Teil IV: Lie - Gruppen

Elemente der Gruppentheorie

Darstellungen von Lie - Gruppen

Gruppen U(1), SU(2), SU(3) und SO(3)

7-z

e,\.e- Q t

F-r ,ge,; -F e4v *".'-fPt*- %u-r^r-e> L,-

(2.3. Jk)(L?r/ - P*^^) , GtN) ?. *k.,

b)'th'noo 1\*-*^- ry"-/e'.*- ) d.'e- Y.; g-L

t/&1- c46k"a; A, $---K.

D> JeL*t: ALLUe*t A ! q -> (ag.) ! 6Lk)G--

- (./

"\ d^z rqz^f-<- kt"^fA*". j-[^>G':e^ """A a;i!Q.

. ,4:b JuNC,ERT!4 Qe,et-1".- l-

,!" A!q

aah 4\"-"<-

"*J q\

"l1r\'Ar

+.* t6^. &or-.,, ezrr/+"-{/I

ef*Yo. ft*. G 4-I

d" $o.t W*Z a'4 /4t*''"r (o")')* = c;, )

. U,",d,<e

Ll! G "& d^r ,-*-o'4o'lJ+= LIL , d,L.

1,rl*'ll = l71'll*= L

- iV-t+

/

/_tE_G!,.ppEN

(,.. free-J- 4L;rt ?,e'.al-- 4,- fuf*'

de- -&u*4U 'L -tr2.2"*'- *^t(h'""*{n)

ler L lV i

-y'--Wy-- . .

"rf* VA* ppe p-tk eo&e *,'--bf.oU h (ry

do f,,\e- GRuPPe q = {g,a,k,

}fu ftQ*-dL rye44""^f"^,

"

, Ju-,'/-

erJ ! G ,\f a-r,& da- V*"aJt

6A!Ct/ A.fr+9q f-,u7*a<E ' C64 ) k

3) E) fd7-te/"^n-^* : % eSr.\l|<f( e- e e *-7

"f =

A"=/ ef a,*.4 e= -t_ bn>.":-aar*l

+) Evewr trQu"u#. !4 j ! q qeltz7<* frt G

dk ?-'f = ftft-"= e

o

o ./u *al.B a /-Vur.t- . '

/->"0^* A4ke4,-'a*^ C t G - j4^''t o, F e C,U

g,4eG 4> xfr*FleC

( Z; flqk/2"2*-cAo'-u)&-e.,

% - /.-""t /*'fJt^ 4ua'*

Y 1 v /

/ ra.+ /,r--r-@*Uon ^ ot-"A 4h*-n OPERAfORE V )/

7-s

2. Ze;U*-Q :- t

t -

?lt *-/r--- #'^{-v*n

l-)* : U*: !"o : ()-1-

9g=-nnq,u7qeR

(5,"/- 144- Ez1{."*/I-'*VU. ,r# 6>(=a'(*)

<-\r. a('r A=>)

9\!YN=Sfs=ls'Zb u-ur6* Gr.^ff.,

rylotL{ d,*"L

ut^; {& }r r Ar - i4q-K+e-<*

U = e-xp [rZ"^X^]a

,n-,V de{ lUI= L

o(a : ree-i le- ?*n'nle- *".* a = Ar2' . . ' ,N?t

do.: !a " k(e;h" , .L fu^ /24 '*&- *tr;rt**er^a,"

l/

U= It+rlt^{+ Oki)q

X" ; Ervudz T6."*

(Ge.tveR+tOEeatl

dr-r C,-ff 'S

U Qv)

tr> d-e GereR,+ToRep 4v S u (Ar) bYdr-," ur+L

/*,t- *tGe?t?.4 :

x^,xnl : 'f*0.4.f*rt@

q> S U (!{) 4>t ehe- Le - 6-!,, ppe

{

/ oLc

T- 2q

o Jte"torTEsa4e qw*,*1l-

4 ! G -bF+ /tilka"A , teec,1 fte{

-A*: -/'

jro*; /-t"/- Qp"**rc'^ d;n* !etetr4ZdutuCteRfI

- / 1 ^

;" /"Qe'.dtr 6; dan /eZ(n-.-.(/ v

u : e*p [;1] = I + i(, + {l'. {l=*..,u

Tn(q 4- !-,\ j"'^{a-4"1rt-

?*O

-'!A * !o

1,4.* = wf [-;1"J = e"p G;1]

