Mesure de l'angle gamma du triangle d'unitarité de la matrice CKM dans les désintégrations B-> D*K auprès de l'expérience BaBar
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(4) γ
(5) !
(6) "#$ %'&
(7) )(*+ ,-. .
(8) / /
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(19) ,/N * $< 03 γ RR @C L ? kl>; .my /00#x!5#I%+0#x0#98:>; *C!5;98+ # B± → D(∗)K (∗)± L LsL LsL=LsL L LsL LsL LsL=L LsLVEKE @C L @ uv;9K7i "!56@|bc @|g D7%+g ib* L LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsLVE G @C L @ L ? ~o 128+) smy 91=6%'!5;98+ L LsL=L LsL LsL LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsLVE S @C L @ L @ np0/0 smy 91=6%'!5;98+ L LsL=L LsL LsL LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsLVE H @C L @ LDC k/0 7Y8+>; #ii%'!Z1;9Kpib* LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsLVE I @C L L E dx0#96%:;C!5; #xs%'!<1;9K7<ib* L LsL LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsL @C LDC uv@ ;9K7 sS;lp7p|hU68+;9|np 8{ Uhsn L LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsL FKF>J? @C LD@C L ? ~o98:/8: JL=LsL L LsL LsL LsL=L LsL LsL LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsL F>? @C LD@C L @ S! #G "!59;98+/6%:8+V0#x170# D∗K LsL LsL L LsL=LsL L LsL LsL LsL=L LsL LsL LsL L=LsL FK@ @C LD@C LDC dx0#96%:;C!5; # LsL L LsL LsL LsL=L LsL LsL LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsL F C @C L E uv;9K7 =i8:98|U *# #914!5@|np5P.|z 6 "!5 g = n q L LsL L LsL LsL=LsL L LsL LsL LsL=L LsL F E @C L E L ? x12 %:8:;960#V0#I0#98:7; *C!5;98+ #U@6 D &;9 8+#I/0 9 #L LsL LsL=L LsL LsL LsL L=LsL F E @C@C LL EE LLD@C dx 70#9 6 %:;C !5; #x00s#U%'/E!<!519; 0#9;98+K7<0# L nLs L LsLsL L LsLsL L L=L=LsLsL L LsLsL L L L LsLsL L LsLsL=L=LsLsL L L L LsLsL L LsLsL L LsLsL=L=L L LsLsLL FKF @C L F uv0#96 0#G8:@8: 0/; 0#= sin(2β + γ) L=L LsL LsL LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsL F#FKFS @C L G 1=R8:"!58+#C f0#G1;9Kp0# L LsL=L LsL LsL LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsL GKJ @C L S /%:6#98+ &LsL LsL=LsL L LsL LsL LsL=L LsL LsL LsL L=LsL LsL L LsL LsL=LsL L LsL LsL LsL=L LsL GKJ R %5
