• Keine Ergebnisse gefunden

Aspects of solitons in noncommutative field theories : the modified Ward model

N/A
N/A
Protected

Academic year: 2022

Aktie "Aspects of solitons in noncommutative field theories : the modified Ward model"

Copied!
124
0
0

Wird geladen.... (Jetzt Volltext ansehen)

Volltext

(1)

! " #$ %'&() *,+"-/.0

1

*2+43 )5765) 8 #/.9) 3 :)"

1

;=<>@?$ABDCEF/GIHKJL0JMONIBQP2ERJISTAU5ERJIVWFXGI>T?ZYS\[]^VWF?$AB

_

<IJJ$MOB`VaA?@b VWH9STAHU cIAVed->TVWfg>IVihABj]^VkJL0Jml5E>I>T<h/AB

fGIBQnBTHoE>TpGI>Tpq?$Ar]

_

B-E/?$Ar]

s<FTJT<B@?$ABDt5ERJIGIB^uvVa]w]wA>T]wxjSE/MJIA>

syBzjB`ABz>ERJ{z

p/A>`ASIUVapIJTAsyVa]w]wAB|JERJIVW<>

}o~€

ƒ‚k„9…‡†‰ˆŠI‹oŒŽ†o‘“’•”—–



Šj’•‘“’Ž˜™š’



›

’Žœ

~

˜™’



–^ ž ŸR†$¡¢

›

¢0‘ž¤£o¥€¥y‚

§¦

…i’



šœ0¢9˜

›

¨©–

09~¤}

’ª˜ƒ«w¬€¬®­

(2)

! "$#%&'$(

"*),+- )/.10'2435687 09

"*),+- )/.10'2435687 09

"*),+- );:<&'6>=68?209#@

A2B )<CED$ #>F/ AG$G$H

(3)

&9$7

6

AG$G$H F ) -0905&9$7 D$ )

AG$H

& 7 $&687> "7&'?

&9 &'097 0'# 35&9F& !&D$ &#" $ED$

%/0'&

A(*)$G,+EH

ED

( #2

-./.10324.656798/#-:<;58=>1?55A@'BA.60C:ED.

(4)
(5)
(6)
(7)

!"$#&%(')#!"*+ , -.,/&/102"3$54.,/6#&798:. ;<$3$$#6"*!),"=>*.*!"*-2?3$ 6A@

,*3B-!"CD#E*3B-.FHGH$>I/?"/?"3B-." J?7HC*,!E3BJ.JK*,/6#63$ L**.3$ 4M*.0$*I.C:.-3"$$#

=N,3$4(O 4(#PCD#E*3B-.!QR/6, 46.*)3BSATE3B** -.1,"=U GV-**.,"=+*,$3$*, O*,$03B, 3$ XW7BJ

.,3B.G,(*)C*,C:3B.*/?"3$ O08ZY[.-,"=U3$ L* *3$10JZ#Q

\ ]3B&.*3B&*2"^C:.-*&,"=5*,$03B, &*,_X.`D07"3B, 6,"=/6,3B, 6*,O , -.,/&/a02"3$

W7BJX.,3B."*<3$ D.3$4L"*.Jb3$ cJ2"3$dQeF)/?"3$ K=N,E-0,"=US" 7"$#E*3BG3$$f8:5, c<3$ L*4(2"8B

-*3$2"+,)/6,EJZ3gW7.Jh0 3$2"#O3$4(/?P/6,EJUG3$

U (n)

@d"$0.JOW7BJ153$ L*,EJZ0-..Ji8D#cjk"*Jl" Jh3$*

, -.,/&/102"3$^ATZ* 3B, MG*F"8,mF/6 L3B, .J& Gn*,$03B, +"3B2(QU^=C7"3B-0"93$ L**.

3$ i3BS-., L*ATZ1"*M*,K0)&`D0.3B, l,"=I2"83$$3$o#i,"=I2"3B-&*,$3$*, a" Jh6"CC$3B-!"83$$3$p#l,"=H

*,"@p-!"$B.J;JZ3"87"3B-<"CC*,-*K*,aI</6!" H*,a"CC*,.TZ3$/?"*3$/6A@pJC J D*,$03B, +8D#&4.,DJ.3B-

/6,3B, b3$ b)/6,EJZ0$3fC7-.S,"=92"3B-S*,$03B, mQ

R=N*1*,/663$ L*,EJZ0-*,#l*/?"qEa, k , -.,/&/a02"3$D3$o#r" Ji3$*1*,B63$ lCD#E3B-.1GH?84(3$ l3$

?*.-., Jl-27"C*a8D#i*D3BG3$ 4O6873B-63$ 4(*.JZ3B L*M,"= , -.,/&/102"3$P4.,/6#s) ..J.J

=N,&,06" 7"$#E*3B!Qlt<.E3$ 4l*aX*2"4(u9GHK ]C*,E-...J]*,OC*.2 L&Pjk"*J]/6,DJF*,4

G3$l3$*) , -.,/&/102"3$PATZ* 3B, k" Ji4(3$?P0 3gW7.JlATZC:,(3$3B, l,"=I3$*SqD ,mG k2"3B-&*,$03B, !Q

eF3B+/6,EJduD+C3$/6^AT"/&CB,"=" ;"$/6,(vf,* Dw@d3$ L("3" LFW7BJ&.,#M3$

1+2

JZ3$/6 3B, !u 7*2">E3$0.G3B-*b/?"q<3$*" 7"$#E3BFGI,LG3$B(QIxy3$*,"=U"$>3$3B3$ D*4(2"8B)L05"$B,!G3$ 4

=N,SC:,!GI[=z0du{GH$IJB,C.J>u{*.-* 3B`D0.a*,K4 2"*?*,$3$*, k2,$03B, !Q?RFS?2"/6&3$/6&.*

=N!"0*<3$ D*2-3B, O"/6, 4&/bQ+x0/6,*(u<-.,/&/a02"3$1-.,0 L*C7","={5jk"*JK/6,DJ

78. c3$ L.3$4L"*.Jc3$ X4(*!"J2"3$0-2b7"/?" L#K*.0$*5"*S.("3$"8B)=|,-.,/&C7"3B*, fQ

}

ATZ!uZS`D0.3B, b,"=92"83$$3$p#P=|,<C*.* L*.JO2"3B-)*,$03B, 3B-., *3BJ*.J>QeF3B2"83$$3$o#

3Ba4,m .J]8L#lK`D07JZ2"3B-;=N,/~,"=^;0-07"3B, !u+G3B-2fu90C:, ]-., -. D2"3$ 4l, ];-!*

,"=JZ3"4, 7"

U (1)

*,$03B, mu3BaATEC$3B-3$$#i-.,/&C0*.J>QPjl?,mG€7"1?-., 3BJ*.Js2,$03B, a"*

2"8B)G3$3$ O&-.2"3$ O08*.-*,^,"=UC,(23$8B)-., ZW4(02"3B, !u 7"/6$#K)4(22/?" 3" c, .!u7" J

8.-.,/6M0 2"8BM0C:, l/18:.JJZ3$ 4b/3$ D*,X=z0$U0 3$2"#h3$4(/?P/6,EJdQaxy3$ 7"$$#ufGIM*/?"q

, c*,/6<C:,(*3$8B<4 2"$3$wm"3B, h,"=y.*<*.*0$*!Q

eF3B^08ZY.-3BF=|,$B,!GI.J>u‚=N*83BA=9*D3BGƒ,"=y3$/6A@pJC: J L5jk"*JX/6,EJ*,$3$*, !uZ8D#P

"CC$3B-!"3B, c,"=SJZ3"87"3B-S"CC*,-*fuIC*,C,(*.JK8D#;„h" L*, fuZ*,)2"3B-<*,$03B, !Q0^" 7‚@

$#E3B*,!G7"3B^"CC*,!TE3$/?"3B, O#D3BBJ)2"/6<*.0$*F=|,G3BJ-*,"={ , -.,/&/a02"3$

W7BJa.,3B.+{I*/6y3$ F-3B, &-., 3$03$ 4)IJZ3g…:* -.I8:pGH. &.27!F ,A…>.-y,

/6,DJZ0$3*C7-.JZ#D 7"/&3B-.mQ9†D3$ -.^AT-93$/6A@pJC: J LI*,$03B, +*,<jk"*J6/6,DJ7"*FqE ,!G fu

GH&"* i"8B*,K-.,/&C7"**.0$*),"=a/6,DJZ0$39C7-.&"CC*,.TZ3$/?"3B, k/&CB,m#.JhG3$_

AT-F3$/6587!D3B,F3$ K

U (1)

-!*(uZ,!G3$ 46a87JP/?"*-*3$ 4?,"={.*^oGH,Qjl5AT/&C$3g=N#6.*

JZ3B*-*C7" -3B.^G3$b2,/6S3$/&CB)/a0$3g@p*,$3$*, OAT"/&CB." JX4(3$<C:,(*3$8B<*!2, =|,/bQ

‡ 6,04(6GI8:$3BF7"+FGH,qMC*.* D*.J&3$ &3B9.3B+*.J+*,/6F$3$4(LH, &F3B*0,"=

, -.,/&/102"3$M*,$3$*, !u7*)*/?"3$ 5&,(^,"=90 *,$.Jb`D0.3B, G,(2S3$ L.*3$4L"3B, _*./6

*GI"*JZ3$ 4Q9jl^#*,10/&/?"3$w!(uD3$ ?,0,C3$ 3B, fuE/6,(+04 DH, .F"+ J;,"=:^-*7"C**

, b)2"83$$3$p#b" 7"$#Z3B^" Jb, cSJZ3"87"3B-a"CC*,-*fQ

ˆ #DGI,*J!‰ } , -.,/&/102"3$&†E,$3$*, !u†D2"83$$3$o#KR 7"$#Z3B!uRJZ3"87"3B-1RCC*,.TZ3$/?"3B,

(8)
(9)

\ kJ kBw.* k0 4A="%('L"* s7"a3B-2kJZ3B& 3B-2Lq,/&/102"3$<.,/63B6w.0k3$ /~"qD3$

x,**-*D0 4483BK*,!GI,J6„h"/?"3$qn"B;"0-* J+L#Z3$qr DpG3B-2q$;0 Jn3$ *8.*, J*K3$

J1t,E-* 4(3BCD#E*3$qlw.0l3$ L**.22" L* r 0 .0$2"* l4A=!Q;t3Bw.0l413$ *8.*, J*

JZ3B ‡ TZ3B* w_ 0 ˆ 2* , =|BJZ.,*3B*-* v(*0 4 fu^J* ‡ 3$4 *-*7‚=N* <4 2" J

"qD0$B^x7,**-*D0 4P3$ J>Q

3BK,$3B4 JcR83$M8:A=|**M*3B-*n/&3$&**-23B.J R5CqE* nJZ3B.*6v(0 4 fQ]t3B83

* sJa, kjk"*Jr3$ 4A=*;3$ L*4(2"8BK-23$2"B;,EJ1/6,EJZ3gWw.3B*K†D3$4(/?O„_,DJ$H/&3$

U (n)

@

GH3$4 sx7BJ fuy2,!G3B6J.*2 i 3B-*Lq,/&/a02"3$

‡

GH3$*0 4uU3$/ D0/ 0 2*.