)^4 /"frt z,t* ?,t = e-yp [-;( J ""f [;4 J - -L

u*d L( ?_(*: ebp [r{J e"p[;( J : IL

ot flIrn-v , &- bc(,".^e"-^ ,,,g.-( 4" "9, ,/'b,p().- (^*-r-- \,/1F{a &r -{AJ-rr^

/)

-

^+

4 ; r ' = 4n' .^.J

) -L

T-dua+.^d-vre- dx"f &v

Arr:4:,

&Vr ou&e&,^^-*/"- >eny'(

JY-3L

--7 A.Zaaa-u,1,.u-e,q/'.4 at* eA f{NbkHt14t"L"l rh &r

4 !.vew, &,Zut i***u

q> eff So(s) kx{>+ 54<,f,reLLu'q,

*nL avSV dt^ & exe-< &; rtur".-*'*( o p*u%"!*-,'.1 !.vew,, &,zu*t i*** ['+;L\lvul^.*,^>rpLavSV dt ^&

Lr: *=Or - xra=, +r= 4)-=, - *r?, t : xpr - x"2e

+ % Ak [tr,+iJ= !iotk

o Jer g"/*d*h,^f,",(- B#.*/ar

,h-T '\ | /

Lt = ;tti

'--t '4 ^ ,.

. ' f f.At L:(l I l)-J 2

uzr -3 /

r-P ex AZ*

o J*U+qye: /e" L.- fl16sry,4 XGI 4Ler &Ff A

e,^./Li^& -ftu J;f*"{^a.,^na >6"" do--UarUA"^

dJ- ,L-CP^pPF* 'C

't' e,u.-ar h*f.q 4-e- E,hzte/e,,.*.-4

? &z C,--f;p^ SurzS ,^4 ,SoCs1

6"A br..- qru- 7zuvavznn e- Lu-

"kf6*. ).- $," -,'-d 3-,Uo$U,up-Jt- ty"^fu; {er Q'o,^-{"^ -

-att4a..-'/. E ,^*L*- &U & gc.,.-, Ct4---/ -okA,*,pJ- -i= &S.

I

- 2 1 - 1 ' a

I L, , Lj J=;t!r|*L,Jr-L;,^^fa^U _

A+-a-_

J!-3q

e)'/ r.^

( Gr-u-ff d^r- ]rc1 e.je-y,

ilf"u ?

)- = l.tu E

f

/ ,nV

x

-f-

t J)^

olol

" )

nth

n / )

11(2. fp d-z

nr^^Y v

! -Lzs

S/t1^k\1* ke.-/a*,{-- ft<(

'i - t"r=: 3l- (grt t->-e1

, J.., = !' J t'k "i)'k

d^r- -}re1 *J* -!, 3 -diea-tv'-&o-P"',^1

Q*ffo f .:Tlb-

o.Wo^a/&r4 3*3 -h$*e+

N = {. :: )

q^^> t-E = 6,'4r=7-'*-4 de,t'fr=7

c ?N

*>.pLy L'.ttr &u z-AeA+< ",&-

J*ZbhkeP T

Z-sin?

tu?

o

t";.ftA,^.A J*4 *y_ t ?"k/o\n", d,1,.