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(39) o w %:8'!5>; %+<;9K0 )1ZU7;9K=$3/00#s/0 >;9C!58:7; 0# K7Dp#98+j760#E,P /0 12 %+< H %+20#s#9Dp1| ;9 9 8+50#2;P$3Ot0#9;9 "C!E! Y/0 |g# ; %+12Z #498:!5/8: J#Z2%gOt0Z@8 "/E%'!5!5;9;98:8+Y7 8: ; 30#9%';9! ;98:K>0; *9,c8+j76j78o6"8:!512> J;98+*j7# 62(j>60<#2%g/ OQ! K"/!5;9128+ [#E#CLo X8:;s%:%+08:>#2Y* !5>98';2!5>; ; 43#C12 68+# #i0#%+0$ # ;9C!5#.my 914!5;98+ #s@6v * 6 B<~\ 8:/E!5 *LPuw!58+#x%+ H %+0#i#9Dp1;998+0#U#OQ!59 ; = "! #V%'$@Lzhs!5#U%+0#s!5@| 00# ?IKFKJ , !5 f;u 8:%:%+#=6;98:%:8+# ) >;%+0#=#9Dp1;998+0#=i- !56 ! q2R 5;98:=%+0#=;9K0 98+0#=s- !56 *,cj768 J91;9; >;xs0/98: i%+0#S8:7; C! /;98+ #x7;9 i "!59;98+/6%+0# p%+0#G#9D71;998+0#S J91;9; >;VU#C%+0/;98+ I6 -96q;9 )0#< 0#9;9 8:7;< %+ 8+#2! @128+# #98:R%+0#< "!59128; 6; 0#</0%:%+0#<8:14!5 *8:"!5R%+0#Z J 6<0/98: 3%+0#8:>; C! /;98+ # 7;9 "!59;98+/6%+0#ELF 0/8 B91;=ZC!51%+Z R%+)14Z%'!(; 9128:"!5;98+ r0#s8:>; C! /;98+ #=my ! | 17;C!5%+0#x$</0%:68J0#S#9Dp1;998+0#G# 6#.|g; !57;I/00#8:>; C! /;98+ #E,%'!=my /0V0#S8:7; C! /;98+ #E,@%+i/0 >; 76 "!59;98+/6%+0#i;x%'!14! # # 0#I "!59;98+/6%+0#i;C!5>;i0#x "!5C!51);9 0#U%:8:R 0#U %'!;9K0 98+*L"o@c,z%+0#x "!59;98+/6%+0# # 7;V/E!5C! /; 98+# 00#V "!5V0#x 1=R 0#Uj>6"!57;98+j>60#U/0 # 9Y0#V!5#I; 6; 0#U 6 /09;C!58:0#x8:7; C! /;98+ #ELP 0# %+ 8+#U/0 #C9Y !5;98+ w# >;i! # # p/8+00#U$40#V# D71;998+0#V %:8+00#s$2%gOt0# "! /0|g; 12 #E,"14!58+#i$4%gOt;C!5;x8:>; 9 0#x "!59;98+/6%+0#x;x$Z%'!<"!5;96 i%+6 #I8:>; C! /;98+ #L. . . . +-"$.%0/ 1 ! ) ' 8! ) !Y% !#" ). SO(3, 1). bN!</0 #9;996/;98+ PO 6i;9K0 98+s/%'! # #98+j76i 6(j>6"!57;98+j>6 $K"!56; s9 *8+ 8:;G;9 i8:>Y*!598'!5>; %'!5;98| Yp8+#9; 2%:%+ 8:;I / ;9 8:>Y*!598'!5>; "!5G%+i * 6 J s~\ 8:/E!5 *,zj768P0#9;G%gOt# 1R%+=0#G8+# 1;998+0#V %gOt0#9 "! /0|g; 12 #U uf8:EI#9p8gLkl%c0#9;I/0 12 J*# 2 0#G;9C!5#9%'!5;98+ # 0#G ;C!5;98+ #Vi%gOt0# "! /0*,j>68Jmy 91>;V%+i * 6 B SO(3) 0# # Isbc >;9 s%'!< "?p8+ fs%gOt0#9 "! /0 ;I@6& 7Y # 1>;i@6(; 12 #EL n78@%gOt 2/0 #98+) 62j>6"! @98:Y0/; 6 x , ;\# <;9C!5#.my 91 x "!56# D71;998+S@6< * 6 BSS~F 8:/E!5 *, %'!