\

D**.**.!Q

\ 8:.*, J*G3BJZ/6 aG3${0 J Mx2"4 1 7-2J9†D2"83$$3$!"UJy2"3B*-2 av(0 4 0 J1JUR Z@

GH JZ87"q3$<J.^*,M4 7" D* hJZ3"87"3B*-2 "04L" 4.5"0Z=+JZ3B.*(Q+vfw.**)"CC*,.TZ3$/&3B^w!3$2"8Z@

#" 4(3$41v(0 4 OJZ0*-*O3$ S4.,DJ"3B*-2%$FGH4(0 4;3$/ „O,EJZ0$2"0/ J^*2"3B*-* cv&(0 4 fQ

}

-2l3$ 3$4 i3$ Z=* J ljl,* hw.0

}

3B-*Dq,/&/102"3$E3$!"10 J_3$*%^,$Ba3$ iJ'+L#Z3$qu

C!* L3B* &G3$U3$/w.GH3$* ˆ "C3$*JZ3BI ,pGH JZ3$4 (50 JZ"4 &Jy 3B-2Lq,/&/a02"3$ <.,/6A@

3B(uL0/ " *-*$3B*) JKJHjk"*JZ/6,DJ$f"BH 7"w.0;B,* Dw.3$ L("3" L*.„_,EJ$73$

1+2

3$/6 3B,

3$ w.0Z=* fQ

3Bc*2"3B*-* v(0 4 J?w.043$4 +$IGI4(0 44(B3B-*D0 4 GH*J " JZ3B.*

†D*$B63$ 4 JkJZ3BqE03B!Q <jk"*JZ/6,DJ$H8.*3$w.S3$ 3$4&8/6q *GH* ‡ 3$4 *-*7‚=N* fu{JZ3B

3$ K4(, JZ$3B-*.- L***0-*L0 4r3$ L**.22" L?*2-*3$ 2* fQ/0 JZ3B.* n4?J.** \ L*4(2‚@

83$$3$!"!uJZ3B10 ^A…>qD3$ae.-* 3$q hw.0

ˆ

, 0qE3B, h, O*,$3$*, 3B*-2 _v&(0 4 cw.010+[=4(0 4

*$$!uJZ0*-*;JZ3BF3$/ ,$3B4 J ;x7"$3$ 8:.*, J*^"0-2&GH.-2*$G3$q J<†E,$3$*, ?4 3B+GH[@

J fQ ",8a3$ 7"01"08: kJZ3B6wm"$*3B-2 s3$/ q,/&/102"3$ ]x7"$Hw.3B$* 2^.0$2"*63$

JZ3$*qD* 30+4(B3B-2c/&3$^J

‡

48 3B** c3$ 1-^ D**0-*D0 4;3$/ 3B-2Lq,/&/a02"3$ lx7"$dQ

\ / R^ 2-*$0*PJZ3BqE03B* G3$KJZ3B_†D2"83$$3$!"KJ?C!* D3B* *2"3B*-* v(0 4 fQ

3B.*

G3$*J JZ0*-* JZ3BX`L07JZ2"3B*-2cx,/ =6JZ3BO†D40 4 8.*3$/&/1!uFJZ3BX3B-* 835$F2-*D0 4s,

JZ3"4, 7"B

U (1)

v(0 4 KATEC$3$w.3$H" 48: ?62!Qyj 3$+w!3$4 fuEJ*HJZ3B0 L***0-*L* Xv(*0 4 ?3$

3$ / 8:.3$/&/1* (- L*2"0/ "$B/74($3B-* ˆ , ZW4(02"3B, b2"83$3$ J>Q 3B.*8-^ D*2"0/ 3B*

JZ0*-*sJZ3B<2*/?" *-2 ˆ , ZW4(02"3B, r448: l0 JiJ* ‡ 3$ D80 4_3$ lJ),$B!" JZ3$4

0 3$!"*†D3$4(/?)„_,DJ$Z=9w.063$ \ 2"83$$3$!"J+ L**C*.-* J Kv(0 4 fQ†E-*$3B*)($3B-2;w!3$4

G3$^3$ 3$4)/74($3B-* ‡ GH3$*0 4 hJZ3B.*.0$2"*1"0Z=Q

‡ 3$ qE0w! "**$$0 4aJ8:q" D* 6w!3$2"8#" 4(3$4 Xv(0 4 6=|,$4(J" ?JZ3B^R^ DGH JZ0 4

J, n„h" L*, n3$ 4A=* JZ3"87"3B*-* } "0 4l"0Z=5J,$3B4 JX„_,DJ$dQ 3B.*(- Z@

**0-*D0 4c6*)3B-2i3$ i4(B3B-*Sjl3B26"0Z=F3$ MGI3$2"0S4(49)*

ˆ

**M, i 3B-2Lq,/&/a02"3$

xBJZ.,3B 8:2"4 fQ 5w!3$2"8#" 4(3$4v(*0 4 ?J.9jk"*JZ/6,EJ$B+8:q" D3$ J>u(q: 6G3$

JZ3B ‡ 48 3B*2HJyJZ0*-*4A=* M„_,EJZ0$2"0/?"CC*,.TZ3$/?"3B, 6/&3${J AT"qE* .0$2"* 14(B3g@

-* X0 J;W J X3$ 52-*B.-*D*%;8*3$ 3$/&/a0 4&3$/

U (1)

x7"$dQ 3B.*.FG3$*JX" P3$ 3$4 b„_**,$3g@

*, <(0 4 c2" *-27"0$3B-*D!u0 JbG3$48 X/74($3B-2 ‡ qE6"0 4 b=^JZ3B.* 3BqE*C7" wQ

8LGI,JZ3Bh,$3B4 JsR^8:3$b3$ 3$4lR5CqE*_ 3B-*Dq,/&/102"3$k†E,$3$*, VqE6"* q" fu

8B3$8 ]E3BB?3$ D**.*2" L*Xx2"4 r0 D8!" DpGI,*!Q ‡ 3$ 302"/&/6 *$$0 4lJ!uU3$ r0 ** nR0Z@

4 fu{G3B-2L3$4* =90 qE*?W J3B-2r"/

‡

JPJ

ˆ

"C3$*Iw.0M†D2"83$$3$!"*0 D**0-*D0 4k0 Jlw.0

JZ3"87"3B*-* } "0 4Q

†E-*"4(GH,*(‰ } 3B-*Lq,/&/a02"3$?†E,$3$*, fu†D2"83$$3$!"*0 D**0-*D0 4u>RJZ3"87"3B*-* } "0 4

(10)
(11)

!"#$&%'"!)(*%'+,-,. /10 2

A )+

+-$#@2&'7 $&',2 6 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

A )A $,82 !$#@2&'! 8 &9$! 6$#%# 82&9D & ) ) ) ) ) ) B

3 45678:9;7%&+,-,./< 8>=@?A8:"< :B

) )+

35C &9#%&',0 &9$#@ #@ 0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

+EH

) )A D &FE, $0'&92 &'#%#% 0 6$#%# 82&9D>1G &' ) ) ) ) ) ) +H

) ))

0'2&9$ %6$#%# 82&9D 60'IE,09 7 $ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

A

J KLMKN9L/"O<PQ/8M%&+,-,./< 8MKNLR,S./8:9 3

T )+ ' #@& 6E,$2&9$! F ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )A

T )A 9$09&'$, ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )$H

T ))

0''&UE62&'$ $09&9$ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

)$B

T )T !&9$ 7 $ 2 $ 9 $0' &9$ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) T A

B KV$9LW0YXZ90/"<L"![\QL]Q/8^%&+,-,./< 8Z456/8_9 B

` )+ 0(#@?$ 7ba6 ,2&9$ ,0<& ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ` +

` )A F2&9$!

U (1)

F,268? $ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `B

` )) F2&9$!

