!#"%$'& ∗
(*),+.-0/213- 4657989:<;>=@?BADC E%CF/HGI5J=#-0/K8L+,:ME<NPO † Q ;SR,-T5.U ?VC R W<?YXZX[?]\]^F/_^a`
b /HGI5C R98cNd8 Qfe ?Y^F/[- E%Cg/KGh5J=i-j/k8l+9:nm]RgXZXporqaqts
∗ u9vwjw0x9y{z|v~}€‚ƒw.„{…‡†ˆ‰vŠŒ‹
† …0…j…z|vT}‚ƒw.„Ž…0†ˆ‰vŠ‹
! "$#%&' () * +,-# .*/102354
6 7 8:9&; ;$<9 =>7@??A7B CEDGF CED H9JIK8H9LM;NCAO9*P CQ; CR8 <9J?H=TS*?U;V7 IW95X
OCY9$B B <Z); <C[F@< 9JD9JI\F^] H*<]_8MC[L+8 B Z8 ?`CEa'9 CED ;$<9+8:9 ZbDLCY9JD);
;NCEc 9+8d e <CE?A9 Df7^; Z_8 L)7Uc c 7@Db?g] S_8:9h Z_8 Df7_B c ZbD)] i
c 7jh*9JI,D)k ;V7j7@?l8 ?mCna'9 ;$<9 o9]bDbc ZbD H%Z+;$< CED);V9F@IpZb?qZbDh ;$<f9
D7B Df9GZI,?g] =r7sI\F7^;T;V9JD tqLJ<+B CEDGF9I u LM;NCY7UD v I>CEDGLCEHb?A9 B 9Iw9
ZO%ZCE?[Zbx*?R9bd 6 <f9 L)7Uc c 7@D 8y;zZI{;NCnDF H7|CED); =>7I ZbD)] ;$<97sIW9$;}X
C[LZ*?sh|CR8~L^S_88MCY7UD B Z8 S8S%Z*??€] ;$<bIW7@SGF@< H9JI;$S*I,x%Z+;NC[7@D ;$<97sI]
7sI i}t c Z);$I>Cl‚ ;$<97sI])kNd
6 f< 9 H7B 9JI 7J= ƒ IW7@SbH 6 <97sI] x9]7@DGh t„ …†‡ =>7I ˆ C[F@<
9Df9IWF^] ;$<97sI] B Z8 …>;$<9JD ;$<bIW7@SGF@< ‰ŠZ)OG7I t„ …‹s‡Œ‡ B Z_8
 S8y; x9F|CEDbDJCEDGF ;V7 x9 Z*HbHIW9LCmZ);V9hŽd 6 <f9
D99h*9h ;V7
Ω−
 ?? CED ;$<f9 x%ZI]7UD h*9LJS*Hb?A9$; …~f‘ H%ZI;NC[LJ?A9+8’‡ B Z_8 =>7@SbDh CED
f“*”‹ŽP 6 <9 ƒ 9??•X5– Z*DbD —˜B 9%C[F ™@S%ZI,a …šZGL)9‡ C[h*9Z_8 9$‚*CR8y;V9h
xbS+; B 9JIW9 =,ZI =TIW7@c L)7Uc Hb?A9&;V9J?g] ZLL)9H+;V9_h›d 6 <9JIW9 B Z_8 D7
9&‚H9JI>CEc 9JD);zZ*?9&OCœh9DL9 Z*DGh ;$<9 L)7U?A7I C[h*9Z_8 ;$<%Z+; Zb??A7B 9h
;$<bIW99 ™S%ZI,a&8 ;V7 c Zba'9 Z D*SGL^?A97@D 7@Db?g] x9FsZ*D ;V7 ;zZba'9 =r7sI,c
CED f“*”Jž i Ÿ dUe d1ƒ IW99JDbx9JI\F›P v <)]f8d ¡ 9&Od ¢"9$;T;)d ‹›P¤£b“¥
…~f“*”Jž|‡ k ¦ Zb?RL^Sb?[Z);NCY7UD c 9$;$<7hJ8 B 9JIW9 HI>CEc Cl;NCQO9d o7I ;$<f9
c 78y; HZI;+P"L7@SbH*?`CEDF L7@D_8y;zZ*D); HJ9I;$S*I,x%Z+;NCY7UD B Z8 ;$<9 7@Db?g]
