Modello Standard e Gravitazione in geometria non...
129
Transcript of Modello Standard e Gravitazione in geometria non...
W
! :
n! p+ e! + ve
SU(3)"SU(2)"U(1)
SU(3)"SU(2)"U(1)
#
SU(2)"U(1)
("e, e), (", µ), (", #)
!5 !5 = 1 !5 = !1
fqfn
n = 1, 2, 3lf
12
u1 u2 u3d1 d2 d3
"ee
$ SU(2)
34
c1 c2 c3s1 s2 s3
"µµ
$ SU(2)
56
t1 t2 t3b1 b2 b3
"!#
$ SU(2)
%!SU(3)
(u, d), (s, c), (t, b)
qfn
!(f = 1, 2, .., 6)
(n = 1, 2, 3)
lf"(f = 1, 2, ..., 6)
qfn (qR)fn (qL)fn
SU(3)
SU(3)c
SU(2)
eL "e SU(2)ISU(2)
U(1) U(1)
SU(3) " SU(2) " U(1)
LMS = LB + LF
LB = #1
4Fµ"F
µ" # 1
4W a
µ"Waµ" # 1
4Ga
µ"Gaµ"
LF =#
i
iRi$ ·DRi + iLi$ ·DLi
LB W Z LF
SU(2)"U(1)
LH
LH = Dµ%†Dµ%#m2%†%# &(%†%)2 +Ge(L%Re +Re%
†L) +G"(L%R" +R"%†L)
SU(2)"U(1)
SU(
SU( Gµa (a = 1, ..., 8)
SU(2) Wµi (i = 1, 2, 3)
U(1) Bµ
µ
'
(µ' ((µ + ie µ)'
e '.
e = 0
',
'
µ
jµ
ejµ µ
U(1)
#SU(2), SU(3)#
'i
SU(N)‡
'i(x) #! Sij(x)'j(x) ,
Sij SU(N) #a
[#a, # b] = ifabc# c.
S
Sij(x) = (ei#a(x)!a)ij & '
!i = (ei#
a(x)!a)ij'j
)a(x)
(µ'i
((µS)
(µ'i #! (µ'"i = S((µ') + ((µS)'
Dµ
Dµ
Aµ(x)
Dµ = (µ # igAµ
Aµ(x) ' Aaµ(x)#
a.
A!µ(x) = #
i
g((µS)S
!1 + SAµ(x)S!1
‡
Dµ
(Dµ')" = (µ'
" # igA!µ'
"
= S((µ') + ((µS)' # igA!µS'
= S(Dµ').
!*' = ig)a#a'
*Aaµ = # i
g(µ)a + fabc)bAc
µ
U(1) SU(2) SU(3)
U(1) #
# = #1, S = e!i#(x)
U(1) : '" = e!i#(x)'
e g
U(1) : Dµ = (µ + ieAµ
(µS = i((µ))ei# Aµ = Aµ# = #Aµ
A!µ = Aµ +
1
e(µ).
SU(N) = SU(2) #a
2" 2 #a = +a/2 a = 1, 2, 3
+1 =
$0 1
1 0
%, +2 =
$0 #ii 0
%, +3 =
$1 0
0 #1
%
&+i
2,+j
2
'= i,ijk
+
2
k, i, j, k = 1, 2, 3.
S = ei!·"/2
SU(2) : '" = ei!·"/2'
Dµ' = (µ' #i
2g! · µ'
(µS = i2(+ · (µ))S )i
!µ = µ # " " µ +
1
g(µ".
SU(3) #a = &a/2
3" 3
&1 =
(
)*0 1 0
1 0 0
0 0 0
+
,- , &2 =
(
)*0 #i 0
i 0 0
0 0 0
+
,- , &3 =
(
)*1 0 0
0 #1 0
0 0 0
+
,- ,
&4 =
(
)*0 0 1
0 0 0
1 0 0
+
,- , &5 =
(
)*0 0 #i0 0 0
i 0 0
+
,- ,
&6 =
(
)*0 0 0
0 0 1
0 1 0
+
,- , &7 =
(
)*0 0 0
0 0 #i0 i 0
+
,- ,
&8 =
(
)*1 0 0
0 1 0
0 0 #2
+
,- .
&&a2,&b2
'= ifabc
&c2
ifabc
.//0
//1
f123 = 1
f147 = #f156 = f246 = f257 = f345 = #f367 = 1/2
f458 = f678 =#32
S = ei!a
2 #a a
Dµ' = (µ' #i
2g&aAa
µ'
Aaµ
Aµ = Aaµ&a
2=
(
)*A3
µ + 1#3A
8µ A1
µ # iA2µ A4
µ # iA5µ
A1µ + iA2
µ #A3µ + 1#
3A8µ A6
µ # iA7µ
A4µ + iA5
µ A6µ + iA7
µ # 2#3A
8µ
+
,-
Aµ Dµ
Gµ"
Gµ" ' i
g[Dµ, D" ]
= (µA" # ("Aµ # ig[Aµ, A" ]
= ((µAa" # ("Aa
µ + gfabcAbµA
c")#
a
Gaµ"
Gµ" ! S Gµ"S!1
(S Gµ"S!1S Gµ"S!1) = (Gµ"G
µ")
S =
ˆd4x
2#1
4Gµ"G
µ"
3=
ˆd4x
2#1
4Ga
µ"Gaµ"
3.