<#9Dp1;998+i/0 # 9YU%+0#G@8+#9;C!5/00#S8:@8:; 0#98:14!5%+0# dx dx = dx dx ,p;S%gOt J6;1 7;9 E j76Z ? L@ dx dx = dx dx ⇔ x = Λ x + a x et g Λ Λ = g g 0#9;s%+Z; # 6s1;998+j76*Lcbc0# 14!5;998+/00# Λ my 91>; %+Z * 6 J22bc 7;9,N/0 12 J*# 2 SO(3) , 0# # V<bc >;9,P%'! "?p8+ wPOt0#9 "! /0Z;i@6f 7Y # 1>;=@6v; 12 #ELBbc0#i;9C!5#9%'!5;98+ #i# 7; /0 7; >60#i!5#G%+s; 91 a L ~F 6 R; 8:i6! /;98+ w8:>Y*!598'!5>; #C 6#V%+< * 6 JZ<~\ 8:/E!5 *,P8:%o#96 ;i< @ Z6<# 8:; %'!5 *C!5 *8+&#C/E!5%'!58: (# 6#<%+ * 6 J(bc >;9,o/E!52%gO 8:7Y !598'!5/0 "!5R2;9C!5# %'!5;98+ 0#9; E!5%:8+#C0(#98:1Z| %+17;=f8:7; *C!5>; %gOQ! /;98+ A# 6V; 6;i%gOt0#9 "! /02<u 8:I#9p8S S = Ld x LBuf;9; #sZ/0H ; < J 6 %gO 8:#9;C!57;Z%'!& "?@8+ rPOt0# "! /0 { 6r "!598:; ;=%+ >Y # 17;@6q; 12 #E,\j768# 7;;9C!58:; 0#Z!5#<%'! # 0/;98+ ? DL C@LV6r# 8:]@6r * 6 B3bc >;9,%gOt# 1R%+(0#< ;C!5;98+ #2 SO(3) ,0# # 3bc5| bc 7 ;937;9;< 2 %+ 6* ,"#= ; p@68:; #smy 91LB > ;< *6 q6#C B I6-.#. |g *6 s 66 B H {%+ 4/0>; ;9C9!512%\8:"&!5 >K>; D@det#98+j76Λ==0#G1 ",F!5!59;9 8+ J/6%+%+0#E *Lz o f6 J4P;E , %+0#G R@-9; #Sj>68J @ SO(3, >;I%+0#1) 0# 7;C!5;98+ #x@6 * 6 JVibc 7;9 U B91;9; 7;IU/0 # ;9968: 0#Z8:7Y !598'!57; ## 6#Z/03 * 6 B*,j768# 7;Z6;98:%+0#Z B 6<%'!v/0 #9;996/;98+ 0#ZbN!5 *C!5 *8+#4 %'!f;9K0 98+ j76"!5>;98+j760#V/ K"!512 #U0# j76%+#I; 6;V0/0 6%+*L J6;I1 >;9 E j76s%gOQ!5%: )R =bP8+ SO(3, 1) 0#9;s0j768:Y !5%+7; $36@?v!5%: )R 0# ZbP8+Z SU (2) J%:%+Z0#9;s /Z/E!5C! /; 98+# 0Z "!5s6@?v128|7;98+ # 6 6@?(>;98+ # j ; j , >;V%'!Z# 121 0#9;x!5 J%+0s%+ #9 8:&i%'!2 0# >;C!5;98+ ] %+ #9 8:(Y*!56; 0 #98 j = j = 0 Lz %'!2/0 9 0# B $<%'!2 0# >;C!5;98+ f;998:Y78'!5%+ SO(3, 1) %+v# 8: Y !56; #98 j = 0 ; j = 0# 7;C!5;98+ e@8:; , 0 6 # 8 j = ; j = 0 0# >;C!5;98+ w@8:; 0, L bc0#Z14!5;998+/00#2 ;9C!5#.my 914!5;98+ , 0 ; 0, # 7;Z 0#9 B0/;98:Y17; M (θ, φ) = e ; Lbc0#3 R@-.; #3j>68i# (;9C!5#9m{ 917;3 "!5 M ; M #C >; 0#3#9 8:6 #E, R@-9; # M (θ, φ) = e $ 6@?[/0 12 B*#C!57; 0#/0 12 %+?0#ELFhi!5#%+0# ;9K0 98+0#8:>Y*!598'!5>; 0#Z "!5= "!598:; *,N%+0#6@?A 0# >;C!5;98+ # #EOt0/CK"!5 >;E,z;%+0#G6@?;Dp J0#s# 8:6 #S8:>; 9Yp8+>;VVmgY! X0 	Dp1;998+j>6*L (%+0#S * 6 J !5%+ # # 6#Gm{ 91 PO 6fj>6"! @96 %+;E,6 R8+#9 8:6= ? DL C ψ ψ= .
(40).
(41). .
(42). .
(43). µ. 0µ. µ. µ. . . 0µ. µ. 0 µ. 0α. α β β. µ. 0µ. α. . 0 µ. αβ. α µ. β ν. µν. µν. α. 4. . .
(44).
(45). 2. 1. 2. 1 2. ~ ~ ~ σ ~ σ −φ· +iθ· 2 2. 1. 2. 1. 1 2. 2. 1 2. 1 2. 1 2. 1. 1 2. 1. 1. L. ψR. 2. 1 2. 2. ~ ~ ~ σ ~ σ +φ· +iθ· 2 2.