;V7j7U?§ZO%ZCE?[Zbxb?A9 ;V7 ;$I] ;V7 F9&; ™@SZbD);NCl;zZ+;NCAO9 Z*D_8yB 9JIK8 =šIW7Uc
™@SZbD);$S*c  9?Rh ;$<97I]+d t – Z);$I>Cl‚ ;$<97sI] B Z_8 aGCEDFŽd ¦ S*I¨X
IW9JD); Z*?©F9xIªZP*Z c Cl‚;$S*IW9 7^=q8«]bc c 9$;$I] Z*DGh 8:7Uc 9 h^]bD%Z*c C[L8
B Z8 CED ;$<9 B CEDF8 …~¬ Z*c x*S ­ ¢ŠS*I>CY9 f“b”UG‡&d
¬ Zbc xbS ?[ZbS*DGL^<9h ;$<f9 8y;$SGh%] 7^= 8&H7UD+;zZbD97US8 8«]bc c 9$;$I]
xIw9GZ*aGCEDGF 7^= ZbD CED);V9JI,D%Z*?ŽFIW7US*H ;$<bIW7@SGF< <CR8 B 7sI,a 7UD ;$<f9
¦ t c 7h*9J?…~f“*”b‘‡ ZbDh ;$<9 ¬ Zbc xbSP7UD%Z&XN¢Z_8MCEDCY7 c 
7 h9?
B Cl;$< CnD+;V9IpZGLM;NC[7@D
h
2
g (ψ̄ψ) − (ψ̄γ5 ψ)
2
i
6 <JC©8 CED);V9JIªZGLM;NCY7UD CED CQ;«8Œ9?g= B Z8 h"CR8y;$SbI,xJCEDGF …šZ_8 B Z8 ;$<f9
>= 7@SbI =>9JI,c CY7UD CED+;V9IªZL$;NCY7@D ;V7 h9)8:LJI>CEx9 B 9GZ*a HIW7L)9)8V8Œ9)8’‡
x9LZ*S_8:9 HJ9I;$S*I,x%Z+;NCY7UD ;$<f97I] CED HIW7jh@SGL)9+8 Z 8Œ9I>CY9+8 7J= CEDX
g
LJIw9GZ8MCEDGF@?g] h"CQOG9JI\F*9JD); ;V9JI,c 8 ;$<%Z+; LfZbD*D7%; x9 IW9Df7sIc Z*?`C)9hŽd
¬ Zbc xbS&X¢ 8y;$SGh"CY9_h ;$<CR8 c 7h*9J?jxN] CEc H78MCEDF Z L7@D_8y;$IpZCED);
;$<%Z+; 8:99c 9_h ;V7 x9 CEDL7@D_8MCR8y;V9JD); B CQ;$< Cl;«8 8«]bc c 9$;$I] ZbDh
;$<9JD =r7sIc S*?œZ+;V9_h Z D9$B …VD7^; L7@SbHb?mCEDGF L7@D_8y;zZ*D); HJ9I;$S*I,x%Z&X
;NCY7UD ;$<97I]^‡ ?A9GZh|CEDGF 7I\h9I ZbH*HIw7_‚CEc Z+;NC[7@Dd 6 <f9%CEI IW9+8Sb? ;«8
8<7B 9_h ;$<%Z); ;$<9JIW9 B Z_8 Z )9JIW7 c Z8V8 L7@c HJ78MCl;V9 H%ZI;NC[L^?R9
9&‚LCQ;V9h xN]
2 6 <JCR8 C©8 D7B LZ*??R9h Z ƒ 7@?©hJ8y;V7UDf9 xJ78:7@Dd
(ψ̄ψ)
e 9 LZ*D F*9$; EC D_8MC[F<); ;V7 <f9?H SbDGh9I 8y;zZbDh ;$<f9 C[h*9Z_8 7J=
8H7@D);zZbDf97@S_8 8«b
] c c 9&;$I] x IW9ZbaCEDGF CED ™@SZbD);$S*c  9J?Rh ;$<97sI]
x5] 8y;zZI;NCEDGF B Cl;$< 85Cnc Hb?A9 |
h C
9IW9JD);NCmZ*?9™@S%Z+;NCY7UD_8d ¦ 7@D_8MC[h9I
d3 y
dy
g 3 + m2
=J .