U (2)

F,268? $ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) +

` )T #@#@4 $>a,6,2&' 0& ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

$A

c de$,S8+fhgZ8:ij8:8:k4568:9lKA9$" 2)2

H )+ 35&9 4 # &9# 0' $09&'$, ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

H )A &9#% / #>09&<C$09&9$ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) B,+

2 XkVL>Xmii_?/L,SnKA9Logm0//,-Lp" q)B

)+ D 2 $Nr!&'F,2&'6 '687 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) B$H

)A 0'&$ #% 09& ,6 F 0'&' &9$#@%#@ 0 $09&9$ ) ) ) ) ) ) ) ) ) ) ) ) B$H

)) D 09& 616s 6$&'! tC6C$09&$ 62&9 ) ) ) ) ) ) ) ) ) ) ) ) ) H$G

)T $# ,2&$>C&97 1G6 $09&'$,2 6 $6 0'&' ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) H)

X %',S8u6L+9Ive/8 w2

)+ :"&9$D209 $ 7!! &' ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) H

)A 62&'0' $ tC 6C$09&9$ 6 &9 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) H

(12)

/V9$L+" $L

X\/_[\98_/R,S8:" ?/

vO6$p9L, &8 ?$L

(13)

7 F_s 6#@2 2 7& 7 & & IC 090D$09/> F&! &9$5&'6<&9 7 &'60

7&'6

(

,#% 0 60''&'60E09 7 $ (

C7&9687 7,'F C MC&97 C 2 &9 &'

26 7%&96$ 2&'$ $6 $#%#>2&D &9 &9 7%06 tC 6)@ 7&

C 6/6 7,6 F$7 F/$<#@ 0< ED0;0 @09% 1 #>F/</&0,

D09#% &'@7& bC7&9687 6$# ,6 C&9% 6 # '09&'62&9$ &'@#@27#@2&96

6 C 090 ' 7&96 )

eC&90'0 ,&'6 0'-1G2# 0' <$6#@# 8 &D$OE, 0' 7 $ ( 7 6 60'09

#% 0 ( $&9D 6$6 & D &'WC 2 &9>? C 2 F $&'D$&' &9 $#%

C 0 6$6.C&7 7 82F&'09& &9 $09&'$, 7%'G &9#@2&9$ "&9#%

/ %$0'&&9 7 #%C $? 7 2&2F,2&96 '687 )

nU AB\p/A>TxA <M2>T<>Tx<UUG-JE0J$VihVkJ\[

7'1F /0 '10< 6

+H T #"<+EG$H (+EG(+ AH%$

72 7 6 %9 ,6%&0

7$09 F/@#@ &FE, 21D$ #@090<&826>F 7@#% $ 6 $6 $#%#>2&D &9

7% ,2& 0;6 $&9,2)9 , '0 ( 7-&' ,2&20 6$#%# 82&9D & &D$ $09

2%7,'"F/%6 $#%#>&96 @F & F/ &9 09 &' 0 ( $ &917,2 $F0' #

2"&'_E,& 03 &' #@ F/ #% 0'&9$ 7&*C $)% 6 6$&9 6 6$ - =$68? &FC

"Hp`&$ &' 0 &9 6 D 7& &9 09&( 09& .$ 7 &'#% 2 7 0'2

D$ ,090 lC7kC17 $ #@ &9$,'/ )

7&"&9 ;$6#@# 8 &D & & &9# 0'&9

(

D4 &9#%&'0'4 76&'60' ,2 & 2&9$

7,' 6 ( F 7 &9&' 7 6 $&',

x i

$ ,6 &9#% 7 #%&&' /2

ˆ

x i

( 2& &' & F 20'$ F

x ˆ i , x ˆ j

= i θ ij ,

6 0 D209 & #%#@&'6 #@2&G

θ = (θ ij )

&9#@ &9$ 097 ( , ) C!$6 $#%#>2&D &9 7 *)( 6 &9D0< &96 #%&'>$ 6&9#@ 7-2&9$

/$&9 !F/6 $#% $&909 ),+

C! D$ 709 7& 09&' 7$7 7,' F/ D$ 7 &'' &96090 F 7 &<

6 & &9 7 $090'C&9$ & C 6 &9047 0 BG rE*C&97 7 &9$ &9>C$?

2 0&9 $ "+ ) (+ T (+EH&$ 7,217 &' $6 $#%#>2&D$ ,26 6 &9D 8F0'

-/.10%2435%6%78243:9;7<9;=4=>6?3@24A*7CBDAEBF0%GIHJA*KFLMA;NPO*A*7<Q:G/6%RM9;7&7TS%HU0%AEBFKD24GWVCBDAXV%GZY&7%G[BF0%GW3DG:\D]^6?9;7_BD6%Ra`b3@5&9^cZG(312>7

9aKF24d*A*KFA*6?3eH19gfh9;7?VCV6&i%i?G(Vj0%243:3kBF6&V24GW3e\@5?A^2>7_BF=>G(3D3Jd*GWA*REGZBFKDf^`mln3@GWGMo>p^p(qWr!sut

(14)

&'6'&' 2&9$@F 7 &'6 &

(

&9 ) &'6 !7

( 7 $!^) $ <7,D$F &9D$ &'

7 , E,09 7$&' 6$#%# 82&9D 6) !&&9# &9F09 @09& 0905 0'D2

6-&9 7&D2' !6$1G -2&C Is !$&D$ #@ 1G2# 0')

. 12 7120< &962$6#@# 8 &D$bE,09 7 $&9 & 6 D &'090 7 $ 6$

&9 $ >$6#@# 8 &D$1$

"<+`%$ (

C7&9687 6 F/-&D$ $# &9 7 $

"<+EB?$

)

6 ( &9 7 $ ED &' 7 D &96 $ $6 $#%#>2&D &9 &9 ,26 &'#%2) &'F/

& ( WG &910'&9&'' &9>7&&96&9$ ( C F09 %ED$ "<+EGp`$ 7,2 / *

/ &917 $ &'% 6 $ 8

B

E,09 F,68? $$%6 F/$#>0'2 &' # "2/-

E,09 7 $ $6 $#%#>2&D$,6 &'#% ? &'76 60'09 &'F/6C &

09/ 09&'#%&) &' F 7&-2,0< & ( #@ F^ C $?&'D$&' 7 02&'

F tC &' 7 $ $6 $#%#>2&D &9 ) %@$%7, %$6 $#%#>2&D$&E,09

7 $&9 2 &97 & # 710< &' &' 7 @7 090'C&9 ,

F0' # 2&&9 &' &9 7 $ &9$ &' 6$#%# 82&9DkE09 7&9 ) 7&

C $6 $#%#>2&D$ E, 0' 7$4 7'

(

$1G2# 0'

( F # 0' $ ,@8687$ 6$

,2&'$

"A(;A +6( + A+`$

27 & 0'2 %+ F2 $ D &'WC 1))

"`'),(

T

H?$

)

&' 7>F/ED$ #@ &9$ &'6 &' ( &''-2 $6$#%#> 82&9DOE,09 7$&'-7,D

#@ 7 ' 217 $&9607 &'6

(

bC7&'6877C $0< C #%&9$ 7

2# >6 $6>6$#%# 82&9D 7 $ 5D &9 2))

"B),( + A(` ( ++( H,( +'+ )&$

)

D $ WG 7,,&D$ D &9C 1$6#@# 8 &D & 2 & 609&'62&9$ &9 7&96 6 ( $

WG# 09

( F &'

"+ (;H+H( ++EH (;B

T ( T ` (H),(;AH ()`'),( ++EA%$

)

7 &' 6 &'$ 2,7 &96 /$6 $#%#>2&D$$ $#% &' E, 0' 7 >7 " )

6 &D$ C 1! 0'$<$ 6 &92 & F 0'&9 D ED &'

&9F09 $ C 2 7109&'#@21$0( & &9 ,# E,09 7 $ mC&7 $D & $)

<HV JT<>T] Ve>D>T<>Tx<UUGJERJIVkh/A MOVaAH?QJISTA<Bo[

E>T?QJ$S`A U <? VWMOVaAr?QbEB`? U </?$AH

7 , &960' 6/6 $6 $#%#>2&D$ E, 0' 7 @7,27&7 & & D & 7

, $6 $#%#>2&D$!$09&$! 7&9/ &9 )

$&' ME09 7$4

(

0'&$&'6 $09&'$,

(

$ $09&$1$ 7$ (

%,090 E,

"+G)?$ '982F0' E,& ' $09&9$ 7!_ , &9$2 #%&9$ $#%-#%&'?C? &' 6

&9$ 6 &',0) ! ( 7 # 8F0' @7 $#%C72D2 F!66&'0

7, 0'&$, #%$ #@&90< 09 # '&9F09 6$0909&&'$ 2 #@ 7 F/ &'4

6-WG &'6 0') 7$$$7 $0'& $09&'$, 7&9$ &'

& #%&D22 F 7 &' F&! &9$'/6!&9\E, 0'@7 $&9 C7&'687 7 # &'4

&' $F_s 6 $-#@27#@2&'6 & ,1 7&'6 & 09&'?2$ D &9C >))

"<+EG)((BH,( +'+

T

$

)

%,2$# 7&9 226 &D$ ' WG6 $09&'$, E,09 7$&' F%7 # 0D$ ( 0'&$,

0 0 @6 6 &20$0' &9 7&96 16 &D &0 60'' &960 $09&'$, $>C7&9687 7 ,27

&'$05#@ F/1WG ,2()

D $0< $#@2&9$ 7% 0< &9 ,6 09, WC &9 $ 09& 2&9$

7 &'' !E09 7$&'$ $6 $#%#>2&D$!,6 ) 687'E, 0' 7$&'?)( $0

(15)

#% 7 #@2&9$4C 0'0 ? C $09&9$ $09&'$,"F &90< C / "2;$6 $#%#>

2&D$$0'& ( )) "T A (p`')&$ ) 7& >&<#% $#%&9 C7%7!6 $#%#>2&D$

09&9#@& &909% @7 $ ( F 6, 7 0'&$ 6$E$ &9$ 27 $6 F 6$#%

&'$0'>&9 7& 09&'#%&) #%&9$'s >1G# 0'

( C 7

U (1)

&9$#@ #% 0

&9 tC &9#%&') /$ 06 &' 7 $&' 6 F 4C 6\C 'E

&'687 60'' @2%$0'& $09&9$ ) &' # 0 &9#%&' ( 7 %$0<

82&'6 $0' &9$ 4(#@&96 &9 1G &'$#@ #% 0

"+ A6) ( + AH%$

$ 7&

1G ,&9$% "6)?$ $> &6 &9$ $09&9$ F>0 7 6 D &'0904!&'>

^ ,2&9$-$ $6$#%#> 82&9D

1,2

) 26 ( 7>090 6$ 7 6 D &90'04 &9$

4 # &

(

&9 , &960' 2$

(

$&D$ F 7 $0' &9$ 7 &'$#@1 #% 04^ 2&9$1

#%&' ) #@? 6$6 !$D &'$,-&6 &'$ ( C 7,2!#>0& $09&9$-&9 $

6 $#%#>2&D$ 6 0'&9 tC 6 &'#%,&9$,0 &9$#@%#@ 0 ! 24&9 ' 82&'6 + G F2

&9&' + A F2 IC&7 @6 $

B

E, 0' F,68? $$ "E$A&$

)