' #! 'S†(x)
Gµ"
S =
ˆd4x'(i$ ·D #m)'
S = SB + SF
=
ˆd4x
&'(i$ ·D #m)' # 1
4Gµ"G
µ"
'
U(1) SU(2) SU(3)
U(1)
G ' F
Fµ" = (µA" # ("Aµ,
U(1)
L = '(i$µDµ #m)' # 1
4Fµ"F
µ"
= i'$µ((µ + ieAµ)' #m'' # 1
4Fµ"F
µ" ,
' Aµ
SU(2) +i
W aµ" = (µW
a" # ("W a
µ + g,abcW bµW
c"
W µ" = (µW " # ("W µ + gW µ "W " .
SU(2)
L = i'$µ((µ #i
2g !·W µ)' #m'' # 1
4W µ"W
µ"
W µ.
SU(3)
Gaµ" = (µA
a" # ("Aa
µ + gfabcAbµA
c" a, b, c = 1, ..., 8
SU(3)
L = i'$µ((µ #i
2g #·Aµ)' #m'' # 1
4Gµ"G
µ"
Aµ.
H = ##
i,j
Jij ·
Tc
F
F > Fc
F > Fc F < Fc
%
L = ((µ%$)((µ%)#m2%$%# &(%$%)2
= ((µ%$)((µ%)# V (%, %$)
U(1)
U(1) : %! ei!% (! cost.)
%0 V
(V
(%= m2%$ + 2&%$(%$%) = 0 .
m2 > 0 %0 = 0
U(1) m2 < 0 % = 0
%0
|%0|2 = #m2
2&' a2
|%| = a % = 0
|%0| = a %
|%0| = | (0|%|0) | = a .
V %1 # %2 % =
(%1 + i%2)/*2 V |%| = a
U(1)
%1 %0 '$
a
0
%
%
%"
%
%" = %# a
(0|%"|0) = 0
% = a+(%"1 + i%"2)*
2
L =1
2((µ%1)
2 +1
2((µ%2)
2 # 2&a2%21 + cubici+ quartici
%2 %1%2
m1 =*4&a2
L = ((µ%$)((µ%)#m2%$%# &(%$%)2
U(1)
U(1) : %! ei!(x)%
Aµ
L = ((µ # ieAµ)%$((µ + ieAµ)%#m2%$%# &(%$%)2 # 1
4Fµ"F
µ" .
m2 > 0 U(1)
m2 < 0
|%| = a %"1 %"2
%(x) = a+%"1(x) + i%"2(x)*
2.
L = #1
4Fµ"F
µ" + e2a2AµAµ +
1
2((µ%1)
2 +1
2((µ%2)
2 + .
+*2eaAµ(µ%2 # 2&a2%21 + cubici+ quartici .
%2 Aµ(µ%2Aµ %2
% ! %2 = 0
L = #1
4Fµ"F
µ" + e2a2AµAµ +
1
2((µ%1)
2 # 2&a2%21 + cubici+ quartici .
Aµ %1%2
Aµ.
W±
m = 0
' i,'i
e, µ, #
µ #
SU(2)"U(1) LB
LF
L0 = LB + LF
LB = #1
4W a
µ"Waµ" # 1
4Fµ"F
µ"
LF = iRe$ ·DRe + iR"$ ·DR" + iL$ ·DL
L '$"LeL
%
Iw = 12 I3w("e) = 1
2 I3w(eL) = #12
Re ' (eR R" ' ("R)
Iw = 0 I3w = 0
SU(2)
SU(2) :
./0
/1
L! ei2!·"L
Re ! Re
R" ! R"
U(1) Y
Y = #1 L Y = #2 Re Y = 0 R"
Q = I3w +Y w
2
L Re R" U(1)
U(1) :
./0
/1
L! ei2$L
Re ! ei$Re
R" ! R"
Q ("e, eL) (0,#1) ("R) (eR)
0 #1 SU(2) U(1)
W aµ" = (µW
a" # ("W a
µ + g,abcW bµW
c"
Fµ" = (µB" # ("Bµ
DµRe = ((µ + ig"Bµ)Re
DµR" = (µR"
DµL = ((µ +i
2g"Bµ #
i
2g+iW
iµ)L
% '$%+%0
%
1/2 +1
L = L0 + LH
LH = Dµ%†Dµ%#m2%†%# &(%†%)2 +Ge(L%Re +Re%
†L) +G"(L%R" +R"%†L)
Dµ% = ((µ #i
2g"Bµ #
i
2g+iW
iµ)%
% :
(%)0 =$
0
-/*2
%
W aµ Bµ
Aµ Zµ
W±µ
Zµ =gW 3
µ + g"Bµ
(g2 + g"2)1/2' )WW 3
µ + )WBµ
Aµ =gBµ # g"W 3
µ
(g2 + g"2)1/2' # )WW 3
µ + )WBµ
W±µ =
1*2(W 1
µ ± iW 2µ)
)W
)W 'g
(g2 + g"2)12
M2W1
= M2W2
=g2-2
2
M2Z =
M2W
)WMA = 0
Me = Ge-/*2
M" = G"-/*2
M2H = 2&-2
Me M" MW Mz
MA
MH -
GF = g2/2M2W
*2 (%)0 = - =
4*2GF
5!1/2+ 246 .