dJ
dJ
M# $bf3N4
*
6 <bIw99 6 f< 9] LfZbD x9 =r7US*DGh =šIW7Uc Z nC D+;V9_FIªZb?%IW9HIW9+8:9JDX
;zZ);NCY7UD … )9JWI 7 h|CEc 9JD_85C[7@D%Z*?Šo"9&]bDbc ZbD %H Z+;$< CED+;V9_FIªZb?[‡
•
Z=
Z
Dφ e
−g
φ 4 m2
4 − 2
φ2+Jφ
•
6 7 8’Z);NCR8«= ] ;$<f9 h"C
9JIW9JD);NCmZb? 9™@SZ);NCY7UDP ;$<JC©8 CED+;V9_FIªZb?
c S8y; xJ9 9O%Zb?S%Z+;V9_h 7JO9I H%Z+;$<_8 B <C[L^< h*7 D7^; L7@DX
;$I>CExbS);V9 Z); ;$<9 9JDGh H7|CnD+;«8d ; C©8 9Z_8«] ;V7 8&<7B ;$<Z);
;$<9JIW9 ZIw9 ‹ Z*??A7B 9h CEDGh*9JH9JDGh9D+; H%Z+;$<_8 CED ;$<9 L)7Uc X
Hb?A9&‚ H*?[ZbD9bd
„ 8:9 ;$<9 h9  DJCl;NC[7@D
y(J) =
dZ
.
dJ
; =>7@??A7B 8 ;$<%Z+;
8’Z+;NCR8  9+8 ;$<9 7I>C[F|CED%Z*? h"C
9JIW9JD);NCmZb?
y
_9 ™S%Z);NCY7@Dd ¬§7%;V9
u 8V8Œ7LCmZ+;V9_h B Cl;$< ;$<f9 ;$<bIW99 CED);V95X
F@IªZ+;NC[7@D HZ);$<_8)P ;$<f9 CED);V9F@IpZb? <%Z8 ‹ 8y;zZ);NCY7UDZI] HJ7jCED);«8
;$<%Z); L)7sI,Iw9)8HJ7UDGh ;V7 ;$<9 ;$<*IW99 8:7U?S);NCY7@D_8 7^=U;$<f9 7I>C[F|CED%Z*?
h|C
9IW9JD);NCmZ*?9™@S%Z+;NCY7UDd
6 <9+8:9 ZIW9
?R7jLZ+;V9_h Z);
•
(J = 0)
•
v
u
u −m2
φ = 0, φ = ±{t
}.