(46) . , !5 B%+#9 8:6I >!56/CK*,"; ψ ,"!5 B%+=#9 8:6x@ 8:;E,"#Ci;9C!5#.m{ 917;V 0# B0/;98:Y17;x "!5 M o; 2 Y*!5%+ / #IKP*,>O 6#98 %'!V# "D7!5198:; ;9G98+c Ot0#9 ;SO(3, "! #o 0#91) J0L /; 0*,*/0 121G/*Ot0#9;o%+S/E! #X!5#%gO 8:>; C! /;98+ 4my!58:R%+ Y 8:X#C0/;98+ ? L F L @ ,@!5%+ #%+0#6@?;Dp J0#V#9 8:6 #-9 6>; "! #%+I11V H %+U!5#%'! ;9K0 98+*,7/*Ot0#9;.|l$ |@8: Uj76cO 8:%+# #C >;I "! #I/0 6 %+0#Vi%'!Z11s14!58+) <!56@?(!56;9 0#V/ K"!512 #xi%'!2;9K0 98+*L x= 98+; 8:7;998:# )0j76 my !51>;C!5%+20#V "!59;98+/6%+0#s0#9;x%+6i#9 8:cLzk%F B6;i;9 =128+#Uf/0 9 0#9| J !5/0s!0Y0/i/0%:68P0#@8P >; 0#S 0#C>;C!5;98+ #I SO(3, 1) LpV8:#98g,7%+V/CK"!512 (6;98:%:8+# V B 6S0/98: /CK"! j>6G;lD7 JGG "!59;98+/6%+GY*!i# ;9C!5#.my 91@8P 121>; "!5o6I;9C!5#.m{ 914!5;98+ 4 IGbc >;9< %+0#i "!59;98+/6%+0#s<# 8: 0 ,Jj>68o @ 7; %'! 0#C>;C!5;98+ A;998:Yp8'!5%+Z SO(3, 1) ,P#C >;=! # # p/8+00# $6q/CK"!512 q# /E!5%'!58: bc >;9 Φ(x) j768g,c# 6# %gOQ! /;98+ rPO 62;9C!5#9m{ 914!5;98+ Λ ∈ SO(3, 1) , Y78+7; → Φ (x) = Φ(x) %+0# "!59;98+/6Φ(x) %+0#SU#9 8: # 7;S! # #C7/8+00#S$=0#/ K"!512 #GVR8+#9 8:6 # {8:;98+ DL C o J 6%+0# ;9K06@ ?f98+!506#X;98: >0Y*#U!59/E8'!5!5; >; 0 #98+ "0!5#o," "%+0!5#V98: "; !5*9L7;9hi8+/!56%+0#X#U%+I >!5/E6! #X/ K0G#i;9K;V0 @9 8+ 08:#; 0#E ,J! # 8:># pY*!5/8+908'!50#V>ψ; $40# "0#V!5X# "8:!598:; 6* ,?#V >4!56@8+/C# K;908:# *6 ψ ;x@ 8:; # ψ L %+0# "!59;98+/6%+0# w#9 8: 1 ,Gj>68s @ 7; %'!q 0# >;C!5;98+ m{ !517;C!5%+A SO(3, 1) ,I# 7; ! # # p/8+00#x$<6&; # 6V!57;98+#9Dp1;998+j>6*L . ψL M2. R. 1.
(47). 0.
(48). 1 2. L.
(49). R. . . +-"$.%0/ 1 ! ) '1 ) / %c! ). C P T. k7; 0# # #.|g 6#14!58:>; "!57;V$=%'!= "!598:; ;S!563 >Y #C1>;x@6; 12 # ,p#9D71;998+0#S@63 * 6 B IbN >;9Vj76 6#!EY #o128+# 0#XG/0H ; I!5P#X%'!U "!59;98+G 0/00>; *,p!58:#98j>6cOQ$UT6I;9 8+#98+)1I#9Dp1;998+ @8+# / ); *,J%'!3/0 -.6 >!58+# A2/ K"!59 C LJ 0#i;9 8+# # D71;998+0#s# 7;i%:8+00# $4%gOt0#C# /0Z11ZZ%'!4;9K0 98+ j76"!5>;98+j76S0#F/CK"!512 #E,*$G;9C!0Y #\%+;9K0 )1 CP T LhiX %:6#,%+ 0#9 B0/; 6=%'!xYp8+ %'!5;98+ </00#F# D71;998+0# 7;P-. 66Z H %+my !51>;C!5%"!5#o%gOt;96G;%'!i/%'! # # 8"/E!5;98+ 0#8:>; C! /;98+ #E,/0 121I 6#%+Y9 # !5#x%'!Z# 0/;98+ ? L F L @ L 465 I584. =0K OQP0D 6KLT8HI:#=fD 6?Y?YT8HI:#=
