

is an unbroken interval in

~

.

sl(3,c)

~

~

(as a point in the Euclidean

is also a weight of

is a weight of

~ I

~ . The set

-

81

-

A~ong the weights of an irreducible representation I there is just one with ~-~(I3) maximal. We call this the highest weight of

~ .

All the weights of an irreducible representation be obtained from the highest

weight of

~

r

can

by repeated applications

of property ~ . The highest weight determines

If

~

J

T

up to equivalence.

is the highest weight of an irreducible representation

then

p =

2 < ~,=

-

>

< a,a >

and

q -

2 < ~,,8 •

< #,8 •

are non-negative integers. For each pair (p,q)

of non-negative integers

there is a (by ~ unique) irreducible representation weight

~

~=(p+q 2

such that

2 < ~,a > < =,a >

= p ,

~(p,q)

2 < ~ # • = q , < ~,~ •

1 (p - q) ) , using the coordinates ' 2~3

V=

with highest

so that >IR2

defined above.

Example s.

i) (1,0)

The representation

~(1,1)

which is the same as the root

of the adjoint representation. Thus

has highest weight ~ t h

coordinates

a + ~

which is the highest weight

;(1,1)

is equivalent to the adjoint

representation and the weight diagram is:

-

-1

I ~."%"

--•

0

•

82

I, 2

1

I

•

-

1

a

I

=) ~, ~p=...,~t~o.

÷(3,0) ~

~..t

•

,.~t

(~, ~).

By repeated, application of property 2 we obtain the weight diagram:

3

-I

I

o

I

3 ....

-(IP2# ~

me . o 0

-~+#

. (z+#~

2

~

1 ~(I 3)

-

83-

Particle assignments.

According to the theory developed above a quantum mechanical system associated with an irreducible representation of the direct product

(IR4@SL(2,C))

× SU(3)

ducible representation of s

will be associated with an irre-

IR4~SL(2,C)

, giving a mass

say , and an irreducible representation of

of weights

~S'""~n

SU(3)

will have electric charge

m

and same spin

1 [wj(Q)

and

3rd

s ,

and a spin

which gives a set

say. The system then consists of

each with the same mass

m

and the

n jth

particles particle

component of isospin

1 [~j(I3)

In pratice however the theory is applied to systems of particles whose masses are not equal, although some are of the same order of magnitude. We give some of these systems below.

The first list gives some particles which are associated with the adjoint representation of

SU(3) . We use the conventional names

or symbols for these particles. Masses are given in MEV units. Each particle is listed under the corresponding weight (root in this case).

-85-

-8

Weights:

0"

0'

Representation of

R4(~SL(2,¢) :

neutron proton

i) baryons mass

= 1130 + 1

Z°

r.+

~o

A°

192

spin -~"

il) anti-baryons

--O

0

+

rO

~+

mass = 1130 ~ 192 I spin

iii) pseudo scalar

mtJ

~nt i

)ro-

~q&u

~o

con

K°

+

K*

m ~

K

~o

mesons mass = 315 + 182 spin 0 +

iv) vector mesons

K ¢°

o

P

mass = 800 spin I v) 2 nd r-meson-

O

+

N~*

N* *

Y;* *

'1

Y;.

0 O

nucleon resonance mass = 1600 spin 3 O

vi) 3 r d = - m e s o n O

nucleon resonance mass = 1688 spin ~ 2

-

85

-

The second list gives some particles associated with the 1o dimensional representation

~(3,0)

of

SU(3) .

I

Weights~

-D

2=+~'

N*

2 N*

0

Representation of R

:

i) meson-baryon

N*

y*

y,

re s onance mass = 1460 + 2:~3 spin ~ 2 ii) 4 th ~-mesonnueleon resonanc~ mass = 1922 spin l 2

This information is taken from Gourdin [ 2 1 ]

.

-

86

-

Index of Terms. With page of first occurence. Adjoint representation of a Lie group

energy

25

energy spectrum

adjoint representation of a Lie algebra anti-unitary

24

equivalent measure

45

32

5

factor set i 12, 13

angular momentum

24

fibre of Hilbert bundle

automcrphism of projective Hilbert space

future

I

G-isomorphism

Borel function

43

4

base space of Hilbert bundle

Borel set

63

44

43

45

helicity

45

71

highest weight 81 homogeneous Lorentz transformation

causal automorphism

Hilbert bundle causal invariance

43

6 Hilbert G-bundle

causality

I

character

48

44

induced representation

character group (dual)

48

cohomology of a Lie algebra

27

d/latation

2

47

invariant of a Lie group 13

invariant measure

31

53

invariant measure class derivation

46

isomorphism of Hilbert bundles isotropy group Dirac bundle

43

49

73 isospin

Dirac wave equation

77

75

dual (character group)

48

lifting of a projective representation linear momentum

electric charge

76 little group

elementary relativistic free particle

2

2

19

49

24

9

-

mass

37

measure

-

restricted inhomogeneous Lorentz group 45

rest frame

measure class

45

non-zero mass

65

representation

Minkowski structure

orbit

87

S

roots

64

8

79

section of Hilbert bundle

49

semi-direct product of groups

orbital angular momentum

67

Lorentz group

3

semi-direct sum of Lie algebras

orthocronous homogeneous

symmetric algebra

21

28

2 spin

67

spin angular momentum past

43

67

S

projection of a Hilbert tensor algebra bundle

26

43 time-like

projective representation

S

8 total space of Hilbert bundle

43

physical interpretation of a symmetry group

21

transition probability

polarisation states

66

translation

2

trivial factor set Radon-Nikodym derivative

13

45

regular semi-direct product relativistic invarlance

4

51

universal enveloping algebra

7

relativistic observables

24

wave function

restricted Lorentz group

2

weights

78

weight vectors

zero

mass

66

78

27

6

-

88

-

Bibliography.

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[3]

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[4]

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[5]

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[8]

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[9]

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Th~orie des Groupes de Lie, Vol 3, Hermann

[lo]

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Fundamental Concepts of Algebra, Academic Press 1956

[11]

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[12]

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[13]

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[14]

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Mesure de Haar, Hermann 1963 Homological Algebra, Princeton 1956

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Lie Groups for Physicists, Benjamin 1966

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89

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[16]

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r17]

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[21]

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[22]

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F24]

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r25]

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r26]

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[28]

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[29]

G. Racah, Group theory and spectroscopy, Springer Tracts in Modern Physics 37

[3o]

J.P. Serre, Lie Algebras and Lie Groups, Benjamin 1965

r31]

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[32]

I. Segal, Mathematical Problems of Relativistic Physics, Am. Mmth. Soc. 1963

r33]

E. Spanier, Algebraic Topology, McGraw-Hill 1966

221-265, 195i

-

9o

-

M.H. Stone, On one-parameter unitary groups in Hilbert space, Ann. Math. 33, 643-648, 1932

[35]

N. Steenrod, The Topology of Fibre Bundles, Princeton 1951

r36]

A.S. Wightman, L'xnvariance dans la mecaraque quantzque relativiste zn Relations de Dxspersxon et partxcules Elementaxres, Hermann 196o

[37]

E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. of Math. 40, 149-2o4, 1939

r.38]

E.P. Wigner, Group Theory, Academic Press 1959

r39]

E. Wigner and V. Bargmann, Group theoretical discussion of relativistic wave equations, Proc. Nat. Acad. Sci. ~ , 211-223, 1948

r .o]

E.C. geeman, Causality implies the Lorentz group, Jour. Math. Physics ~, 490-493, 1964

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