7 6&9F/ $6$#%#> 82&9D 1G #% 0

(

0 ? C ' 2 #% 0

(

C&9090 F/7#@&9 &96(7& 7 & ) : D 7$$7#2' 6 (7&#% 0 092,

C 090 "E$A( ') ( T (H,( + A (`H( +` ( H&$ ( 7 & 6ED &' 7&7 &6 82&'0<

D #@ 605WG#%&9,2&9$ ) GJIH9VW>TA

7 $ 09&' 7& 7 &!&!' $090'C

82&' ( C!$&D$ F&'56 6$ $%7!F,'&'6 2 $6$#%#> 82&9D-$ $#% 4

&9 7 6$ 687,6

(

C7mC %F/7 7 4C 6 'eC 0'0 6>7 2$

#>02&' E, 0' 7 $&9 $ 7 D 0 $#% 6# 0'1G 02)% 7& WG '&&9$ &9

6 0' -7>&9 6 &9$ 7 D 20< 0 #@' '!#%&'2$-F/ tC 7 tC &96 )

, 7 #%$

(

C EG 6 D&9$ $ 7@67 82 7C 7C ,0'2&9$

8 &9$ 09& $ 7 # '&82 /2)

7,6 7@&9609 $#%@$20 #@? $ 09C &9#%&'$0&'# #% 0>

7C17C +&'68?)r17 $ # &96 7 1G & 6%<$&D &'0 $09&9$- 7 2)

7 ED &9 WE,&&9$ 7 &' -&'$F&90'& &' 6 7 #% 0

6> WG# 09 2 &'$F09kE, 0' 7 $ &9

1+2

&'#% &9$ ) F,'&'6 ' &9 >

&6 F/ $ 7&9&' 7 6,- 7>$6#@# 8 &D$%0'# ( C7OC> 6&9F/ 7

#% &FE, #% 0

( F 27 &9 7 8 6 >'eC 090 '>&9 7 /2 &'6)

6 $6 0'@7& 266 F 7C&' 7C 7&1#% 0"6 F/ 02 60' E,09 7 $&9

D &'%7 &'# 09#% 2&9$ 7 &9$#@1 #% 056 $ 2&''!%3 2$$1#>0&09&' )

7 82&'6 $0' &9$" 7 $6 $#%#>2&D$ #% 0,2 &6 &9%687,'"$ )

! &9 6 &'-7F, ,$'#2&' 0'&' 7&9 9 F/$

( C D &9

6 $# 09 - 6&&'$ 2 F/09& $09&9$ ) &&9$@ &9%$F 0'&' $09&9$

(

'

C 090( C -20 #>F/@7!&6,&9$&' 7#%C $? 2(&9$@7 %$&D$!WG# 09

24 * 9$09&'$,)

8&909 ,0<&@ 7F&90'& 72&96$09&'$,@&6 & 7 62&9

&' 7^E7 687' ) & F,' $ 7.C?

"A T $

$$ 7ZC&7 >)+-#@&'

(16)

. ) 35 687 0'() ! &9 7 ! &'

( &)),7 ,2&96 $# 7ba,6, &9$

(

C &'& &9-&9 D2&2 F,26 ( 7 090'C&9 $!@6 $# &9&' 7ba,6 ,2&'$,

&' &)( 6 ) #%$ 7,2 0'&$, 8F0'&C&97&' 7 &'/6 &9D

'#@&' F@F 6$#% 8F0' /$ # F &9 7# &9 7 0'0 &9 &9$#@

#% 0 ) ,$@7 6' 2 &$,0

U(1)

9 $09&9$ ( 7 /6# 7 & &

WG 09&'6 &90 6$# ( ,0< &96090 ' C 0'0' #@ &96090< ( 7#@ 09& 7 0'&$,

1&9 &UE 2#@ &9 @#@ ) ,7#@

(

C 7 a,6 2&9$ F0' # &9

7 $,2F 0'&'>6' $ 7 WG# 09 7 $6 $#%#>2&D$

U (2)

#@ 0 ) #%#@& &9 7

687&9 D 02 7! 2(7-687,' ( 0'&50'2 $0D$ &'$, &9 D &9()

,$0 26# &$>C&7 7 E,&'&9 7 E,0 687,' C D &9C 09$ &'#%

/ 0'&$,&9687,6 &G ) 9&' 7!6$/$&9>09&99 #

C$&D$ $#% &'# 09 1G# 0'> $09&9$ 0'# #% D &9'C&7 _ ,0 6$,82 D$ 0'6&

&' 7 D $002) :<# 09 &' 7 &' 2 && &9F09 $ 2 #%$

6$# 09&962 # 0& 0'&$, C&7 &&'6 D$ 0' 6&&9 (

$#% C7&'687 WG 7&9F&62 &9)9

$,20< ( 7 F/09&2 0'&'2#@ 7 )" 1G # 09&9 @7& ( C -&0 %7

F274 0'&9? F/7,D &' 7 2&&9

U (1)

tC 6 $0'& )

687,' D!C D$2 $ 09D 4!7 '09&'62&9$ 2 D $Nr &'F, &96',2687

7 $6 $#%#>2&D$ #% 0) 7@ &' 09 7ED F F09&7 $$ 7

C&97 D )/10'C . );35687 09 &9 "H%$ ) 6 &'$ )+ $&D$-%$ 0 D D &9C 7

0< &' &9) 6 F ,20 &9 7@#@ 09&9,6 F 0'&' &9$#@ #% 0

r

$09&$

(

&'& &'@&' #% 7 "70' #@&$0'

(

,#%0< 7

r

74 #@#%& /C

-7 6$# 09WG 0'2) "70' #% &96 &%6$# ]C &'D &'$2 7 09&'#%&@

02$% #@090 #>,0 & 86 ) 7&1,F0' , 6&9 #% 09&<C,6 6s 6$&'

$ 7 62&' tC#>0& $0'& 09# &909&9 %&D$ 20

90

62&' $0'1$

72 $ 6$090'&&9$ ( &' 6 $' 7 WG6$$0' &9$ )% -$ C /&'F09 '$

7&!#@&#@2687 6 0'' 7 687,' IC&7 $#% 6$#%#%$ 716 $ &

6 09&'

$ 7 2&2F,2&96 '687 &' 7 $,F/09&2 &9)

<>-hA>JIVW<>T]

7$7$ !7 F,_ 1687,' lC # 09A 71$090'C&96 $D$ &'

?@0'

µ, ν, λ, . . . ∈ { 0, 1, 2 }

7 6 $&9,2

(x µ ) := (t, x, y)

&'7

(1+2)

&'#%,&9$,0

D

&9?C? & ,6

1,2

( C7-7-#% &96-&8?2 6

η = (η µν ) = diag( − 1, +1, +1)

7<28090 2 &4 #@#%&9635 D & &D &8!,$7,' 7/

ε 012 = 1

) 1,20( C %7 3 2&' 09

i, j, . . . ∈ { 1, 2 }

$ 7 &96 &' 7 , &'0

6 $&', )- - 7, -7 > 0 , $27 / ( 2))/ #% 2>7

#% 0'&5&9 687'

)

,$7 &9 6&9$%/7 #% &UE@$09&' &'$#@1#% 0

(

%WG &9$ 2/7 6

&9#%

1+3

&9#@ &9$"&" ( C7 76 $&9,2 "2 F

(x a ) := (t, x, y, ρ)

C&97

a, b, c, . . . ∈ { 0, 1, 2, 3 }

7820'0< 2 &4 #@#%&96 $4& E, kC&7

ε 0123 = 1

)

&'6 @C 8?2-6 &' &) 6$WG ( &9 & 6 $D$&' $ 8 &9$ 7

# 0'

tr

$76 D< $$ ,6 C7&'09

Tr H

76 ED$ 47!&'09F/

(17)

,6

H

7>FF D & &9$

Tr := tr Tr H = Tr H tr

& $!7>6 $#>F&', &9$ "F/7 7 ' 2&'$,)

, 7 #%$

(

&' 20'&9C&7 $6 $#%#>2&D$ E,09 7 $ ,090 6$

$#% 282&'$0!#>F&9$&&9 &96 7 #*6$#%# 82&9D & 0 &'&'62

E09 ? &' D20' &' $#%!$,F/09&' $$ ,26)< #@?2-WE,&&9$ & &96 C!

70'2 &2&9$ '*$,F/09& 2%66 2 D 72&9$*$6$#%#> 82&9D

$090< $ $6 $#%#>2&D$ 6 6 )

(18)
(19)

# ?

$ ,0< &9YE,09 7$&' $ $6$#%#> 82&9D ,6 mC D &9C 7 &'0

F'&96 2"$6 $#%#>2&D$>$$#% &9 7&!687,' ) 7 WG /&&9$MC&90'0 F &96

#@2&0 $ 7!F_ 2,0< & ) D &'09 &6 &9$17&*C&9

&9 0'2&'$ kE, 0' 7$4 6

(

$ WG# 09

(

F/ &' 7 D &9C

"<++EH(/AH (`6) ( ++ A$

)

z syAM<BTU5E0J$Va<>T] <M JISTA <B`? VW>EB€[B\</? GTxJ

> 7C $6$#%#> 82&9D & &'OE,09 7$4 $# # 7 # &960*/$&'

2D &9C

( C 620'0

"H)?$

7 #@? F0910!&9&' F 7 #%&9,0C $? 09

(

27 &968? D$ C #@

(

C7&'687 $$70< 2 7,27 $#@&'60 / &9

$&D$ ,26

X

6&9 0@F/6' F%7/&' 7!09$ F

A = (C ( X ), · )

2#@ 7 6 &9$;+ $

X

$$7 IC&97/$&9 tC& &9&'$ #>09& 09&962&'$() C!$6#@# 8 &D & 6>7 F/&9 6 &9 @E, 0' 7$&' $>7 ,6

X

F 0'6

&9 7!$&',2 @09$F

A

F $#% 09$F

A ?

( C7 7 6 (tC >6&9$

&026 F @$6#@# 8 &D$ 6,,090< 6090' 7 4C 6

f · g −→ f ? g,

f, g ∈ C ( X ) .