: ? + O(") , c*r?= 7- "0(?")S;,. f : ? + O(?t,

&,W* (4 - ex7 tr"q + ET. " ""Tl,n"^) de,^ CerveR*TO4ettr db- SOCa) :

f =c-/-:fS:;l ,7^=7-=fool -r='T--f" 'o)

/.L= 'l^- Lo t o ) .. /2: ,' L:rZ: ) ' /= ue-

L:"":).

Ji.*. b7"ao,. qL-,-a Le - ftLCeG(a a1#

[t;,T]= /roT

l it -e

Z+',,--d-^+-"L/- -*> ot- &I(' -^4 t-''"(-r'' !',-r* SUt^21I I

tn: *--t- -l

' / q )

TA-,- ./ ) J3 :

: 1lA*-)

C _ J-: J:* T:* J.-

Ic,J;]:e: [J-,

+ E, 1ez 2u-{cJ'^4'1,_/

frff

T- l\o^2

!= t;n-2: ' .-1,14 lA"-)

J=k lAr^2: et1rv) lArunLl>

o

a

-bK^ J'- T.-: rt" T: > o f7t

) -nn? > o

&--{r,a n)t +t^a.v) n^-ra-(ew ry14- t ( w'n^.qr< : J^ )

J; lA,h**:j>:e

EF J- q lL j>: [r'- ]='- +LJ+,f-l) I)i')

: (A-j"- j>lAi>= .> .,> A:1'i*t1

a

-f* : J; *i q

: + tt;J- + -r-qJ-? J:

J-]:2J. , LJ*,:rs]: +L

J.

fJ* ,

-u-+

3, }t- Ge,.p?e S UQ)

Jo-t/<//u^?e,4- fu- CR,.PPE S UCz; .> e-vzrz-

lnt *rrX^ . !{/n;,^-{u in<-dt-!66 .}*- {oA(ug

/*x )u" eouc,iw 2 o"n7 a.,'- T.-+,v{x-= (r,x-1 [i]l

rD- Gnr.RAlvReU , Xo= * Cq: t-2 2,2)

Lt - j4^ 6,^ge-.-

, c : (: :), %= ft:,)%:

Psu

[ : : )

tun- dn^

swLY:!:*l # + % =o , Sf q%:24u

-f\. I

/ te- ALC!"KAr:

n . , ' t

d/6. k\^>t<n"-. ( -

: - l - / l = t - =. L * Z 3 j

E-> ?'e-' t'< ?e!-u)"2^/-/*t

d,l,

Av'423

qu{%

I-31: - )

Teb-

s-273

t-= - l _abe_

- t " '

L 4

Otu, T

.F=-Qt v ' l7Z

fq % : i!oLc

%

{oo,or}- q%*u4%:2U

fr_-7

e2*----J <-r* 7<"aour- =\onfu-k---1ou &" SUP;v -

--+

-|T'"tl

/ -

i,n r f o' a)-3*-a,> -R*={;^c<-:

IJM> s:M I M -

C;r't %t J'Q,*,. I dt"t> | iul-r>lj"L")

1*i=n)2-

j= t

j=z

: LL) S;'1'l*tr

:

I^

f o 1 --D.--gte-'tf

Lz ' - ) 4=

E3l -/

*-^f&

r A*ar WL+ J ${

;i

[zi*t) -!4-(4 fb]t-

L&yra

LA

y-

L

_1-

[z]

tzJ

B)

@ TZJ

co f3f

e3)

usw

: lzl= r+7= 15)

?-(? -

tLf

LzJ

Ls) @ Al

@

@

T-e

n--{-.{cn#"'- +%a J- | A *} b'\

, -

J_ /A w^i^)=: o rg %^;n--J

n =. q d-a,, --..,,-7

h;/-L -&*-{c*'4 zn}F L

f t n \ ' '

j I l^); | 'v1 :1,J-7)"' ' j ) 4'\*':t/*aj

."ktJ'r-/<" &,^',{"/*-U,, ^ -{ li = Z, tu :