SU(2)" U(1)
$"eLeL
%,
$"µLµL
%,
$"!L#L
%
(eR), (µR), (#R)
("eR), ("µR), ("!R)
W± Z A
SU(2)
$uLdL
%,
$cLsL
%,
$tLbL
%
IW = 1/2 Y = 1/3
U(1)
(uR), (cR), (tR)
(dR), (sR), (bR)
IW = 0 Y = 4/3 uR, cR, tRY = #2/3 dR, sR, bR
Ge G"
Ya
U(1) Y
Q(u, c, t) = 2/3, Q(d, s, b) = 1/3
SU(3)
SU(3) SU(3)f
a = u, d, s, c, t, b
(
)))))))*
u1 u2 u3
d1 d2 d3
s1 s2 s3
c1 c2 c3
t1 t2 t3
b1 b2 b3
+
,,,,,,,-
R,W,B
SU(3)c
L = #1
4Ga
µ"Gaµ" +
6#
i=1
'i(i$ ·D #mi)'i
Dµ = ((µ # i2g&
aAaµ) Ga
µ" = ((µAa" # ("Aa
µ + gfabcAbµA
c") i
&a
SU(3)
SU(3)
q ! Uq, q =
(
)*q1
q2
q3
+
,-
q U
SU(3) SO(3)
SO(3)
SO(3)
SU(3)
SU(3)C SU(2)W U(1)Y Q
uL F F 1/3 2/3dL F F 1/3 #1/3uR F S 4/3 2/3dR F S 2/3 #1/3"L S F #1 0eL S F #1 #1"R S S 0 0eR S S #2 #1
FS
[3(generazioni left)" 2(isospin debole)"3(colore) + 6(particelle right) " 3(colore)] " 2(spin) = 72
[3(generazioni left)" 2(isospin debole) + 6(particelle right)]" 2(spin) = 24
H = H ,H
HL = (C - C - C )6 78 9quark left
, (C - C - C)6 78 9leptoni left
HR = ((C, C)-C - C )6 78 9quark right
, (C, C)-C - C)6 78 9leptoni right
N = 3
H$"ee
%
L
$"!#
%
L
$"µµ
%
L
$ui
di
%
L
$ci
si
%
L
$ti
bi
%
L(ui)R (ci)R (ti)R (e)R (µR) (#)R(di)R (si)R (bi)R ("e)R ("µ)R ("! )R
i
H = H ,H ,H ,H
H = (C - C - C )6 78 9antiquark right
, (C - C - C)6 78 9antileptoni right
HcL = ((C, C)-C - C )6 78 9
antiquark left
, (C, C)-C - C)6 78 9antileptoni left
H , H CH ,H
H ,H = (H ,H )c .
·c
'c ' C' T = C('†$0)T = C$0'$ = i$2'$
C =i$2$0
('L)c = C((PL')
†$0)T = C('†$0PR)T = C('PR)
T = CP TR'
T = PRC' T = ('c)R .
$"ceec
%
R
$"c!# c
%
R
$"cµµc
%
R
$ucidci
%
R
$ccisci
%
R
$tcibci
%
R(uci )L (cci )L (tci )L (ec)L (µc)L (# c)L(dci )L (sci )L (bci )L ("ce)L ("cµ)L ("c! )L
G" . Ge
'L,R‡
'cL,R = ('c)R,L CPT
SU(2)
$"Le!L
%CPT%!
$("c)Re+R
%.
SU(2)
R L
CPT
NRCPT%! N c
L .
‡ L R !5
! · p
"L NR
LD = #mD:"LNR +NR"L
;= #mD""
" ' "L + NR
"L, "cL, NR, N cR
L = L"+LN "LNR
"I = 1/2 "L1/2 NR 0
SU(2)
LY ukawa = #h"4"e e
5
L
$%0
%!
%NR + h.c.
mD = h"v/*2 h" v = 246
h" ! 10!11 m" / 1
he / 10!6
'$ = ' = 'c
'c
$µ(i(µ # eAµ #m)' = 0 ( # e)
$µ(i(µ + eAµ #m)'c = 0 ( + e)
PL,R ' 1%%5
2
PL,R('L + 'R) = PL,R[('c)L + ('c)R]PL,R#!
!'L = ('c)L = ('R)c ' 'c
R
'R = ('c)R = ('L)c ' 'cL
' = 'L + 'cL .
"cL
"L"L"L = "cL"
cL, = 0
Lmass = #1
2mL"" = #1
2mL("
cL"L + "L"
cL)
" = "L + "cL" = "c "L SU(2)
"I = 1 mL
NR
"R"R = "cR"cR, = 0
Lmass = #1
2mR(N
cRNR +NRN
cR) ,
SU(2) "I = 0
LM = #1
2mL"L"
cL #
1
2mRNRN
cR + h.c.