: D 7$7 7& 6 !$ &'6090 D &9$0 6 $#%#>2&D &9 &&90905 &9!' 6 &2&9D & $)

7 ,2#@ 80 0 &9$ 7 09$F

A ?

&@7 ( 2@0''09 60'0< ( $&D$ F 7

8 6 $#%#>2$F/ tC 6 $&'2

x µ

X

[x µ , x ν ] ? := x µ ? x ν − x ν ? x µ = i C µν ,

A )+

C7

C := (C µν )

&> 02 &4 #@#%&96 ? tC OE,09 &'#%,&9$

(

0927

) 2

#%& &9 7 $6 $#%#>2&D &9 ) 7&687,$1 09$F 7,' ' &9616$_ 6

C7&'687$#%bC&'0905F &9 71$0'09C&' 0& )

7 @WG & 1G &9D #%$ 2 09&2%$ $20 #@2&9$1&9 7@F/ED$

! & 7&9620;'09&'62&9$ ) 47 71G "+ ($H+H($B T ( T `(+H)?$

6 17&9 ) 09&9mC&97 7 $ 046'

(

C!C&9090 C 6 $6 2%$

#%$ 6&FE,6 &9 &9# 0'WG# 09 ",$6#@# 8 &D$ $#@2&'$,4&9>7 $090'C&9)

,$ 7& ( C # 09 WG ,,&9$

C

&9 # 7 6 $&9,2

x µ

'

C µν = θ µν + f µν ρ x ρ + R µν ρτ x ρ x τ + . . . .

A )A

-_BD2cuL 2>7%d BFA 3DRmA A*BD0 N6%7&cZBD24A*7&3hHU2>=4= 3DA*REGZBF2>REG(3X5%KFA O*GjBDA i G BFA A KDG(3kBFKD2c BF2>O^G<A*KMBFA_A d^G/7%GWKF9*=nt G HU2>=4=

BD0%GWKDG/NA*KFG:NA^=>=4A H 3kBu9;7&V&9;KuVX5%0 f3@2cZ23@BF3 5%KF9^c BF24c/G 9*7&VXi Gb3@A^RmGWHU0&9 B13@=4A*5&5_fMHU2>BD0jA*6&KJ7&A;BF9;BD24A*7h9*=>H19gf3e24RE5%=4f_247%d

BD0&9;BH Gb9;=4=4A H NA*K9*=>= N6&7&c BF2>A^7&3Uf 2>GW=4V%2>7%dEHJG/=4=TVGZY?7%GWVjG5%KFGW3F3D2>A^7&3J2>7jBD0&GIcZA*7_BFG BcZA*7?3@2VG/KFGWVTt

(20)

z

<>T]¤JE>JQ?$AM<B`U5ERJIVa<>T] E>T?mB`A BjA]wA>JERJIVa<>`] <M

>`<>Tx<UUGJERJIVihVkJ-[

CC &9 71$090'C&9 C 6&97 8 6 $#%#>2$ @F/ 7 /6 &20 $#

[x µ , x ν ] ? := i θ µν ,

A ))

$ $#%@6$ 8 ?WC 4 #%#% &96 0<$6 $#%#>2&D &9

θ := (θ µν )

) C2%7,2

7&@09$F&96 6 &@7&9 F@ !& F/ 09$FMC7&9687 &!C 0'0? C $#

,# #% 687,&96 ( C7 &9!6 !'!09$ F $ 7 &9&'! #@#@ 8)

7& 4C 6 &'# 09&'eC7, & 0'0<

"E ( B,( AH%$

6 $# &9$

A θ

709$F

A = (C ( X ), · )

( C7 %709$F

A θ

&)( $#

A

&9 7# 0&0'&96 &9$

0<C 0) 7 0'&9#%&9

θ → 0

C 6ED$ !7 6$#%#> 82&9D 09$F

A

2&9 )

L deQ8

N8:8p[O9LNf _09 i

&&9$,090< 687 &'

X =

2d , d ∈

&' 0' 7 '&9 1G2# 0' 824 $#@

09$F' &9 6$/$&9 $6$#%#> 82&9D ,6 (

,#% 0

θ

$#% 0% ,6

2d θ

( 6 7 , &960'66

d=1

( 7 6 60'09 D 20 02)

A ) +

2d θ

7

θ

#% 0 ,6

2d

C7&'687 &

2d θ

& E F7 $#@2&' 7

09$F #% 7 6 &' $

2d

D & 7 C $0' D 20 6 ( $#%&9#@>20

60909 D A$0 6 ( C7&'687 &$&D$ $

f, g ∈ C (

2d )

6 "T H,(B+H?$

f ? g := f e 2 i µ θ µν ν g .

A )T

7 7 F/ED$% 8 6 & E, D &' 7- $&$ F& D6 $ E09

θ µν ∂µ ⊗ ∂ν

&96

F 7 $6 $#%#>2&D &9

θ

) C2 7, 7& 6$#%# 82&9D & #% 62

0FC -F/1F$7@ 6&'605$#

θ

F #@ $ 7$$,0 2$#@2&9$

O

'

θ = O T M n

i=1

0 θ i

− θ i 0

O =: O T θ

O , θ i

≥ 0 .

,$ 2&9$,0 &9# 0'&96& >CbC&'090 0FC '#% 7,2

θ

& 7 &'6 02#

θ

)

A ) A

! #"$%"'&'()*+

7%$6 $#%#>2&D$ 0'

(

0 ? C '7

D $0 0'

(

&D$1$# 7 2F ED$%1G

# 09 ' 6&'0<6'% tC &'#% &9$ ) 7 7,D$@6 $&9,2

x, y

C&97 $&D &20

4 6$#%#> 82$

[x, y] ? := i θ ,

A )`

$ 6 $ 8"0 ,2#@

θ

C7&9687

(

C&7$ "0'' ; 09&9 (

C6#@ F/ '&&D$)

: D$ 7$$7 7 6$#%# 82&9D 02 C&90'0 F &9#@ &'# /$6 &' #@' 7&

7& C lC&9090 ( $<C ( &968? 7!#%$$ 0*&9

2d θ

2 6&'0'& 0 ( C7

6 4$)

(21)

@,0< 17 / &9 7 8 6

A )T C 1G , &9 $#% 6 &9$

f

2

g

&'

f ? g = f e 2 i µ θ µν ν g

= f · g + 2 i θ µν ∂ µ f ∂ ν g + . . .

= f · g +

0 &D22&D$

2 6 7,2 7 &'$0 - 6 %2tC W , &9 $F09 6 &'% %

687$< 06 #% 7<$&' 6 C&7 79 8 6 &9 6 >F ED (

&))

Z

d 2d x f ? g = Z

d 2d x f · g .

A )H

, 7 #%$

(

&9 C&90'0 ED$ 6$D&9-%&' 6 6$# 09WG 6 &9,2

z i

$

2d

F

z i := x 2 i 1 + i x 2 i ,

$

i ∈ { 1, . . . , d } ,

A )

2 % 609 7 # %&9#%&'$0' 6 $&9,2 D &'

z i −→ a i := z i

√ 2 θ i

,

A )B

θ i 6 = 0

( 687 7,27 F,6&96 8 6 $#%#>2&9$ 0'2&'$ C ,

a i , ¯ a j

? = δ ij .

A )H

,$#>0' A )H

7 , WG 7&9F&-7>&9&9# 6 $ 6 &9$ F/ tC 7 6$&' $6 $#%#>

2&D &9 @7C ,# #%687,2&'6 57,2#%$&96-6&90'0'2$ 7,2 C 1092,0'09 &'

7%F $&9&9 7& 6 &'$() 7& 0'2&9$.C&'090"C F/%WG 09$& 7 F WG &9

7

θ

$#% 20 ,6 F ' 2$ $ #@ !&'09F 62)

ij8u)m( )6,S9LL"<,

#%&9$ @F ED

(

27 C 16$687 $6 $#%#>2&D &9 &'>7 6' /6$ 8

$# &9$ ,#%

θ

&!7 $# 0'2&9$ / $ ,26) $&' 12&' 7 6$#

# 8 &9$ 0 &9$ A )) ) 7& !& F/ 09$F #@ 20 F 0'& F $#%&9@7

6 &9,2 109&9"' 2$"26 &91$%$ $#%_G &90'&' &90'F/ ,26

H

2& &'

A )) ) 7& &) 2&9$-C&90'0 F/&9 6 &9 7&!F,6&9$ & 0'0<

# 687 #%$ 6$D$ &9-$ 64 &'@$!6 $# 8 &9$WG 09&'6 &90)

0'6 &' 7 60'

6 $&9,2 A )

A )B

F 7/ $

ˆ

z i , z ˆ j

ˆ a i , ˆ a j

2& &9

z ˆ i , z ˆ j

= 2 θ i δ ij

ˆ a i , ˆ a j

= δ ij

,

A )+EG

(22)

C%!7, 7- 609 6 $&9,2!/2$ 2js 62&9$ &'7&'0'2&9$ 2$

(

d

&9#@ &9$,20,7,#%$&'66&90'0'2 ; 68? ,6 lC7&9687 C!F

H

d =: H

) F'&

$

H

d

& ( ',,0( $&D$ F 60< &9@7 62&'$ / $

ˆ a i

@7 $ 82 $

D26#

| 0 i

| k 1 , . . . , k d i = Y d i=1

ˆ a i k i

√ k i ! | 0 i ,

$

k i ∈

0 ,

A )++

C7

ˆ a i | 0 i = 0

$ 090

i ∈ { 1, . . . , d }

) #2? $ 8 &9$ # &' $ 7)

mC&9090 &' 7& 6&9$ 1G 0'&96&0< &9&962 2$>F

ˆ

F 09D$ 7& $ &' 7

$090'C&9 6&9$ $ 2&9$,20 &9# 09&'6 &9 '09$%' C 6 $D &'6 7,2 7 C&'0905F

@6$,&9$)

%

7 '$PC7 7@2&9$ 7 $6#@# 8 &D & 0'F F 2$ $

&90'F/ ,6 D &9 4!, 0 0$$ 6$# 2&9$ & 72 7<tC#%&9$ 6&&'