" 3 .',: ̂-Qr-d.1.-Lf -J*W'

1*.-a--"- *L*Sucz) i

. ?e*nr - Jr*-Lj'^d'- ft

irc<-aa^e-:Lk-

W=e' '",.r ns/-lcz-1

:

g: i'"'* i''' , J=: i!'* /:''

{^.Ae- &*^{-eL,J"'- { ttlttt} M;T

l-j"'-i"'I s J+Jn'*i"'

I: -kl , 8<^V7,''1 j"*!L

l'/-.,7-)

;i)

I J= a, H=a} ': l7-, '/-7

-Ar^w<-"*'-T 1'*1^ T- :

cf> la,o ) : rt [l'rr r-'L> l'/-'/'-)* l't'-'"-)12---Lt

lLr-L-> : l ' / t . ,-t/--> I ' / t

,* ' /->

n,i; ) J6*_d4.,\q^e A'{- ^ -X Ntt 6 7*^1

9<--'

I T: T** , j '/-= J**-1 > :

I J-r, LL=o) ": A I t'/-,-'L>lf-'/;>- l'/-'/')l'/-'-'-tL

a{< _

7-Y

+, bn- C<uPPe SC,'G)

->cp GiereRATTRe,u ; Xo= +

(o'a,2,..-,e)

lJ : wf [+""+J

d!*, Cett-kLArvN - h^ * 3tq :

:3ll , A':E:r,\=:G:1),

)'+= fnil ' )s:3,T!] , )'

)=- F,;:] ' )o: tG71),> ,L, - ALceR Pa I l,^ , )rl

= 2;

=Gll ,

-{.u. ̂.

( l to .7

) ,

+)u=o , Sp)o)9:2Tou

r> =-?rp"rut<K*J4,"-44 /*,.9 lG),-+.,bc =4SF f!,yJA-o 1l;"fu/ - ?<>r J.,av,,ob*da- %^{*{o^ ,

fi"= = 2 f'+?= " frnn : 2ffr> : 2 f=r,

=

= -z{,'r=-z{rn= : A fo-r =

rt to*:l

T-e

,! 6)-.,,'/';t"'4- &^^ k'!Laf

'" e'

Gel^rtcur,f&t@,8

D Ue) - {s-'.^pytt

f'= -7- :U

7 fede- ;n<dr.+-,t6/c

A*-sc-/,''<fffi_

Ezl

attt---4-

(a

""f&

L @tL

L s,-L

Es)- @--@----(E

ett)

"4< .

ED :4,^r^,*-" d.,^i f-,' $tN bzq, )&^

r\ rta Offi*,-*4-a-.-,o/1

-a- +a_

4* Adt-lrf - Ja'a {"qfi?-'ote

/ G-.Ta

rylt r

: @---G & @-@ : @-"-A

fzZ CzJ

: l=7 @ l1-f

@--^0- - o6*--P-- +

,, ,,

jg- - - -!&A--!-e: g l - - - -!

fz) C=)

r>)

- ?7- -'/- * 2- +>7,2-

GrT

,x* =T*sp-\" 4\ dv Qt^rLr-

-hTe&a'('"xrAry-

f- - - u

lr>= C) , L= l+'>= [t) ba tu> =-LruJ.'reopex,

' - '/ Q-u ;4.tr-k,5

. /

trt , Tt] :

trr , UtJ:

lTt, Vt] :

lT*,

[U*,

lV*,

LT*, V*J

lT*' V-)

lU*, V-J

fr-tt

= ?tt

+rlzur

+llzvt

r_J - ZTt

U -l :312Y -

V-): 3l2Y +

: IT*, U_J :

: -U-

: T-.

lY,Irl - o

lY,Uof:*Uu

LY, Vrf : xvx

Tt=2Ut

Tt : 'ZVt

lU*, V*] : 0

LT*, U*l : Y+

U'r,YJ-o-

SUG ) &utt,<t-A7TK

7{ -7o

o Skcs)- e"ufr:_'+Aouq -

z -

r 4oL-t-

,-gtlcsl - /6n,{"'--,r*" 1

+: , fc,LoJ=oQ : /

i ^o t<L*./ - u--x

v -

t#-692- AfR'*{-or-r. 'v l #

'7i: Li- I t\-'/ t

E

I

ot n't.