L R c
˜ 2" 2
˜ =
$0 0
0 100
%
.˜
M =
$. .
# . .
%$0 0
0 100
%$. # .
. .
%
=
$100 . 100 . .
#100 . . .
%
. / 2 . = 0.99939
. = 0.03490
M =
$0.122 3.488
#3.488 99.878
%
m22
<(100 ,)( ,)
m11 m22
m22 0 m12 > m11
M M
Lmass =
2#1
2mL"L"
cL #
1
2mRNRN
cR + h.c.
3+mD("L"R + "R"L)
= #1
2
4"L N c
R
5M$
"cLNR
%+ h.c.
M '$
mL mD
mD mR
%
M =
$m" 0
0 M
%
Lmass = #1
2
4" N
5M$
"
N
%
" N
"L NR
" N
Mm" M
M ("N)T
("N)T ( "cL NR )
("N)T
M M("N)T ( "cL NR )T
M 1 mR 0 mD > mL 1 0
mRM
mLM
mD m2D =
mRmL mD
mR mL
M $mL # & mD
mD mR # &
%= 0
&± =1
2(mL +mR)±
1
2
=(mL +mR)2 # 4(mLmR #m2
D) .
mLmR = m2D
&+ = mL +mR 'M
&! ' m" = 0
&+$
mL # (mL +mR) mD
mD mR # (mL +mR)
%$"cLNR
%= 0
"cL = (mD/mR)NR
N =
$mDmR
1
%=
mD
mR
$1
0
%+
$0
1
%=
mD
mR"cL +NR .
N = (NR +N cR) +
mD
mR("L + "cL) .
&!
" = ("L + "cL)#mD
mR(NR +N c
R) .
mR 0 mD
N M
NR
N " m" / 0
"L "cLmL mR mD
mL 1 0 ( )
mR 1M
(mR 0 mD)
mR mD
Mm" = 0
M =
$0 0
0 M
%
("N)T
mR 0 mD
U(1)
U(1)
"L ! ei&"L .
"cL = C"TL
LMmass =
1
2m4"TLC†"L + "†LC"
$L
5,
LMmass #!
1
2m4e2i&"TLC†"L + e!2i&"†LC"
$L
5.
L = ±1
U(1)
[E]3/2 [E]
L5 =g
M:'TL+2%
;C† :%T+2'L
;+ h.c.
+2 M
g 'L =
$"LeL
%% =
$%+
%0
%
% =
$%+
%0
%
################!1*2
$0
v +H
%,
L5 "L
LMmass =
1
2
gv2
M "TLC†"L + h.c. .
L5
[E]5
[ gM ] = [E]!1
SU(N)
xµ
xµ = xµ(x") , µ, " = 1, ..., 4
dxµ (/(xµ
dxµ #! dxµ =(xµ
(x"dx" ' G!1(x)dx"
(
(xµ#! (
(xµ=
(x"
(xµ(
(x"' G(x)
(
(x"
x = x(x)
G!1(x)
G(x)
G(x) G!1(x)
G(x)
%(x) = %(x) scalare
Aµ(x) =(x"
(xµA"(x)
Aµ(x) =
(xµ
(x"A"(x)
Tµ"(x) =
(xµ
(x'(x"
(x$T'$(x) .
SU(N)
GL(4)
4" 4
(µA"
(Aµ
(x"=(x'
(xµ(x$
(x"('A$ +
(x(
(x"(2xµ
(x((x)A)
#*µ" Dµ
DµA" ' (µA" + #*µ"A*
DµA" ' (µA
" # #"µ*A
*
(DµA") =
2(x'
(xµ(x$
(x"
3D'A$
#*µ"(x) =
(x'
(xµ(x$
(x"(x*
(x%#%'$ +
(x(
(xµ(x)
(x"(2x*
(x((x).
[Dµ, D" ]A* = R)µ"*A)
R)µ"* ' (µ#
)"* # ("#
)µ* + #)
"(#(µ* # #)
µ(#("*
A*
R)µ"*A)Sµ" Sµ"
R)µ"**
"*g
µ* ' R
d4x d4x
d4x = ((xµ
(x")d4x
"! = (ei"a(x)#a
)" xµ = xµ(x$)
SU(N) GL(4),
U(x) = (ei"a(x)#a
) G(x) = %x!
%xµ
Aµ !!µ$
Aµ " Aµ = ! ig (#µU)U"1 + UAµ(x)U
"1 !!µ$ " !
!µ$
Gµ$ R&µ$!
Gµ$ = [Dµ, D$ ] R&µ$!A& = [Dµ, D$ ]A!