g
; CR8 9Z_8«] ;V7 9&‚H%Z*DGh Z IW7@SbDGh ;$<9+8:9 8’ZGh*h@?A9 JH 7jCED);«8
V; 7 h"C©8:L7^OG9JI Z8«]c H);V7%;NC[L &9 ‚HIW9)88MCY7UD_8 =>7I 9ZGL^< 7^= ;$<f9
$; <bIW99 8:7U?S+;NC[7@D_8d
o 7I ;$<f9 H%Z+;$< Zb?A7UDF ;$<f9 WI 9Zb?›Z+‚*CR8 … L)7sI,Iw9)8HJ7UDGh"CEDGF ;V7 ;$<f9
"
8y;zZ);NCY7@D%ZI] H7jCED+; Z);
‡ Cl; =r7U??A7_B 8 ;$<%Z+; φ=0
• y(J)
CR8 ZbD 7hbh =TSbDL$;NCY7@D 7^=
J
y(J) = a1 J + a3J 3 + · · ·
C d!9bd
6
<CR8 CR8 ;$<9 =,Zbc Cn?mCmZI 8:7@?S);NCY7UD
y(0) = 0
H9I{;$SbI,xZ);NCY7UD 9$‚HZbD_85C[7@D ZIW7@SbDGh
d
g=0
o"7I ;$<f9 7^;$<f9I HZ);$<_8)P
C©8 Z c CQ‚ 7J= 7h*h
•
y
H7_B 9JIK8 7J= d ¤7^;V9 7UD*?g] ;$<9 8:7U?S+;NC[7@D ;$<%Z+;
CR8 Iw9_FSb?[ZI CED
Z);
d
Z);
J =0
g
g=0
6 <9 7^;$<9JI 8:7U?S+;NC[7@D_8 ZIW9 7J= ;$<9 i8«]bc c 9&;$I]
•
; ]HJ9&k’P xJ9_LfZbS8:9
y(0) =
Z
4
=>7US*DGh xN]
ZbDGh 9&OG9JD
O^Z*DCR8<9+8
xIW9ZbaCnDF
2
−g φ4 − m2 φ2
Dφ φ e
O^Z*DCR8<9+8 CQ= ;$<f9 x7@SbDGhZI] L)7UDh|Cl;NCY7UD8 ZIW9 L^<78:9D CED Z
8«]c c 9&;$I>C[L c ZbD*D9JI 8:7 Z8 ;V7 IW9+8H9L$; ;$<f9 9&OG9JD DZ);$SbIw9
7J= ;$<9 9&‚H7@D9JD);)d
ZGL^a ;V7
S%Z*D);$S*c o CY9J?©h 6 <97sI] •
o 6 CR8 h*9+8~L^I>CEx9_h xN] 7@H9JIªZ);V7I  9J?Rh8PJB <C[L^< 7@x9] ;$<f9
I,Sb?A9+8 7J= SZbD+;$Sbc – 9LJ<ZbDC[L8P"ZbDh Z L)7Uc Hb?A9$;V9 8:9$; 7J=
8y;zZ);V9+8d
>D HZI;NC[L^Sb?[ZI
SbSbc P
|0i
t Cnc Hb?A9 q‚JZ*c
•
tqLZ*?œZI o CY9J?©h
•
P
∃
Z 8y;zZ+;V9 7^= !b ’ 9Df9I\F%]
;$<9 O%ZGLNX
H*?A9
oŠIw99 =r7USbI h|CEc 9D8MCY7UDZb?Š¡ 9J?[Z);NCQOCR8y;NC[L
6 <f97I]P
S=
Z
d4 x
"
(∂µ φ) (∂ µ φ)
2
−
m 2 φ2
2
#
⇒ (−∂ 2 − m2 ) φ = 0 ,
• |0i
h7j9)8 D7%; h*9JH9JDGh 7@D 8&H%ZGL)95X;NCEc 9 H%ZIªZbc 9&;V9JIK8P
⇒ (−∂ 2 − m2 ) h0 |φ(x)| 0i = 0 ,
6 < 9 8y;zZ);V9Jc 9D+; ;$<%Z+; … µ CR8 ;$<f9 9Df9IWF^]X'c 7@c 9JD);$S*c
•
P
7UH9JIªZ);V7Iª‡
C d!9bdP;$<f9 O%ZGL^SbS*c
;V7
P µ |0i = 0 ,
<%Z8 D7 __f# 7I
_
?A9GZh8
h0 |φ(x)| 0i = h0 |φ(0)| 0i ≡ n
C d!9bdP§SbDJCQ=>7I,c Cl; ] S*DGh*9JI ;$IpZbD_8&?œZ+;NCY7UD
⇒ m2 n = 0
•
6 <CR8 c 9ZbD8 ;$<Z); 9%Cl;$<9JI
m2 = 0
7I
n=0
6 <CR8 C©8 Z ;$IrCAOCmZb? 9&‚^Zbc Hb?A9 7^= ƒ 7U?Rh8y;V7@D9!8 …~¬ Zbc xbS!8’‡
•
$; <97sIW9Jc …~¬ Zbc xbSsP 7UD%Z&XN¢Z_8MCEDCY7›P ƒ 7@?©hJ8y;V7UDf9 f“*”  ƒ 7U?RhJ8y;V7UDf9bPŠt¤Zb?[Z*c Pje 9CEDbx9JI\F _“b”† ‡$P
¡ 7USGF<b?g] ¨= Z f Z8V8:7LCmZ);V9h B Cl;$< Z )_$_M
•
J," CED Z IW9?[Z+;NCAOCR8y;NC[L  9J?©h ;$<97sI] h*7j9+8 D7%; *'~#
;$<9 O%ZGL^SbS*c
;$<9 ;$<97sI] <%Z8 +9IW7 c Z8V8 9&‚LCQ;zZ+;NCY7UD_8 d
⇒
Q _ _j354
•
J µ(x) = ∂ µ φ(x)
∴
CQ=
6 <9 L^<%ZI\F*9 CR8
∂µ J µ(x) = ∂ 2φ(x) = −m2 φ
m2 = 0 ⇒ ∂µ J µ(x) = 0 .