(50) . . MN 6;GPOQ!5RB P,p8:7; 0# # #.|g 6#I!56/E! #%+x %:6#G#98:12 %+*,p/0%:68Px%'!10/E!58+j76i/%'! # #98+j76*L7~\ 6G/0%'!@, 0/ #98+ #V6	D@#9; )1i K>D@#98+j76 S 14! / *# /0 8+j76 s14! #C# m , RB8+#C#C!5>;x!56& 98:/8: Jimy !51>;C!5% U%'!@D7"!5128+j76*L 3 J6;/E!5C! /; 98+# I#!s J*#98:;98+ 3!5#%gOt0#9 "! /0i;%+x; 12 #G$=%gOQ!58+iV# 0#/0p @| 00#U#9 "!5;98'!5%+0# ~x ;x ; 12 J %:%+ t L v8:;x%gOt JC!5;98+ vs "!598:; P /0 121=;C!5>;U%'!Z;9C!5#.my 914!5;98+ /CK"!5 E!5>;%+0#/0p 00#POt0#9 "! /0 ~x −~x L*o%:%+I0# ;0j>68:Y*!5%+>; V$i%'!U "?p8+ 3!5#o6128: 8:# 68:Y78+ P[O 6 2 ;C!5;98+ cLNo%:%+2;9C!5#.my 912 /2%gO 8:12 6%+#98+ p~ ≡ m ;s%+Z1 1>;/8:;98+j>6 ~l ≡ ~x ∧ p~ S ? L E& (~x, t) → (−~x, t) C P T. d~ x dt. p ~ → −~ p ~l → ~l. ;9; Z#9Dp1;998+= J91;i/%'! # #CV%+0#UY0/; 6 #iv6@?v/E!5; 98+0#E,J/0 121%+=1 >;9 Z%gOt# 1R%+Z0# %'!5;98+ # ? L E&2N%+0# Y0/; 6 #< J %'!58: 0#j768/ K"!5 7;24#98: *4 "!5= "!598:; *,\;=%+0# Y0/; 6 #2! ?@8'!56@? { 6 # 65|gY0/; 6 # xj768N 0# ; >;U8:>Y*!598'!5>; #EL J6;i;C!5R%:8:V6/%'! # #98"/E!5;98+ v8+>;98+j76</0 /09"!57;i%+0# p@68:; # #C/E!5%'!58: 0#E,j768I J6Y>; ;9 f/%'! # # 0## /E!5%'!58: 0#4#O 8:%+#4# 7;48:>Y*!598'!5>; #3# 6#4 "!598:; p@68:; # /E!5%'!58: ii6@? Y0/; 6 #I! ?@8'!56@? 6(6@?4Y0/; 6 #S J %'!58: 0# 6( # 65|#C/E!5%'!58: 0#I#EO 8:%+#S/ K"!5 7;x #98: * "!5I "!598:; p@68:;x# /E!5%'!58: PO 6&Y0/; 6U! ?p8'!5%c "!5G6 Y0/; 6I B %'!58: L bN!Z "?@8+ @6&; 12 # ;9C!5#9m{ 91=%+s; 12 # ,@%'!58+# #C!5>;x8:>Y*!598'!5>; 0#U%+0#I/0p 00#VPOt0#.| "! /0*,;x "!5I%'$<11 / K"!5T E!57;V%gO 8:12 6%+# 8+ p~ ;G%+ti1 −t1>;U/8:;98+j>6 ~l (%+6x B*#C2 ? LF (~x, t) → (~x, −t) p ~ → −~ p ~l → −~l.
(51) ;9; Z#9Dp1;998+=/0 9 0#9 J w!56f 7Y # 1>; @6v# #V@6v1 6Y1>;ELPbO 8:>Y*!598'!5/0 "!5 T #98: *8"< "8:!58:9;9;98'!58+/%P6@%:8+6<# 0j>/06 Z3%+0 #i ;C7!5/06@0? # # 6#0#s x myp!Y/0X00 # 3#969#8: a 6→ 6b# 1; b7→;I8+a># ;9 8+>j7; 6s%+0$ #i%g1Ot;C1!5;G0#!5#98 #b%+00j># ;s6!5%B98:C%J!50#9 ;4 p!5@6#s8:%g;SOt%+;C !5 ;# @6& 128+x 7/00# # 6#EL bN!A/0 -.6 >!58+# / K"!59 ;9C!5#9m{ 91 %'!A/ K"!59 w%+0/;998+j76 # J*# *Lo%:%+&%'!58+# #C 8:7Y !598'!57; #x%'!Z14! #C# *,@%gO 8:12 6%+#98+ fC;G%+ 1 1>;V/8:;98+j76 @6	D@#9; )1 S L S 465 I5[Z. =0K OQP0D 6KLT8HI:#=fHI: 6K @ST7HI:#=hK#_ K Rb= @bT #T8?A@S=