θ

$#@ 0 ,6 2 _ &D209)< 7 !6 ,090 WG & %&$#%$7&# F tC 2

'&2 09$F "6 &9$1 $6 $#%#>2&D$%0 ,6 % 09$ F 2 09&9

/ $ ! 7 &90'F/ ,6 &9 6 F/ED$)

&9

"++EA&$

7 6 $#%#>2&D$ 09$ F>256 &' $

2d

F/09$$&9>>7 687pC 60'' ( & )2) 6'&9 '&90kC&7 090;7 &' &9D22&9D 2&'E&9 ) :<09#% 57&0'F

#@'/ 6$ &9 09&' / $ F 7 D A$0C 0 #@' "<+ A T (B+H&$ )

6$# 09WG 6 $&9,2 A ) &&$&D$ $ 6&9$

f(¯ z, z)

( C7

z := (z i )

& 6$09096 &9D 2&9$ $ 7 6 $&9,2 ( F

f (¯ z, z) 7→ F (ˆ a , ˆ a) :=

Z

d d p ¯ d d p

(2 π) d f(¯ ˜ p, p) e i (¯ p · ˆ z+p · z ˆ )

=

0< $

f (ˆ a, ˆ a )

A )+ A

C&97 7 ;&9 $#

f(¯ ˜ p, p)

f

$&D$ F

f ˜ (¯ p, p) :=

Z

d d z ¯ d d z

(2 π) d f (¯ z, z) e i (¯ p · z+p · z) ¯ .

A )+ )

7># C 0< $ &' 7 E,&9&'$ F ED> -7>09&'! 2&' 72-#@',

#%$$#%&'0"&' /2$ & #%#@& 1G, &9$

(

687 7,2

(

$1WG# 09

(

ˆ a ˆ a 2

$ #@', 0 $

ˆ a ˆ a 2

= 1 3 (ˆ a ˆ a 2 + ˆ a a ˆ ˆ a + ˆ a 2 ˆ a )

) 7&%6 $WG

F ˆ :=

F (ˆ a , ˆ a)

&-620'09 7 4 # F 04$ 0 4 #>F/$04

f

) 7 D 0C 0#6 7 $$#%

6&9$ 6 $&'2 #>0&09&' F 7 4C 6 ' 2$ 7 !&'09F/ ,6

H

C&7 71,0 /26# /&9&'$ ' 6) % &'1#@? 09 , #@ &9$ 7,2

', >$# 7 0<$ &9 &9 6 2F ED$7 1G & %0 27 /&'F09 $&'

6 &&9$ '&' &! @ 2$1WG &9$ 6 06&9620<6 &'$,)@ 7 %&'

6 F 090'C&9 $- C &'7 -6 &'

w(¯ p, p)

&' 7>&9 $ 2 A )+ A 2)) "+'+EG&$

$ D &9C $ 7& &96 )<C 0 7,2 7$#@09& 2&'$ &9 A )+EA & 687687 7,2

1 7→ ˆ

)

(23)

6 D!7 6E,$2&9$ 6 6&9$

f

$# /2$

F ˆ

C % # 0'

7 &'D>#@'&'% A )+EA C7&9687 &!&9D F

F ˆ 7→ f (¯ z, z) := (i 4 π) d Pf(θ) Z

d d p ¯ d d p

(2 π) 2 d Tr H

F ˆ e i (¯ p · z+p · z) ¯ e i (¯ p · z+p ˆ · ˆ z )

= F ? (¯ z, z) ,

A )+ T

C7

F ?

&D$ #

F

F 0'6&9 7 6 $# '&&9$ F 7 4C 6

Pf(θ)

-7 "7 $6 $#%#>2&D &9 $

θ

C7&9687 $ &9&9D #%&',2 &

&9D F

det θ

7 C& WE, ' ( $WG# 09 ( &9 "EG?$ ) ,7#%$ ( 7 #>F/$0

Tr H

&!, 71% 716 1 D7 !&'09F 62)

17 !7,D$ %F&s 6 &9$ 71$#

(f (¯ z, z), ?) ←→

F ˆ (ˆ a , ˆ a), ·

A )+`

20'09C&'-4&'D &'$2 $F09# $

θ

$#% 0,26D &'-&94^ &9D209"$#>0'2&9$

&9 7 /2$ #@09&# &' 6 72)

&'( %"'&'( &"

$" ' 0<@,0< *E097$&' &97' 2$ $#@09&# & &9&9090/6 @

&9 6 2,09$$ &9D22&9D 1 &9 $2&9$ $ 7 09 D0"/2$ ) 7 D 20<

0<#@' A )+ A

,2F0' , 82@7@0'09C&' #@2&9$ 7%&9&'$

F/ tC 4C 6! / $#@09&#

∂ z ¯ i f −→ 2 1 θ i ˆ

a i , F ˆ

∂ z i f −→ − 2 1 θ i ˆ a i , F ˆ

,

A )+EH

6 C 090 '

Z d d z ¯ d d z

(2 i) d f −→ (2 π) d Pf(θ) Tr H ( ˆ F ) .

A )+A

, 7 #%$

( C >#@ &' 6 7 7#%&9& 3 '0'6 2$

− ∆

C7&'687 &E, D &'

∆ ˆ F :=

a,

a , F ˆ

= a ,

a, F ˆ

.

A )+EB

?0 &, F 6 &'$,!2

a

$

a

$0<$)

#@#@& &' ( C $ 71$090'C&9&'6&9$,4 $ 7 ,&&9$ F tC 8 6

2% 2$ $#>0'2&9$%$ 7

θ

$#% @0 ,6 !#% & @F 7

D

A$0C 0,#@'

824 6 ' 2$

6 $&',

¯

z i , z j ↔ √

2 θ i ˆ a i , p

2 θ j ˆ a j

/2

&D22&D$

∂ z ¯ i , ∂ z j ↔ 2 1 θ i ad(ˆ a i ), − √ 1

2 θ j ad(ˆ a j )

6 $#%#>2$

&90

R d d z ¯ d d z

(2 i) d ↔ (2 π) d Pf(θ) Tr H

6 )

C!7,2 &'

d d ¯ z d d z (2 i) d

C C&909050 #@?1,157 #%$16 $D$ &' 82&'

d 2d z

17 2#@ 7&')

(24)

" "$%" ! #&$"$ " $ %"'&

2&&9$ 7 2$ #%&9$ 7,2#%$&9616 &'090'2$ 82 (

7 WG &!7 /6 &0

60'' 2&9

H

C7&9687 #@ & $2 &' )" 7

(

6 6090' 6$7 2 0' 2

&'# /$"0' &9%$&6 &9$ 2,7 $6#@# 8 &D$&9$#@1#@ 0$ 7 /2$0'D$ 0

7 D$ 7$7 &9 6 &9$ ) 7&1

(

C%60'096!$#% 7 F,6&96%26

F/$47# '2 ' $7 0&"&9>7& 7 & ) #%$ &90' @&6 &9$6 (

$-1G# 0' ( F $ &' "H+H&$ )$ 2&&9$ ( $#% /6 &0&84 ' 2$-!

C7&9687&96 02&'< 8 &9$ &9 76$# 09WG 02 0< &' 7 !&'09F ,6

,2 F 7 67% 82 )

7 &'0' 7#%$&96 6&90'0'2$ , 68? 6

H

7 6$7 8

| α i

>E,

' &9$,8 7%&97&902&' /2

ˆ

a

C&7 &9$ D209

α ∈

62 F/!C&

'0'&96$# F&'2&9$ F,6& 82

| α i :=

X ∞ k=0

α k

√ k! | k i = e α ˆ a | 0 i = e 1 2 α¯ α e α ˆ a α ¯ a ˆ | 0 i .

A )+H

C17,2 7&WE,&&9$ &687687 7,2

| α i

&!$#20'& @&9 F!27

h α | α 0 i = e αα ¯ 0 ,

A )AG

& ))(

h α | α i = exp(α α) ¯

) ,$# 7 $#>0' A )+H 7!6&9$%;7 6 &9$ 2@&97&902&'

/ $ !

| α i

6 F/1 6 &' 0'&'@&9#%#2

ˆ

a | α i = α | α i

ˆ a | α i = e α ˆ a | 1 i = ∂

∂α | α i .

A )A +

, 7 #%$ ( 716$7 2 A )+H $# ED$ 6 $# 0' F,'&

H

C&97

&9$ 4&9

ˆ

=

Z dαd ¯ α

2πi e α α ¯ | α ih α | .

A )A$A

.-,/6&'0 &9 4&' 7 6 $WG 6 $7 8 & $09<&' $6 $#%#>2&D$ E,09

7 $ C&90'0 F 7,$# &9$ 6 $7 82 >7 &9 #@2&9$

#%&'2&9@20'2&9$! 8 &9$&' 7 6$# 09WG 0'2)

,$ 7 E, !47 ( WE, 7 7&9 !/2

D (β)

$ $#%

β ∈

F

D ˆ (β) := e β ˆ a β ¯ ˆ a = ⇒ D ˆ (β) ˆ a D ˆ (β) = ˆ a − β , ¯

A )A')

$# C7&'687&C E, 7,2

D ˆ (β)

&9 20'26 $7 2-6

D ˆ (β) | α i = e β(β+ ¯ 1 2 α) | α + β i .

A )A T

!282&'$,!246 $78 F$#% 052$09

ϑ

( $ 7 7 7 ( &'6 F

R ˆ (ϑ) := e i ˆ a ϑ ˆ a = ⇒ R ˆ (ϑ) ˆ a R ˆ (ϑ) = e ˆ a ,

A )A+`

(25)

687 7,2

R ˆ (ϑ) | α i = | e α i .

A )AH

7&84 $#@2&' 0 & 7 ( &9 $$$09&9F ED@4

#@2&' ( $&D$ $ $#% 6 $# 0'1G ,#%

β

F

S ˆ (β) := e 1 2 ( ¯ β ˆ a 2 β a 2 ) ,

A )A

C7&'687 &'# 09&'

S ˆ (β) ˆ a S ˆ (β) = ˆ a cosh( | β | ) β

| β | + ˆ a sinh( | β | ) β

| β | .

