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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
$ 090i ∈ { 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)
( C7z := (z i )
& 6$09096 &9D 2&9$ $ 7 6 $&9,2 ( Ff (¯ 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
$&D$ Ff ˜ (¯ 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-#@',
#%$$#%&'0"&' /2$ & #%#@& 1G, &9$
(
687 7,2
(
$1WG# 09
(
ˆ a † ˆ a 2
$ #@', 0 $
ˆ a † ˆ a 2
= 1 3 (ˆ a † ˆ a 2 + ˆ a a ˆ † ˆ a + ˆ a 2 ˆ a † )
) 7&%6 $WGF ˆ :=
F (ˆ a † , ˆ a)
&-620'09 7 4 # F 04$ 0 4 #>F/$04f
) 7 D 0C 0#6 7 $$#%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&$$ D &9C $ 7& &96 )<C 0 7,2 7$#@09& 2&'$ &9 A )+EA & 687687 7,2
1 7→ ˆ
)6 D!7 6E,$2&9$ 6 6&9$
f
$# /2$F ˆ
C % # 0'7 &'D>#@'&'% A )+EA C7&9687 &!&9D F
F ˆ 7→ f (¯ z, z) := (i 4 π) d Pf(θ) Z
d d p ¯ d d p
(2 π) 2 d Tr H
F ˆ e − i (¯ p · z+p · z) ¯ e i (¯ p · z+p ˆ · ˆ z † )
= F ? (¯ z, z) ,
A )+ T
C7
F ?
&D$ #
F
F 0'6&9 7 6 $# '&&9$ F 7 4C 6Pf(θ)
-7 "7 $6 $#%#>2&D &9 $
θ
C7&9687 $ &9&9D #%&',2 &&9D F
√
det θ
7 C& WE, ' ( $WG# 09 ( &9 "EG?$ ) ,7#%$ ( 7 #>F/$0Tr H
&!, 71% 716 1 D7 !&'09F 62)17 !7,D$ %F&s 6 &9$ 71$#
(f (¯ z, z), ?) ←→
F ˆ (ˆ a † , ˆ a), ·
A )+`20'09C&'-4&'D &'$2 $F09# $
θ
$#% 0,26D &'-&94^ &9D209"$#>0'2&9$&9 7 /2$ #@09&# &' 6 72)
&'( %"'&'( &"
$" ' 0<@,0< *E097$&' &97' 2$ $#@09&# & &9&9090/6 @
&9 6 2,09$$ &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! / $#@09&#
∂ z ¯ i f −→ √ 2 1 θ i ˆ
a i , F ˆ
∂ z i f −→ − √ 2 1 θ i ˆ a i † , F ˆ
,
A )+EH6 C 090 '
Z d d z ¯ d d z
(2 i) d f −→ (2 π) d Pf(θ) Tr H ( ˆ F ) .
A )+A, 7 #%$
( 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 7D
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&')
" "$%" ! #&$"$ " $ %"'&
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$ D209α ∈
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 )+HC17,2 7&WE,&&9$ &687687 7,2
| α i
&!$#20'& @&9 F!27h α | α 0 i = e αα ¯ 0 ,
A )AG& ))(
h α | α i = exp(α α) ¯
) ,$# 7 $#>0' A )+H 7!6&9$%;7 6 &9$ 2@&97&902&'/ $ !
| α i
6 F/1 6 &' 0'&'@&9#%#2ˆ
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)
$ $#%β ∈
FD ˆ (β) := e β ˆ a † − β ¯ ˆ a = ⇒ D ˆ (β) ˆ a † D ˆ † (β) = ˆ a † − β , ¯
A )A')$# C7&'687&C E, 7,2
D ˆ (β)
&9 20'26 $7 2-6D ˆ (β) | α i = e − β(β+ ¯ 1 2 α) | α + β i .
A )A T!282&'$,!246 $78 F$#% 052$09
ϑ
( $ 7 7 7 ( &'6 FR ˆ (ϑ) := e i ˆ a † ϑ ˆ a = ⇒ R ˆ (ϑ) ˆ a † R ˆ † (ϑ) = e iϑ ˆ a † ,
A )A+`687 7,2
R ˆ (ϑ) | α i = | e iϑ α i .
A )AH7&84 $#@2&' 0 & 7 ( &9 $$$09&9F ED@4
#@2&' ( $&D$ $ $#% 6 $# 0'1G ,#%
β
FS ˆ (β) := e 1 2 ( ¯ β ˆ a 2 − β a † 2 ) ,
A )AC7&'687 &'# 09&'
S ˆ (β) ˆ a † S ˆ † (β) = ˆ a † cosh( | β | ) β
| β | + ˆ a sinh( | β | ) β
| β | .
A )AB
#
#@&9, &'# &9 E, 0' 7 &!kE, &D &0 0'&' 7 _ , &9$
#%&' $#% $&9D #% 0 ) !$ , 0< ( 7 ^ 2&9$@&&' &9 09&&96 #@ 0
2 ,090 7&'$70 $09&' $09D &9 7 # &9 0'0 09&9 & &96&9F0'
'?;) &@7 $ , 0 0' $? $ /&'F0'&9# 0'&FE,6 &9$\C7&'687 #%&9$7 20'09C $
E&9 2109' F/60'' 0'&')! #@ 6' ( 7&1& &9 &9F092 7
1G & #@ ,2&9600 C&' &, &9$>C7 7 6 $&' $09&'$, 0 7,D$
2 &9 8 &9$ &' # ' $09$$&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
$09&9 &'$#@ #% 0 7
(
C$$<7F/ ?