(52) . . . . #98+ #(0# 914!58+#(6v "!59;98+/6%+ v14! # # ;(POt9 *8+A# 6 4#C!5121>;mg!58:R%+v B 6(j76 0/ ;9; "!59;98+/6%+&# 68:Y %+0#%+ 8+#2 %'!v10/E!58+j76(j76"!5>;9m8+j76 %'!5;98:Y78+#9; *L ;9; "!59;98+/6%+ 0#9;!5%+ # /E!5C! /; 98+# 0Z "!5U6m{ /;98+ POt ψ(~x, t) ,z 6w "!5U%+=Y0/; 6sPOt;C!5; |ψi !5#i%gOt0#9 "! /0<<TV8:%:RJ9;E, ;IY98" %gOt0j>6"!5;98+ v=np/ K o¨@8: = ? LG ∂|ψi i~ = H|ψi ∂t n78%gOt JC!5; 6 O 0# >; Z6<#9Dp1;998+*,z%gOQ! /;98+ O < 8:;U "! #U7;9C! + /CK"!5 1>;=!5# 6s10# 6 *,;x 8:;I /sY98"< ? LS |hψ|O O|ψ i| = |hψ|ψ i| /*Ot0#9;.|l$ |@8: ? LH hψ|O O|ψ i = hψ|ψ i 6 hψ|O O|ψ i = hψ|ψ i ? LI 0#9;x@8:;G68:;C!58: #EO 8:%PY98"s%'!Z 128+) %'!5;98+ c,";x!5>;98|g68:;C!58: <#EO 8:%PY98"=%'!Z# 0/0 *L O C P T. . †. . 0 2. 0 2. †. 0. 0. †. 0. 0 ∗. (;$<%'"!. bcsTUP!5128:%:; 8+ H s%gOt0j76"!5;98+ np/CK o¨@8: V#EOt0/98:; H=. . |P~ |2 ~ + V (X) 2m. ? L ? J . 0#9;s%gOt JC!5; 6 B*# 8:;98+ 2; %gOt JC!5; 6=8:12 6%+#98+ cLFn78 0# ;s6# %:6;98+ [2%gOt0j76"!5;98+ Xnp~/ K o¨@8: E,\!5%+ #=%'! "!598:; 30#9;=P~/0 #C9Y04#98 P|ψ, ti 0#9;<!56# |ψ,# 8X6ti4# %:6;98+ cL ;9; 4/0 @8:;98+ ]0#9; #C!5;98+#9my!58:; v#98 Lkl%xmg!56;4 J 63/0%'!rj76 # 8:;3# D71;998+j76 "!53 "!598:; *,Sj>6 P # 8:; 6 JC!5; 6x6[P,8:;C!58:H] P= 0= P S;G%:8:E!58: P iPV =(X)iL # 6 "# !59;98:LZbN@!Z6AY*!5 %+98:6I/8:1 B ED4/0 9 0i#9 J%gOt BC!5!5; /0*6,\V BA*# B8:;98+6 ; ;9 6Y 8:;x/0/CK" 12!51 V7;s%+0#9#8: * R # "9!5Y G!5R "%+!509#<8:; # 4 /0 12 J 9; >; ~ P X ? L ?K? ~ ~ hψ|PXP|ψi = −hψ|X|ψi /0 j768c0#9;x! # # 6 s#98X ? L ?@ ~ = −X ~ PXP 6 smyY! X0 &0j768:Y !5%+7; 4 ? L ? C ~ P} = 0 {X, {a, b} = ab + ba 0# >; =%gOt BC!5; 6UPOQ!5>;98|/0 121=6;C!5;98+ cLNn78N%gOt /0 #98+) 6f;C!5;I |~xi X ? L ? E& ~ xi = −PX|~ ~ xi = −~xP|~xi XP|~ 0#9;GY0/; 6I X~ iY*!5%+6I −~x L f! P|~xi ? L ?F P|~xi = e | − ~xi . . †. −1. −1. . iδ. . #( ' . ,-$# <>$#% ! ' -,. ) 8 "!A' . - !%
(53) ' ! /:.. P. 2.