A )AB

(26)
(27)

#

#@&9, &'# &9 E, 0' 7 &!kE, &D &0 0'&' 7 _ , &9$

#%&' $#% $&9D #% 0 ) !$ , 0< ( 7 ^ 2&9$@&&' &9 09&&96 #@ 0

2 ,090 7&'$70 $09&' $09D &9 7 # &9 0'0 09&9 & &96&9F0'

'?;) &@7 $ , 0 0' $? $ /&'F0'&9# 0'&FE,6 &9$\C7&'687 #%&9$7 20'09C $

E&9 2109' F/60'' 0'&')! #@ 6' ( 7&1& &9 &9F092 7

1G & #@ ,2&9600 C&' &, &9$>C7 7 6 $&' $09&'$, 0 7,D$

2 &9 8 &9$ &' # ' $09$$&9620m ,&&9 ) mC&'090 096 &', 7&%&9 &9 7

0'09C&' 6 &9$&C7PC C&9090 6 $&9 &'#@&'0< $09&$&96 $09&9$ ( $ $09&9$ $

7$ ( C7&9687

(

$ $! (

WE, 'eE,& ' $09&9$ #%&'?C? &'

E09 7$4 0'0< &&'@$# $#% 0'$&'60 $#%)+

&'090-7 6# 0' 60'' &FE,62&' 2 7 $/ F,2&9D $09&9$ #@ oC 090 F/

2

(

&9 &'# /&'F0'

(

8'?;)4 7

(

$<#%&'$76$,&97 $ $ &9# 0'& &'

7 $F0' # ) -09$ 70'&9 7 $@^) $ 7EDF/ &9 D &' 7 ,

09C &9#%&'$0 #% 0 6 # &9#%&'0' &9 &' 7&'7 &'#%,&9$,0

#% 0 &' 7<7' "7257 #@ F <#%$<'&90168F0'2) 7 $9 ? 8

09&&96ME, 0' 7$&' F #% 2 7 0'C &9#%&',0 #% 0 $ C $09 0'0<

_ &9 71 D>6 #2 2 712 717&9$7 &'#% &9$,05#% 0 /

6$ '&9F09) &96 &' 7& 7&@C%2 &9#2&90 &' &' 6 0' &960E,09 7 $&9 7

?2 2kC\C ?2 & 7 1G & 6 $&D &20$0' &9$> 7 _ , &9$

#%&' )

$# F&9&' 7 tC #%&'$ 68? ( C @7 , $ 2,0< $0'&&'6 $09

&'> ,&'6 0' 09C &9#%&'$0;E, 0' 7$ &9620#% 0

(

7 6 60'09 687&'0

0'&9%&'$#@@#@ 0 ( &9-WG &'$ 7 $6#@# 8 &D$ 09# ) ! F&'Wa

D &9C&9 76$#%#> 82&9D%&'bC 6 7$6 $#%#>2&D$!66!60'& &' 0

7 0'2&9$ 57 6$#%# 82&9D$ 2 #% 0/ 6$#%# 82&9D 602 E, 0' 7$&')

$@$$&9 &9 82&'0"09 $#%@ 6 $1F68? $$ &9)@ 7

2,0< &12"$/ F,2&9D $&9D &'0E, 0' 6 $E,2&' &9 $204&1 $09 F_s 6 &'

E09 7 $ 7 1G&% &9 2#@ 09&92 $ &9) ,$ $ ED$ D &'WC $

7 &96 2 $09&$- &' 8$-&9 7>6 $#%#>2&D$ &9 &9 , &960'!$ 7

-_BDKF24cZBD=4fa3D5?G(9;L 247%d[BD0%23 c/A*7&V2>BD24A*7EKu9 BF0%G/KeV%GZY&7%G(3 HU0&9;B 243eL_7&A HU7m9^3 \@3DA*=42>BF9;KFfIH19gO*G(3` S 2>7?3kBFGW9*VmA*N\D3@A^=>2>BDA^7&3k`uS

3@247&cZGJBF0%GU=49;B@BFG/K 5%0&KF9^3@GJA;N,BFG/7EKDG/NG/Ku3'BDAb9 REA*KFG13@BDKF247%d*GW7^B VG/Y&7%2>BD24A*7M9*3WS NA^K G%9;RE5%=4G*S;2478o>p(rt GZBWS^H G1HU24=>=&7%A;B

RE9*L*G1BF0%243e3@A^RmGWHU0&9 Be3D6%iBF=>GV%243FcZKF2>RE247&9 BF2>A^7E0%G/KFG9;7&VaBF0_6?3 9*V0&G/KFG1BDAIcZA^RERmA^7E5%KF9^c BF24c/GJ247M5%0 f3@2c/3 =>2>BDGWKF9;BD6%KFG*t

O*GW7 BF0%A*6&d*0 BD0%GWKDG G243@BWS1A*Nbc/A*6%Ku3DG*SJA;BD0&G/KXKFGW9^3@A^7&9;i%=4GCNGW9 BF6%KFGW3X9^3MG*td&tec/A*7NA^KDRM9*=2>7 O 9;KF29;7&c/GjA*N BF0%G

9*c BF2>A^7TS%9*3Df RE5BDA*BD2c:NKFG/G(VA*R A*KcZA^7Y&7%GWREG/7_BUA;N'NG/KFRm24A*7?316%5?A^782>7_BDKFAV6&c/2>7%dmBF0%G/R o>p(r t

(28)

$09&9 &'$#@ #% 0 7

(

C$$<7F/ ?

"<+EG)( + ))(B$H?$

) D

$ 60

(

&'# 0 F 7>D26 &' 7 "6$#%# 82&9DOE, 0' 7 ( $/ F,2&9D

$09&'$, 6&D$ @0'2 2 &' &' 7 $6#@# 8 &D$ &' ( 'lC 0'0) ,$- 0

6-$ 7 &96 $6 $#%#>2&D$!$09&9$! &9$ C 7

"T A ( T ()H T (`')( T H,(*+`(*H?$ 2 6 !7 &' )

z cI<u ? VWU A>`]^VW<>EH]^VapU5EXU </?$AH]

7>#% 0-2 &'#@4 &9 - 7 $ 09&' &9$#@#% 0 ( 620 E, 0' 7

$&9 $

(1+d)

&9#@ &9$,20Aa;2 ,6 &9#% C&97>E09 ? &' D209 &9 &9#@& $

#@&$0' WE, F 0& 6&9$ 71$#

S[Φ] :=

Z

d 1+d x 1 2 G ab ∂ µ Φ aµ Φ b − V (Φ)

,

) )+

C7

G

7 #@&'6 $ 7 82$ ,26%

V

& #@ /&'0 ) 7 8 &9$

$ 09&' > - 7 $# <7 #% &96% #% 0 & ,090 6090' $09&' &

7 6 $# $!

G

/ $ 7bE09

Φ

&90*)

. 7 #%#@&9 WG# 09 /!$09&9&9$#@-#% 0 &7;C 090

O(n)

&9$#@ #% 0 ' E>&' 6 F

090 D 3 D "T G&$ C77\E, 0',

$&D$ F

n

6$# /$ 20!&9D$ 6 7 0'$2$&' & &9# 0

L =

∂ µ Φ aµ Φ a

) &'6 $ 0'& &9$ 7 3 $$&' &687&9 D F 0'09C&' $$, 8$ #2&90' ( 2)) $

U (n)

D20'YE,09 ( &909&' 7 &82 &9$#@ 687&90 #% 0 C&97

L = Tr ∂ µ Φ µ Φ

) . FD &9$0 7%#% 01 s $09$F,0

O(n)

$

U(n)

#%#@4 (

/6 &9D0< ( mC7&9687 7 &'1,2#@ )>.-&'6 02 &'# /$6 $ 0'2 1'09&962&'$,

C&'090 F/>7 &'6&9$ "7 687&904#@ 0 7#@&&'ME, 0',

Φ = Φ

C7&9687 0'09C -

Φ =

− 2P

C&7 7#%&9& s 6

P

#@ ?

r

) 7 6s 6 $

P

&0

,#% & F61"7 0906 $E,$2&'$ ,6 OC&7 &9#%&'$ &9D F &-?;)

7 090 6s 6 $ &9&90 ^ &9D209

P

( 2 7 #%? ( & 60'09 7

'#@&'

(

&!&$&D$ F 7 6

Gr(P ) = U (n)

U(im P ) × U (ker P ) ,

) )A

C7

U( X ) := {U ∈ GL( X ) | U = U }

7 220'0&84 09&' $#@1

&9$

X

) 7$ ( 687 &9D

P

( 7 687 7 #%&&'

Φ

( F/09$ 6 8&9

'#@&' ( 2>7 6 ;090 7#%&&'

Φ

6$# / &9 1&s $&'4&9$

64

#@&' )> 7 &96 &'$ 7#@&&'

Φ

6 7 &9 $'#2&' &9$#@

#% 0 ( C7-6E,$2&9$ ,6 &,2#% & F s 6$

P

) 4&9,090< ( &9NC 6 $&9

$0< s 6 $ ?

1

C@09 C&97 7 1 620'09

P n 1

#% 0 ")$G,( T +'( ),+'(9+ ))&$

C7&9687 %F_s 6! WG &D$ &9 7&9 C )" 7 1& #>687 #%$ F/$2&' F/$

7 #% 0 %2 7 &' /&'%F C >C&9090 % &9 7& 7 &@ 7 6$

6 2%$ $2042 7 /6&FE,6 .C7&9687 C&9090 F &9 6 &9 7 1G

6 &9$ )

7mBD0&GU=>2>BDGWKF9;BD6%KFG1A*7%GU6&3D6&9*=>=4fIG/7?cZA*6&7^BFG/Ku3'BD0%GU7&A;BF9;BD24A*7

Gr(k, n)

NA*K BD0%G3DGZB A*N 9;=4=

k

cZA^RE5%=>G

V24REG/7&3D24A*7&9*=

3D6%i&3D5&9*c/GW31A*N

n

t

(29)

3 LL gZ8u66L " deQ8u)8_,

7 #% 0 &9 6 2F ED$ # ' $$7 $&9D WG 09&'6 &9 $09&9$ ( 09' &'

09C &'#%,&9$

(

2 7 D ' WG# 09 $ 2 #%$ 20'&&'6 7&9$7 4

&'#%,&9$,0!#% 0 ) C D 7 0' ( 7 #% 0 2 $&9D &'0$09&9$@&'

2F& >&'#% &9$ ) 7! &96 &' %7 #>F/ 090'C &'#%,&9$<$"$&9D &'0

$09&9$ WG & 2#@7$# F +- &968?