Q≡
Z
3
0
d x ∂ φ(~
x, t)
6 < CR8 L^<%ZI\F9 …>B <C[L^< CR8 ZGLM;$S%Z*??€] 7@Db?g] h*9  D9h B <9JD Z*HX
•
H9GZI>CEDGF CED L)7Uc c S+;zZ);NCY7UD IW9?[Z+;NC[7@D_8 Z_8 =r7U??A7_B 8 ‡«h*7|9)8
D7^; *'~# ;$<f9 O^ZLJS*Sbc oqIW7Uc ;$<9 LfZbD7@DC[LfZb?L)7Uc X
c S+;zZ);NCY7UD IW9J?[Z);NCY7UD8 i
h
i
∂ 0φ(x), φ(y) = δ (3)(x − y)
x0 =y 0
⇒ i [ Q, φ(y) ] = 1
¤7^;V9 ;$<%Z+;
∴
i h0 |[ Q, φ(y) ]| 0i = 1 .
dQ
=0.
dt
>D F9Df9IpZb? ƒ 7U?Rh8y;V7@D9!8 6 <97sIw9c LZ*D x 9 &** V; 7
•
x9 ;$I,S9 S8MCEDGF 9$‚^ZGLM; IW9+8Sb? ;«8 7J=
oU6 B Cl;$< 7@S); S8:9 J7 =
$# H9JI;$S*I,x%Z+;NC[7@D ;V9L^<bDC[™S9+8d
Q f b*U0 * jj 553 4
•
9JI] =r9&B c Z_8V8&?R9+8V8 HZI;NC[L^?A9)8 H*<7%;V7UD C©8 $; <f9 c 7y8 ;
= Zbc CE?mC`ZI 7UDf9d
•
6 < CR8 CR8 B <f9IW9 L)7Uc 9 CED u = ;V9JI  7I,a'9D FZO9 Z ;zZ*?a
…~f“b”†‡ Z); ˆ ZIO^ZI\h P"c ] ;$<f9)8MCR8 ZGh^OCR8:7I PGe Zb? ;V9Iqƒ CE?x9JI;
…~¬§7Ux9J? ¢@Z*SbIW9Z);V9 ¦ <9Jc C©8 ;$I] f“¥‘‡$P 8SFbF*9+8y;V9h ;$<Z);
?A7j7@a Z);  7I,a'9D!8 HIW7UHJ78:9h c 7jh*9J?17J= ­ –
Z
O^ZI>CmZbD+; 7J= ;$<9 ¬ Z*c x*SX 7UDZXN¢Z_8MCEDJC[7 c 7h*9J? B Cl;$< CEDX
;V9IªZL$;NCY7@D
g (ψ̄ γ µ ψ) (ψ̄ γµ ψ) .