(54) f B6;I/CK 8+#98:x%'!2/0 7Y>;98+ ws K"! # e = 1 ,img!YX0 f$2/0 j76 P = 1 ,/0 j768N#98: *8" j76 P 0#9; $Z%'!my 8+#GK9128:;98+v( %:6#xPOt;9 s68:;C!58: *,;V# 0#GY*!5%+6 #G 0#x# 7; ±1 L bXOt BC!5; 6x8:12 6%+#98+ P~ 0#9;G%+ C!5; 6V0#G;9C!5#9%'!5;98+ #x8:@8:; 0#98:14!5%+0# d~x s%gOt0# "! /0 ? L ?G i T (d~x) ' 1 + P~ · d~x ~ VS;9C!5#9%'!5;98+ # 68:Y78+SPO 6G;9C!5#.my 914!5;98+ 4S "!598:; I0# ;X0j>68:Y*!5%+>; x$U6I;9C!5#.m{ 914!5;98+ S "!598:; #968:Yp8+ PO 6s;9C!5# %'!5;98+ !5#I%+s# #x B*# 2 ? L ?QS PT (d~x) = T (−d~x)P / ? L ?H i~ i~ P 1 + P · d~x ≈ 1 − P · d~x P ~ ~ POt [ ? L ?I {P~ , P} = 0 bc<1 1>; /8:;98+j76 0#9;U%+ C!5; 6 0#U ;C!5;98+ #ELP~X68+# j>6<%'!4 "!598:; 2;i%+0#V ;C!5;98+ #s/0 1Z| 16; >;E,J!5%+ #G "!598:; =;G1 J~17;V/8:;98+j76 /0 121=6; 7;i!56# # 8X ? L @KJ [J~, P] = 0 . iδ. 2. . '.P9 * +( %'Q. 9 03 #:(;$ . Nb !w#9Dp1;998+((/ K"!59 CC ;9C!5#.my 91*, %gOQ! /;98+ %gOt JC!5; 6 C ,6 "!59;98+/6%+& #C !5@| ;98: "!59;98+/6%+3Z1114! # #C*,J1 1>;/8:;98+j764; # 8:c,P14!58+#=!0Y0/ 0#s 1=R 0#<j76"!5>;98+j760# {/ K"!59 %+0/;998+j76*, 1=R %+ ; 8+j76G 6R"!59D 8+j76*, F B*#C0#EL bN/0 6 %'!5 S%+0/;9 14!5 *;98+j76G128:8:1=61 !5#x%+sTU!5128:%:; 8+vs%gOt0j>6"!5;98+ f=np/ K o¨@8: x0# ;= ? L @>? |P~ | e ~ ~ ~ + e A ~ + eφ H= − [A · P + P~ · A] 2m 2mc 2mc e 0# ;U%'!/CK"!59 %+0/;998+j762Z%'!3 "!59;98+/6%+*, A~ %+Z J ; >;98+%XY0/; 6; φ %+Z J ; >;98+%X%+0/;998+j76*L H 0#9;xR8+(8:>Y*!598'!5>; {/*Ot0#9;.|l$ |@8: [H, C] = 0S "!5G%'!Z;9C!5#.my 914!5;98+ q ? L @K@ ~ → X ~ X 2. 2. . P~. → P~. e → −e ~ ~ A → −A. ; C 8:;;9 %:8:E!58: *LF~!5<!58:%:%+6 #, C = e ,c; q B6;</ K 8+#98:<%'! K"! # 4>6%:%+34# 9; 3$ !EY 8: LV%+ # C !Z6@?(Y*!5%+6 #G 0# ±1 L C =1 φ → −φ. 2. i2δ. 2. " 9POPN$< 9P"
(55) 0 * " 3. h<OQ!5 )0#I%+s 98:/8: B=s/0 9T 0#9 B !5/0*,"%gOt BC!5; 6 T 8:;GY98"<. ? L @ C. ~ = 0 [T, X] {T, P~ } = 0 {T, J~} = 0. bN! %'!5;98+ /0 121=6;C!5;98+ 0#& JC!5; 6 #& J*#98:;98+ ;(8:12 6%+#98+ 0#9; LGhZOQ!5 )0# %gOt0j76"!5;98+ ? L @ C,\%'!& "!59;98+3 >!56/ K/ K"!5 (#98: *3# 6# T L ] 8:;Z[X ,/(P !E]Y =8: i~δ , /*Ot0#9;.|l$ |@8: j>6 T 8:;x;9 =!57;98|g%:8:E!58: *,z/0sj768c0#9;I%+s/E! #I# 8 T 0#9;x!57;98|g68:;C!58: *L T iT = −i i. j. ij −1.