*#gd4x =
*#g
<#g(x) = (
(xµ
(x")<#g(x)
gµ"
S =
ˆd4x*#gR
S
Rµ" #1
2gµ"R = 0
gµ"g00 ! 1 + %
Rµ" #1
2gµ"R = 0 #! 22% = 0
GL(4)
V ierbein : eaµ(x)
eµa(x)
eaµea" = gµ"
eaµ = gµ"ea"
eaµeaµ = *ab
$a
{$a, $b} = 2-ab
$aeaµ = $µ(x)
{$µ, $"} = 2gµ"(x)
'(x)
Trasformazioni di coordinate : ' ! '
Trasformazioni di Lorentz : ' ! ei+ab(x)(ab'
,ab +abi2 [$a, $b] '
/abµ /ab
µ ! /abµ # (µ,ab
"µ' = ((µ +1
4/abµ +ab)'
(i$µ"µ #m)' = 0
L =# 1
2k2*#g R+ e'(i$µ"µ #m)'
e ' det eaµ =*#g
/abµ
"µea" = (µe
a" + #*
µ"ea* + /ab
µ eb" = 0
#)µ"
/abµ
/abµ =
1
2ea"4(µe
b" # ("ebµ
5+
1
4ea)eb(
:((e
c) # ()ec(
;ecµ # (a3 b) .
$
'(/( + /A)'
Rx p
E = p2
2m + k2x
2.
C&(R )
x p
[x, p] = i"1.
A·$
·$
(ab)$ = b$a$, 1$ = 1
a$$ = a.
A
"x"p 4 "/2
"/2 "a a 5 AA H
0 : A! (H)
H = L (R) '(x)
H = H$ 5 A
2i" ((t# 1(H)
3'(x, t) = 0
/(' = 0
'/(
A /( 5 End(H)
A,H, /()
AH A
/( H
A C C2a + !b 2,! 5 C a, b 5 A A
A " A! A (a, b) 5 A " A!ab 5 A
a(b+ c) = ab+ ac, (a+ b)c = ac+ bc, 6a, b, c 5 A
ab 7= ba
AA $ # algebra $ : A! A
a$$ = a ,
(ab)$ = b$a$
(2a+ !b)$ = 2a$ + !b$
a, b 5 A 2,! 5 C ·
A 8.8 : A! Ra, b 5 A 2 5 C
8a8 4 0, 8a8 = 0 9 a = 0
82a8 = 28a88a+ b8 : 8a8+ 8b88ab8 : 8a88b8
topologia uniforme U
a 5 A
6, > 0, U+(a) = {b 5 A| 8a# b8 < ,}
$#algebra $#algebra
8a$8 = 8a8, 6a 5 A
C$#algebra A $#algebra ‡
8a$a8 = 8a82, 6a 5 A‡
C$ C0(M)
M$
|| f ||& ='
|f(x)| .
C$ B(H)
H || · ||H $
||B|| = {||B3||H |3 5 H, ||3||H : 1} .
H n B(H)
Mn(C) M, n " n M$
M
||B|| = M$M .
C(M)
M C$#C$# C
M C M
C$#‡
C$# C C spazio struttura
C CC$# C
4 : C !C | 6a, b 5 C 4(ab) = 4(a)4(b)
‡
>C caratteri.
4(I) = 1.>C C
C 4n 5 >C4 5 >C a 5 C {4n(a)} 4(a)
C. C >CC >C
>C idealemassimale CC$# C
% 5 >CC = ker(%) , C Ker(%) C.
I CC C/I C ;/I /= C,
C ! C/I % 5 >C
C =C , C < =
$&1
&2
%
0 : C !C
0(a) = &1
% C%(a) = &1
I = {a =
$0
&
%}
• •
C =C , A < =
(
))))))*
&1&2
&n
+
,,,,,,-
• • • • · · · •
c 5 C trasformata di Gel"fand c>C
>c : 4 5 >C ! >c(4) 5 C, >c(4) = 4(c)
>c c. C>C C>C c 5 C.
C C$# c ! >cC in C
4>C5;
||>c||& = ||c|| , 6c 5 C
|| · ||& C4>C5
M
C$ # algebra C(M)#C ( ) M
m 5M %m 5 C ( )
%m : C(M)! C, %m (f ) = f(m)
Im = ker(%m)
C(M) M
%m #C(M)
C(M) Im m 5M
C$
C$
$
AM
>AA (A)
>APrim(A)
E ! M M
E = #(E,M)
C&(M) M.
M
A =C&(M) E C&(M)
E = #(M,E) E EC&(M) Em E m 5 M Em = E/EIm
Im = C(M) Im = {f 5 C&(M) | f(m) = 0}.
m M V (m)
M
M.
M E
0
0 : E !M
0 1(m) fibra m 5 M
m
m M Um 'U(m) =M k %m
%m : Um " Rk ! 0!1(Um)
x 5 Um v 5 Rk
0 > %m(x, v) = x
v !%m(x, v) Rk
0!1(x)
U %
0
U " Rk U
0 : E ! M sezione
s : M ! E 0 > s = identitaM
M
M #(M,E).
m M V (m) = V = cost.
V m M
M"V M.
M = S1, U "R U
S1 U "RS1 " R
A
A C E CA
A" E < (a, -)! a- 5 Eab(-) = a(b-), (a+ b)- = a- + b-, a(- + 5) = a- + a5
-, 5 5 E a, b 5 A.