"`+` ((A$A&$

C7&'687 82 -7,21

6 8&9 ' # &' 76 WG &> 7 8 &96kE,&9 ' 0'&' 7, 7

D26 # & 09*) 35 $090'C 7 09&' >2 "BH&$ 'bC 0'0'>+&'68?)r $&'$&9,0 ,6 > $

&9 0'&0'%#%$ &'0 ) 7& 6$,&9 &9 6&9$ 2 7 $# ) )+ ( C7

(

&9# 0'&96& ( C '#%

Φ

F/ 60 2) 8 &96 $09&9$

Φ

%WG #%& 17 '

6 &'$0

E[Φ] :=

Z

d d x 1 2i Φ ∂ i Φ + V (Φ)

=: E 2 [Φ] + E 0 [Φ] ,

) ))

C7

i, j ∈ { 1, 2, . . . , d }

( &'# 0< &' 72<&9 7$09@&' , &960'"F/ 82&', > ,2&20 609&'

Φ(x) → Φ(λx)

) 7 ' $# !7&! 609&' '

E[Φ] −→ λ 2 d E 2 [Φ] + λ d E 0 [Φ] =: E λ ,

) )T

7 16 6$&' !7>687&'6 " #%&9$09$' F/ D)$ 82&',& 7 6 092

7 6&&9$

0 = ! dE λ

λ=1 = (2 − d) E 2 − d E 0 = ⇒ E 0 = 2 − d

d E 2 ,

) )`

!6 C 7,2 7 F/7ED &9$ 7 6 /,6 6 &090< $ 7 #>F/ 2&'0

&'#%,&9$) &96090< 7 ? &9 &'6 6

E 2

7 &'0 '

E 0

2 F/7

/&9&9D 7$ ) )`

6 0 7$09 $

d = 1

C7'@$

d ≥ 2

&9@&@D &9$0 ()

7 66

d = 2

/ 1G6 &' &','%7 &9& &90'0 '&9F09 6 $&9 7

&9,2&'$ D2&7&' 2&0IC7&9687 #%&9$7 &'090!090'C $ 0'&') 7 C

D27& 7$# &9 60' E09 7 $&9 &< &'609 2&&9$,0, # C&77&9$7 $

&D22&D$%$>7&9$7 C 7 &9D22&9D %&' 7 3 $$&' 2 7 %0 687$

7%6$/$&9 ' &9 mC MC7&9687 20'09C $$09&'$, D$ &9 7%6'% 7&'7

&'#%,&9$)

. &'# /$&9# 09&'62&9$ %7 &9 6&9$ 0927 C620'

θ

D &'@7 824

6 A )T & 7,2&9 7&!$6$#%#> 82&9D &9@7 60'&9 $#@ - ED$

7 2F ED$7 $ # &> &96090 09$ D20'&9

"AH?$

) 7$ 8 &96\E,& ' 6

09&'$,@6 1G&@D$ &' 7&9$7 &9#@ &9$ "T )&$ 7 1G2# 0'

d = 2

C&7 2&0 ) ,$

V ≡ 0

( 7CD$ ( 7 ' 6 7 6$#%#> 82&9D

1G &9$ ) )) 1 7 / A )H +- &'68?)r7$# &'090 '0)

%&@0^C$ 7pC7&90' #%&'$&' 72%F &9 7 &D$ $ WG & 6 0%&4

6 F/ D C 6 0 7& 7 $ # $#%16$,6 &9D 1F 09& &9 7,27

5&9*K@BNKFA*R BF0%243WS BD0%Gm247^BFKDAV6?c BD24A*7<A*N d^9*6%d*GbY?G/=V%3 c/2>KucZ6&R O*GW7_BF3UBF0%G A^i&3@BDKF6&c BF2>A^7&3A*N BD0%23 BD0%GWA*KFG/R i_f<9

V2 G/KFG/7_B:3Fc/9*=>247%dmi?GW0&9gO 2>A^KWt

(30)

6$# 0$ %&9D2&'6/7! ' 2&'0 609&' &9# 0'&9 10'2&'$%F/ tC 7

&)( 47 ' ,090 6!D &9&'0 7# "<+EG)( BH?$ )

d = 1

( $

WG# 09 ( 7 6$&9&'$ ) )` &' 0'

E 2 = E 0

2 7 ' $0' &9$ & 7 , D$ 0

09&-F/ tC ? &' &96 &'056 $&9F&' )

3 $ 8uR)V L9L10

: D$,090 ( $MC $09 20 0'&9? 2,0< &9#@ $0' &9$ 7 /6 &9D

#% 0 2-7,()$ - &9# $ 8< &'$ 2&&9 &' 7& 6$WG 1<6 0' & &' $0' &9$-

>$&D$ #@ 0 &;C77 &9& '&9F09 6$# 09 0< 09D-7-_ ,2&'$, 5#%&'$, $ ( &9

7/C$ ( C7 7 7 #% 0&&9 $F0'2) eE09 7$4 ( 7< # &9F&909&9 ,090

>7-WG &6 2 &'_E,& # F 6$, D22&9$ 0'C ( C7&9687 ( eC&9160''

;#% 0

(

&4&9&9&'60'0<>&' !7 WG & 6; 6 60909%3 6G ,2&' $# 0'2&9$

"E +(+ ),+*$ (

C7 7 _ ,2&' "#%&' & '6$# ,2&'F&'09&9 6&&9$ < -"_ , &9$-$

_G &'09&' ME09 )% WG # 09& 7& 82#%

(

09 F&'Wa 6$,&9

"+ ))&$

' &9# 09

(1+1)

&'#% &9$,0(6' 7 !$C 6 <&9 _ , &9$ $ 7 0(0'&9)< 71^ ,2&9$

#%&9$ &$&D$ F

∂ t q + α q ∂ x q + ∂ x 3 q = 0

C&7

α ∈

.

6 7 0'&9 2$ "A'),(Hp`&$

L := −

x 2 + α 6 q

, A := −

4 ∂ x 3 + α q ∂ x + α 2

∂q

∂x + f

,

C7

f

& F&94 6&9$ /&' $

t

&&9$,0D2&2F0'

k

) 0'6 0'2&9 7 6 $# ,2&9F&90'& 6&&9$ @7 24^ 2&9$

L Ψ = k 2 Ψ , ∂Ψ

∂t = A Ψ ,

$ #@ _G &90'&'4mE,09

Ψ(t, x, k)

α 6 =0

$&D$

∂ t L + L, A

= 0 ,

C7&9687 /$ &', &9 7 WE,&&9$

L

A

( 7 6 60909 3 G ,&9 ( 6 7

$&9$&9,0,^ 2&9$ 2 #%&9$%$ 7lE, 0'

q

) 7,D$!7 WC& 7^ ,2&9$@ #%&'

'1@6$# ,2&'F&'09&9 6$&9&' "09&' ^ 2&9$

(

7 09&9 # 7&-#@ 0 )

D &9C 7!66&9$ F/ tC 71G&6 ( 3 6G ,&9&9F&909&9 %#%&'$

F/ D

( C !C&'090"16 $#%#@ 6&96% 6090" #% 0 &9 $F0'OC7D$ 7 1G &

687 3 6G ,2&'-#>0 2&' ) &'17%!$C 6 <&9 1^ 2&9$

(

7 % 09-

&'# /$!&' $2F0' # &' 7& 2) ,$ WG# 09 ( 7 $09&'&9$#@@#% 05&9

2

&9#@ &9$ ( & ))

d = 1

( &-? C F &'$F09 "+ )')?$ )+- @7 1G & 6 /C 0

&A*Kh9 RM9 BF0%G/RM9;BD2c/9;=4=4f REA*KFG<KD24d*A^KDA^6&3EVGZY&7&2BF2>A^7 A;N[2>7_BDGWd*Ku9;i%24=42Bkf A^7%G<7%GWGWV%3XBDA G/RE5%=4A f 9 =42>B@BD=4G REA*KFG

3D6%iBF=>G/Bkf l!3DG/GEG^td?t o (r!sUNKFA*R HU0%2cu0 HJGEHU2>=4= KFGZNKu9;247 247 BF0%23[NKF9*REG/HJA*KFL89*7&V i?GXc/A*7_BDGW7^BbHU2>BD0 BD0&GENA*=4=>A HU247%d

9 5?9;24K1c/A*7&V%2BF2>A^7Tt

Referenzen

ÄHNLICHE DOKUMENTE

While the supertwistor spaces examined so far naturally yield Penrose-Ward trans- forms for certain self-dual subsectors of super Yang-Mills theories, the superambitwistor space L

The definition of these indices and the description of their behaviour under changes of the triangulation and under gluing will be the main input into the algebraic treatment of

We probed these models via numerical solutions of the null energy, weak energy and strong energy conditions, for f (R) family of gravity theories results are presented at Figs.. 3,

Our research is based on high-precision inter-satellite Ka-band ranging (KBR) observations collected by the Gravity Recovery And Interior Laboratory (GRAIL) mission during the

By this choice, the dromion localized in all directions is changed into a chaotic line soliton, which presents chaotic behavior in the x-direction though still localized

As it was already mentioned in experimental part, the aim of experiments with real weapons was to measure magnetic signature of real weapons and to check the work of portal system

generally accepted mechanism of DNA backbone cleavage with protonation of the nucleobase by the acidic proton of the adjacent 5 = -phosphate group (hence termed as a-B-mechanism) is

These conformal blocks for a four-point correlator are functions of two cross-ratios and the coefficients are those that appear in the operator product expansion of local fields..