(56) 465 I5. . =0K @b] P,_`RUT8=iHI: 6K @bT8HI:#=he#=c?;DU] 6O . . . ?. 121I 6#X%gOQ!EY #- $ Ypj76*,*%'!i10/E!58+j76Vj76"!5>;98+j76x/%'! # # 8+j>6GcOt0# ;X "! #/0 12 "!5;98:R%+U!0Y0/V%'! %'!5;98:Yp8:; < 0#9;9 8:>; *LBIw=0/98: <%+0#V "!59;98+/6%+0#i%+1>;C!58: 0#sv8:>; C! /;98+ c,B%'!;9K0 98+<j76"!5>;98+j76 0#/ K"!512 #I!=; I8:>;9 p@68:; *L@hs!5#/0V/E! @ *,>6V "!59;98+/6%+xcOt0# ; %:6#0/98:; x "!56Gmy /;98+ POt /0 1213[10/E!58+j76 /%'! #C#98+j>6*,c14!58+#= "!56q/ K"!512 ]j76"!5>;98+j764 >;<%'!("!5;96 Bq@6q#9 8:r /0;9; = "!59;98+/6%+# %+ (%'!Z"!5;96 = /0 / K"!512 c,%+0#x JC!5;98+ #xs#9Dp1;998+scOQ!5 *8+# # 7;x "! #I %'!<11 14!58+) *L~!5x/0 # 0j76>;E,%'!Z/0 #9;996/;98+ 0#I JC!5;98+ #Is#9Dp1;998+s 8:;I# Umg!58: V J 6I/ K"! j76s;lD7 J /CK"!512 cL"o%:%+=0#9;G%+ *6 ;Gmg! #9;98+@8+6# *,"!56#C#98P 6#G #V!5#I/0;9; # 0/;98+ j76i%+0#I 0# 6%:;C!5; # G%gOQ! /;98+ C , P , T #96o%+0#X@8P >; #/ K"!512 #E,#!5#J-6# ;98"/E!5;98+ cLxS Yp6G;C!58:%:%+0I0#9;X@8+# B 8:R%+ !5# L\bXOQ! /;98+ 0#Z#9D71;998+0# C , P , T J 6%+0#<@8J 7; #Z;Dp B0#<3/ K"!512 0#9; 0#9610!5#%+ ;C!5R%+E!56F ? L ? L K"!512 F PF P CF C TF T CPF CP CPTF CPT C P T. . †. S + (t, ~x) P + (t, ~x) ψ + (t, ~x) Vµ+ (t, ~x) A+ x) µ (t, ~. †. S + (t, −~x). −P + (t, −~x) γ0 ψ + (t, −~x) V +µ (t, −~x) −A+µ (t, −~x). †. −1. S − (t, ~x). S + (−t, ~x). P − (t, ~x) iγ 2 γ 0 ψ¯−T (t, ~x) −Vµ− (t, ~x) A− x) µ (t, ~. P + (−t, ~x) iγ 1 γ 3 ψ + (−t, ~x) V +µ (−t, ~x) A+µ (−t, ~x). S − (t, −~x). −P − (t, −~x) iγ 2 γ 0 ψ −T † (t, −~x) −V −µ (t, −~x) −A−µ (t, −~x). S − (−t, −~x). −P − (−t, −~x) iγ5 ψ −T † (−t, −x) −Vµ− (−t, −~x) −A− x) µ (−t, −~. L McC!5#9m{ 914!5;98+ PO 6&/CK"!512 "!5S%+0#G#9D71;998+0# , , , ; {U >!56/CK=$ @ 8:; V# %+ ? v? #C!2"!5;96 3"# /E!5%'!58: *,z # 6*# /E!5%'!5F8: *,B#9 8:6 #i=bN C>;9P,BYT0/; CP6s 6 CPY0T/; 6s! ?p8'!5%{ K"!56;x R"! # Lbc0#I; 910# γ # >;G%+0#G14!5;998+/00#UshU8:C! /*L
(57). µ. 465 I5. \^] P,_`RbacOd=. . bN!3;9C!5#.my 914!5;98+ q/0 1=R8:0 , , !36A#9;C!5;96;=; 6;s "!59;98+/6%:8+A;9K0 98+j76"!5>;98+j760# C/ K"!512 #EL"o&P;E,@ 6#IY9 #x!5#GC%'!ZP# 0/;9T8+ ? L F L @ j76i%+0#G#9Dp1;998+0# C , P , T J6Y>;x;9 Y78+ %+00# "!5o%gO 8:7; C! /;98+ mg!58:R%+*L5X2 Y !5/CK*,*8:%?p8+#9; 6SC!58+# 2;9K0 98+j76S 5my Gj>6812 J0/ KG%'!VYp8+ %'!5;98+ V%'! #9Dp1;998+ p/*Ot0#9;%+x;9K0 )1 G ,7j>68B#9;98: 6%+Vj76I; 6; V;9K0 98+Uj>6"!57;98+j>6U0#/CK"!512 # %+p/E!5%+*,B8:>Y*!598'!5>CP; 2T<bc >;9;s!0D>!57;s6CPwTiT !5128:%:; 8+wK9128:;98+A 8:;U;9 <8:7Y !598'!57; Z "!5 CP T LBo C!58+# ]4/0 ;9K0 )1*,F8:%0#9;;9 )0#@8 4/8:%+4/0 /0Y 8:Z64;9K0 98+ 0#</ K"!512 #Z %'!5;98:Yp8+#9; E!5%:8+# ; !5#x%'! j>6%:%+ CP T # C!58:;IYp8+ %+0*,!56# #98c/0 #98+) |g;.| wj>6i%'!Zi!5;96 /0 # 9Y
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