A
E "A < (-, a)! -a 5 E(-)ab = (-a)b, -(a+ b) = -a+ -b, (- + 5)a = -a+ 5a
-, 5 5 E ; a, b 5 A.
A E
(a-)b = a(-b), 6- 5 E , a, b 5 A
{en} EE
#
n
enan
an 5 A {en}E
- 5 E?
n enan
E A
E FE "
F = E , E "
M, N A 1 : M! N& : E ! N & :
E !M 1 > & = &.
id : M %! M
& ? @ 1
& : E #! N
1 > & = &
AE 1 : AN ! E
AN N AA & : E ! AN
id : AN %! AN
& ? @ 1
id : E #! E
1 > & = idE
p 5 EndAAN + MN (A)
N "N A
p = & > 1.
p2 = & > 1 > & > 1 = & > 1 = p p
AN
AN = pAN , (I# p)AN
1 & E pAN
Ep 5 MN (A), p2 = p, E = pAN
E pAN
E Ap
E = {5 = (51, 52, ..., 5N ) ; 5 5 A, p5 = 5}.
M E A =C&(M)
#(E,M) E !M, E
A = C&(M) N p 5 MN (A)
#(E,M) #(E,M) = pAN
C$# A
AE A( , )A : E " E ! A
(-1, -2a)A = (-1, -2)A a
(-1, -2)$A = (-2, -1)A(-, -)A 4 0 , (-, -)A = 09 - = 0 ,
-1, -2, - 5 E , a 5 A ·AA.
- 5 E
||-||A '<
|| (-, -) ||
A E|| · ||A E A -1, -2 5 E
{(-1, -2)A}(-1, -2)A A.
A E
A C$# A
( , )A : E " E #! A, (a, b)A ' a$b , 6a, b 5 A.
A||a||A =
<|| (a, a)A || =
<||a||2 = ||a||.
A = C(M).
(a, b) (x) = a$(x) b(x)
AN (a1, ..., an) 5 AN
((a1, ..., an), (b, ..., bn))A '?
k a$kbk
(a1, ..., an)a ' (a1a, ..., ana)
a, ak, bk 5 A.
||(a1, ..., an)||A ' ||#
k
a$kak||
AN A
#(E,M) A = C(M)
M
E ! M, #(E,M) C(M)
( , )Ep: Ep " Ep ! C Ep
C(M) #(E,M)
(-1, -2) (p) = (-1(p), -2(p))Ep, 6-1, -2 5 #(E,M), p 5M
C$ # .
A B C$#E A( , )A B
B ( , )B
E
(-, 5)B 3 = - (5,3)A , 6-, 5,3 5 E ;
B E( , )A (b-, b-)A : ||b||2 (-, -)A
A E ( , )B(-a, -a)B : ||a||2 (-, -)B
A =MN (C)B = C. MN (C) # C E = CN MN (C) E
C
ub = (u1b, ..., uNb)
au = aijuj
u 5 CN , a 5 MN (C), b 5 C u, v 5 E =
CN
(u, v)C '?
i uivi(u, v)MN (C) ' |u )( v| = uivj
(u, v)MN (C)w = u (v, w)C?
j uivjwj
K(H) B(H) C$#H
T HT
6, > 0, A E = H | ||TE# || < ,.
K(H) B(H)
I
T H. H{%n}n'N
T%n = &n%n &n ! 0 n!B.
T H {%n}n'N {'n}n'NH T
T =#
n(0
µn(T ) |'n )(%n| ,
0 : µj+1 : µj
T = U |T | |T | =*T $T
{µn(T )}|T | {%n} 'n = U%n {µn(T )}
T, µ0(T ) = ||T ||.
T 5 K(H)
µn(T ) n!B.
2 5 R+ 2 T 5 K(H)
µn(T ) = O(n!'), n!B,
AC <B :µn(T )
n': C, 6n 4 1.
L1
T 5 L1
(T ) '#
n
(T 5n, 5n) ,
{5n}n'NT. (T ) =
?&n=0 µn(T ).
L1
C |µn(T ) : C k!1 L1
N!1#
n=0
µn(T ) : C N
L(1,&) T 5 L(1,&)
(T ) =)&
1
N
N!1#
0
µn(T ).
r,(T ) = ,1
N
N!1#
0
µn(T )
6T 4 0, T 5 L(1,&) $N (T ) '1N
?N!10 µn(T )
,
l&(N)
,{$ } 4 , $ 4 .
,{$ } = {$ }, {$ }
,{$1, $1, $2, $2, $3, $3, ...} = ,{$N}.
,{$ } = ,{$ }
{$N}H
/
(A,H, D) AH D = D$
H
(D # &)!1 & /5 R H
[D, a] ' Da# aD 5 B (H), 6a 5 A.
H Z2
$ H $ = $$, $2 = 1,
$D +D$ = 0,
$a# a$ = 0, 6a 5 A
H
H =(1 + $)
2H,(1# $)
2H = HL ,HR
D
D
|&k|!B k !B. (D # &)!1
µk((D # &)!1) ! 0 |&k| = µk(|D|) ! B.