"<+EG)( + ))(B$H?$
) D
$ 60
(
&'# 0 F 7>D26 &' 7 "6$#%# 82&9DOE, 0' 7 ( $/ F,2&9D
$09&'$, 6&D$ @0'2 2 &' &' 7 $6#@# 8 &D$ &' ( 'lC 0'0) ,$- 0
6-$ 7 &96 $6 $#%#>2&D$!$09&9$! &9$ C 7
"T A ( T ()H T (`')( T H,(*+`(*H?$ 2 6 !7 &' )
z cI<u ? VWU A>`]^VW<>EH]^VapU5EXU </?$AH]
7>#% 0-2 &'#@4 &9 - 7 $ 09&' &9$#@#% 0 ( 620 E, 0' 7
$&9 $
(1+d)
&9#@ &9$,20Aa;2 ,6 &9#% C&97>E09 ? &' D209 &9 &9#@& $#@&$0' WE, F 0& 6&9$ 71$#
S[Φ] :=
Z
d 1+d x 1 2 G ab ∂ µ Φ a ∂ µ Φ b − V (Φ)
,
) )+C7
G
7 #@&'6 $ 7 82$ ,26%V
& #@ /&'0 ) 7 8 &9$$ 09&' > - 7 $# <7 #% &96% #% 0 & ,090 6090' $09&' &
7 6 $# $!
G
/ $ 7bE09Φ
&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# 0L =
∂ µ Φ a ∂ µ 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&97L = Tr ∂ µ Φ † ∂ µ Φ
) . FD &9$0 7%#% 01 s $09$F,0O(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 6P
#@ ?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 ) ,
) )AC7
U( X ) := {U ∈ GL( X ) | U = U † }
7 220'0&84 09&' $#@1&9$
X
) 7$ ( 687 &9DP
( 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 1 620'09P 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
t3 LL gZ8u66L " deQ8u)8_,
7 #% 0 &9 6 2F ED$ # ' $$7 $&9D WG 09&'6 &9 $09&9$ ( 09' &'
09C &'#%,&9$
(
2 7 D ' WG# 09 $ 2 #%$ 20'&&'6 7&9$7 4
&'#%,&9$,0!#% 0 ) C D 7 0' ( 7 #% 0 2 $&9D &'0$09&9$@&'
2F& >&'#% &9$ ) 7! &96 &' %7 #>F/ 090'C &'#%,&9$<$"$&9D &'0
$09&9$ WG & 2#@7$# F +- &968?
"`+` ((A$A&$
C7&'687 82 -7,21
6 8&9 ' # &' 76 WG &> 7 8 &96kE,&9 ' 0'&' 7, 7
D26 # & 09*) 35 $090'C 7 09&' >2 "BH&$ 'bC 0'0'>+&'68?)r $&'$&9,0 ,6 > $
&9 0'&0'%#%$ &'0 ) 7& 6$,&9 &9 6&9$ 2 7 $# ) )+ ( C7
(
&9# 0'&96& ( C '#%
Φ
F/ 60 2) 8 &96 $09&9$Φ
%WG #%& 17 '6 &'$0
E[Φ] :=
Z
d d x 1 2 ∂ i Φ ∂ i Φ + V (Φ)
=: E 2 [Φ] + E 0 [Φ] ,
) ))C7
i, j ∈ { 1, 2, . . . , d }
( &'# 0< &' 72<&9 7$09@&' , &960'"F/ 82&', > ,2&20 609&'Φ(x) → Φ(λx)
) 7 ' $# !7&! 609&' 'E[Φ] −→ λ 2 − d E 2 [Φ] + λ − d E 0 [Φ] =: E λ ,
) )T7 16 6$&' !7>687&'6 " #%&9$09$' F/ D)$ 82&',& 7 6 092
7 6&&9$
0 = ! dE λ
dλ
λ=1 = (2 − d) E 2 − d E 0 = ⇒ E 0 = 2 − d
d E 2 ,
) )`!6 C 7,2 7 F/7ED &9$ 7 6 /,6 6 &090< $ 7 #>F/ 2&'0
&'#%,&9$) &96090< 7 ? &9 &'6 6
E 2
7 &'0 'E 0
2 F/7/&9&9D 7$ ) )`
6 0 7$09 $
d = 1
C7'@$d ≥ 2
&9@&@D &9$0 ()7 66
d = 2
/ 1G6 &' &','%7 &9& &90'0 '&9F09 6 $&9 7&9,2&'$ D2&7&' 2&0IC7&9687 #%&9$7 &'090!090'C $ 0'&') 7 C
D27& 7$# &9 60' E09 7 $&9 &< &'609 2&&9$,0, # C&77&9$7 $
&D22&D$%$>7&9$7 C 7 &9D22&9D %&' 7 3 $$&' 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 8246 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
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