" U: t*l e^!r,4 - 'L, n 2 ,

U - f"xotoJ ,

I

' ,Le - rkgt b-e :t t s

[to,4 J: - f*. t.

d7- de'^ &'rr^'t /'*ka-'-{*' {t*' frt-/

. -4^* bo ̂,^ ' {^ft" - Pnfud'* ,

{+^,+r} : f &.

(dot.' 9qu,,'**#'1

eh

. e/*.21s-L: e-rz 2u"yv'u3e ( ?--^s)

dA'eZ- E,9e--u caa'y'.n 1*4-L/

. C*l< l+tR- a&

,7,u'-d T

3

za'= l-

Y- rt+* &tvpeRLAb tz,v Q

Tso sP/tu

" T4o- sa,). "I

"Ll - y;"' f;

SU(3) Kommutator-Relationen

J!-ls

ercs.tcrr- Gn-lvtv- A*"*{n'^ nv SUcs2//

eD Z*tdtd- c/-e"<k {A-,8 qn a-r'- -1

lr^l t,L, t!>

nl2 L.&L k;'(,-f

?4"At-tfJs- td^o&- t4(

71""^ Gc&"^*-f6,o {.re.lu,

I T ,3 ,,> : Z (Ltet't' ly r, ) lt'nt\/> lrrsJt't! y' )I f:r'Yrtt) 4+!

4> &ffe o'r,e-,e -%-q;^d<-- (.'-^'r Rna"Lt --bo," OUV,

lrrtr,-t;,Y> ir,r ffr';:Frr "") l[nh

,.fur s1 - 7*.e+*q

(".,^-^{,{ r.ot- 4 )

T+os,6^4r*z

J^-J. 6. L=1 , E!)

e*.

-n -1> -

!t\t rc-Hr,f *ta e R{k-h.e

9"-tr,=4 ..t ,

'/, t,

[=J,r/ '

* t / -

-%

+s +J

f.d-'.L\e.r^ dee"

-7i 7

tett u dan -e-. ilo'/9=

* -7-f b-/t

+3

cbh#

F^rc

o E apn r+<{{e A-\r0-

CaaarR- epqF*r'--\- l -

4\ su(3)

Jo--/ek,

Jyb{+ faJ

-4;fb# E 3 l

' ^+l-EPh{t r Zl

$r /"* t()

CL{* tr)

e : Et::.t^:

o

%%

4oz/3

3

E> ?et\4:t -

Q^"4- P'*"4- *^d

-Ele&-n("nrLry ou*.Z GG'nu -

2 +o (n)'+o (r)e

->+

Q,^r{- 4**?,*,4 -

fir*{*rJ,.

)out)oa) = 4i3,

CrtAq6)

-LZ

Con;,^--.- -W g** ; Ck):;7

f Cczl

tctt - {b)

.> Day*-ffe-

/-.

o o$ U (3) ?r-d.*J.{obJ>kl(,'--qn-' (,/-

(&-ftcu -k {(^/-- *4 *-trf> G<-l*rAA oLa4vqu^v1,a'* :

/r'^-d'r^^ dr'r--.K'g1 , f&^,*t"ry % )w ***---<d9^^ {zk-,*r:

L3l & t=l : f f_l @ t l)

-f u(z) )' - /

4-9^* Ult-ZrU

4\n-eJ- at-e-{"!<

@* .

c-'t=3 p t=7

@+

hJ

L=J

.r"t..

f tr.l

O*---4c>

r6)