D
D
(A,H, D) n |D|!1
1/n |D|!n
Aˆ
a ' ,( | |! ).
|D|!n a L(1,&)
|D|!n
A. |D|!n
µj : C j!1 j !B C
ˆI = ,| |! = ,
!#
=
µ (| |! ) = ,
!#
=
=
C |D|!n
A
A S(A).
d(%,3) =a'A
{|a(%)# a(3)| : || [D, a] || : 1}, 6%,3 5 S(A) .
A = C2 < a =
$&1 0
0 &2
%D =
$0 m
m 0
%
m m 5 C [D, a]
[D, a] = (&2 # &1)$
0 m
m 0
%
|| [D, a] || : 1
[D, a]†[D, a]
|| [D, a] || = [D, a]†[D, a] = |m|2|&2 # &1|2$
1 0
0 1
%
|| [D, a] || = |m| |&2 # &1| : 1& |&2 # &1| : | 1m= 1, 2
1 · a = &1
2 · a = &2
d(1, 2) = sup{| 1(a)# 2(a) | : || [D, a] || : 1}
= sup{|&1 # &2| : |&1 # &2| : | 1m|}
= | 1m| .
D m ! B & d(1, 2) ! 0 m ! 0 &d(1, 2)!B
ˆa = (a|D|!n)
|D| =*D†D =
$|m| 0
0 |m|
%
ˆa = (a|D|!n) =
$*1
|m|n 0
0 *2|m|n
%=
1
|m|n
$&1 0
0 &2
%=
1
|m|n (&1 + &2)
Y D
(M, g) g
Spin(n)
(A,H, D) n M
A = C&(M) M C
H =L2(M,S) ' M
L2(M,S) g
(',%) =
ˆdµ(g)'(x)%(x) =
ˆdx*#g '(x)%(x)
D / = dxµ/µ
g
f A H
(f ')(x) ' f(x)'(x) , 6f 5 A,' 5 H
(eµa , a = 1, 2, ...n) n M ((µ, µ =
1, 2, ...n) M
{gµ"} {-ab}
gµ" = eµae"b-
ab , -ab = eµae"b gµ".
Cl(M) M x 5M
ClC(T $xM) #(M, Cl(M))
$#
$ : #(M, Cl(M))! B(H),
$(dxµ) ' $µ(x) = $aeµa , µ = 1, ..., n
{$µ(x)} {$a}
{$µ(x), $"(x)} = 2gµ"(x) µ, " = 1, 2, ..., n
{$a, $b} = 2-ab a, b = 1, 2, ..., n
D ' #i$ > "s
D = $µ(x) ((µ + /Sµ ) = $aeµa ((µ + /S
µ )
"s
"sµ = (µ + /s
µ = (µ +1
4/abµ $
a$b
D
D2 = "s +1
4R#)
µ"
R gµ" "s
"s = #gµ"("sµ"s
" # #)µ""s
))
#)µ"
M
# = i!n/2$1 · · · $n
D #D+D# = 0 #2 = 1 #$ = #
i!n/2
(A,H, D) M
M A AM
M
d(p, q) = supf'A
{|f(p)# f(q)| : || [D, f ] || : 1}, 6p, q 5M .
MˆM
f ' c(n) ,(f |D|!n) , 6f 5 A
c(n) ' 2(n![n/2]!1)0n/2n#(n
2).
(C(M), L2(S,M), i$µ(µ) C(M)
M L2(S,M)
M i$µ(µ A = C(R), H =L2(R), D =ddx
x 5 RS(A)
x(f) = f(x), 6x 5 S(A), f 5 C(R).
x y S(A)
d(x, y) = sup{|f(x)# f(y)| : || [ ddx
, f ] || : 1}
= sup{|f(x)# f(y)| : |f "(x)| : 1}= |x# y|
R
J : H! H
(A,H, D) n
J : H! H
1a. J2 = .(n)I1b. JD = ."(n)DJ
1c. J$ = (#)n/2$J
.(n) ."(n)
.(n) = (1, 1,#1,#1,#1,#1, 1, 1) ,."(n) = (1,#1, 1,#1, 1,#1, 1, 1) , n = 0, 1, ..., 7
6a, b 5 A Ab0 : b0 = Jb$J!1
[a, b0] = 0 2.a
[[D, a], b0] = 0 2.b
2a
J HA
a5b ' aJb$J!15 , 6a, b 5 A
a 5 A H Ja$J!1
J C
J' = C' ' $0$2' , 6' 5 H
1# 2
n
(A,H, D) J
n
1/n
a 5 A a [D, a]
*k k * B(H)
*(T ) = [|D|, T ]
H& ' Ck (Dk)
A
J A
$ n
c
$ = 0D(c)
p (n#p)
AM
(A,H, D, J, $) A
M (C&(M), L2(M,S), , C, $5)
(A,H, D, J, $)
(A,H, D)
A.