6 < CR8 8:99Jc 9_h ;V7 xJ9_LfZbS8:9 ;$<9 8«]bc c 9&;$I] ;$<%Z+; CR8
xIW7@a’9JD C©8 ¢"7IW9JD); 8«]c c 9&;$I] X'IW9J?[Z);NCQOCR8y;NC[L CnD)O^ZI>CmZbDL9bd
6 S*I,D_8 7US+; ;$<%Z+;  7sI,a'9JD h"Cœh Cl; P x*S); B CQ;$< 8:7@c 9
•
c 7h"C  LZ+;NCY7UD <CR8 L7@DGL^?S8MCY7UD ;$<Z); ;$<CR8 ;$<f97I] CR8 9™@SJCAOX
Zb?A9D+; ;V7
CR8 L)7I,IW9L$;)d
6 <CR8 CR8 Z 8SbI,HI>CR8:9 CQ= ]7US 7UD*?g] a%Df7_B ) | b
•
›_ J! # P x9_LfZbS8:9 ;$<JCR8 CED+;V9IªZL$;NCY7@D
C©8 B 9J?? a%D7B D ;V7 x9 <7@H9J?R9+8V8?g] h|CQO9I\F9D+; S_8MCEDF ;$<CR8
ZbHbHIW7sZGL^<d
•
 7I,a’9JD c 7jh*9J? IW9+8Sb? ; LZ*D x9 IW9FsZI\h*9h Z_8 Z
6 < 9
•
…Vc 9+8V8«]%‡ r b 7^= HJ9I;$S*I,x%Z+;NCY7UD ;$<97I] …šZ_8
LZ*D ;$<9 ¬ Zbc xbSP 7UD%Z ¢@Z8MCEDCY7 c 7h*9J?[‡$P x*S); B <Z);
]+7@S ZIw9 IW9Zb??g] h7jCEDGF CR8 LZ*?RLJS*?œZ+;NCEDGF CED Z h"C 9JIW9JD);
Hb<%Z8:9 ;$<%Z); L)7I,IW9+8H7UDh8 ;V7 Z*D Zb? ;V9JI,D%Z+;V9 8:7U?S+;NC[7@D ;V7
;$<9 =TS*DGLM;NC[7@D%Z*? h"C
9JIW9JD);NCmZb? 9_™S%Z);NCY7@D_8 7^=
oU6 P L7I¨X
IW9)8&H7UDh|CEDGF ;V7 8y]c c 9$;$I] xIW9ZbaGCEDF x7US*DGhZI] L7@DGh|CgX
;NC[7@D_8 d 6 <CR8 CR8 h"CnIw9_LM;$?g] IW9J?[Z);V9h ;V7 ;$<9 c S*? ;NCnH*?A9 8Œ7U?SX
;NC[7@D_8 7^=§;$<9 7sI\h"CED%ZI] h|C
9JIW9D+;NCmZb?9™@S%Z+;NCY7UD HIW9&OCY7US8?g]
h|CR8~L^S_8V8:9hŽd
6 <9+8:9 x7@SbDGhZI] L)7UDh|Cl;NCY7UD8 ZIw9
•
hj µi = nµ
j µ = ψ̄ γ µ ψ
ZbDGh P›8MCEDGL)9
h
•
Jµ ν , j λ
i
i
µ
λ
ν
ν
λ
µ
∝ g j −g j ,
h
B < 9JIW9 µ ν F*9JD9JIªZ);V9+8 ¢6 `8P
J
6 <CR8 IW9_™SCEIW9+8 ;$<Z);
Jµ ν |0i 6= 0
.
$; <9 8«]bc c 9$;$I] J7 =;$<f9 O%ZGL^SbSbc CR8 xIW7@a’9JD P Z*DGh µ
j |0i
L7@D);zZCED8 Z + 9IW7 c Z 8V8 …,8HCED ‡ HZI{;NCœL^?A9
Z Hb<f7^;V7@Dd
• ∴
9+8HCl;V9 ;$<f9 = ZL$; ;$<%Z+; tqL^<)B CEDF*9JI <ZGh ZIWF@S9h x5] ;$<Z);
N; Cnc 9 ;$<%Z+; ;$<9JIW9 B Z8 D7 h^]bD%Zbc C[LZ*? IW9Z_8:7@D =r7sI ;$<f9
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