A C.$A = ,p$pA
0 A $0A = A $1A
A *a, a 5 A
*(ab) = (*a)b+ a*b , 6a, b 5 A*(2a+ !b) = 2*a+ !*b , 62,! 5 C
*(a) = *(Ia) = (*I)a + I*a &*I = 0.
/ 5 $1A
/ =#
i
ai*bi , ai, bi 5 A.
* : A!$1A ,
A $1A.
p $pA
$pA = $1A$1A · · ·$1A6 78 9p!volte
(a0*a1)(b0*b1) ' a0(*a1)b0(*b1)
= a0*(a1b0)*b1 # a0a1*b0*b1
$pA
/ = a0*a1*a2 · · · *ap , ai 5 A.
$A* * : $pA! $p+1A
*(a0*a1*a2 · · · *ap) ' *a0*a1*a2 · · · *ap .
*2 = 0 ,
*(/1/2) = *(/1)/2 + (#)p/1*/2 , /1 5 $pA, /2 5 $A.
$A* p
Hp($A) ' (* : $pA! $p+1A)/ (* : $p!1A! $pA)
H0($A) = C.
$AA-A
(m : A-A! A) , m(a- b) ' ab.
1 - a # a - 1 a 5 A.?aibi = m(
?ai-bi) = 0
?ai-bi =
?ai(1-bi#bi-1)
* : A! (m : A-A! A) *a ' 1 - a # a - 1
*(ab) = (*a)b+ a(*b). $1A (m : A-A! A)
$1A + ker(m : A-A! A)
*a 3 1- a# a- 1#
ai*bi 3#
ai(1- bi # bi - 1)
$1A (m : A-A! A) *
* : A #!$1A, *a = 1- a# a- 1 .
A = C(M) M
C A-A S(M "M)
Af 5 A (x1, x2) 5 S(M "M)
*f · (x1, x2) ' (1- f # f - 1)(x1, x2) = f(x2)# f(x1).
$1A$pA f p + 1
f(x1, · · · , xk!1, xk+1, · · · , xp+1) = 0.
*f(x1, · · · , xp) 'p+1#
k=1
(#)k!1f(x1, · · · , xk!1, xk+1, · · · , xp+1) .
(A,H, D)
0D : $A #! B(H) , $A = ,p$pA0D(a0*a1 · · · *ap) ' a0[D, a1] · · · [D, ap] , aj 5 A
* [D, ·] A.
(*a)$ ' #*a$ [D, a]$ = #[D, a$] 0D(/)$ = 0D(/$) / 5 $A0
0D($A)
0D(/) = 0 0D(*/) = 0.
0D(/) = 0 0D(*/) 7= 0
0D$A J0 ' ,pJ
p0
Jp0 ' {/ 5 $pA, 0(/) = 0 }
J = J0 + *J0 $A.
A
$DA ' $A/J + 0D($A)/0D(*J0).
$D(A) $A J
$pDA = $pA/Jp.
J *
$DAd : $p
DA #! $p+1D A , d [/] ' [*/] + [0D(*/)]
/ 5 $pA [/] $pDA $p
DA
$pD + 0D($
pA)/0D*((J0 C $p!1A))
/p =#
j
aj0[D , aj1][D , aj2] · · · [D , ajp] , aji 5 A.
.0
1#
j
[D , bj0][D , bj1] · · · [D , bjp!1] , bji 5 A#
bj0j [D , bj1][D , bj2] · · · [D , bjp!1] = 0
@A
B
d
C
D#
j
aj0[D , aj1][D , aj2] · · · [D , ajp]
E
F =
C
D#
j
[D , aj0][D , aj1] · · · [D , ajp]
E
F
A = C, C < a =
$&1IdimHL 0
0 &2IdimHR
%
H = HL,HR < ' =
$'L
'R
%
$ =
$#I HL 0
0 I HR
%
D =
$0 M
M$ 0
%
Y = {1, 2} d(1, 2) = 1! !
M$M
$1A Y " Y
Y " Y (1, 2) (2, 1)
$1Ae e(1) = 1 e(2) = 0 (1 # e)(1) = 0, (1 # e)(2) = 1.
e*e, (1# e)*(1# e) .
e*e(1, 2) = #1 (1# e)*(1# e)(1, 2) = 0
e*e(2, 1) = 0 (1# e)*(1# e)(2, 1) = 0
e*e(2, 1) = e[(1 " e)(2, 1) # (e " 1)(2, 1)] = e[2e(1) # e(2)1] = e(2) = 0
2 5 $1A 2 = &e*e+µ(1#e)*(1#e)&, µ 5 C. * : A! $1A
a 5 A
*a = (&1 # &2)e*e# (&1 # &2)(1# e)*(1# e) = (&1 # &2)*e .
0(e*e) ' e[D, e] =
$e(1)IH1 0
0 e(2)IH2
%$0 [e(2)# e(1)]M
#[e(2)# e(1)]M$ 0
%=
$0 #M0 0
%
0((1# e)*(1# e)) ' (1# e)[D, 1# e] =
$0 0
#M$ 0
%
2 = &e*e+ µ(1# e)*(1# e)
0(2) = &0(e*e) + µ0((1# e)*(1# e)) =
$0 &M
#µM$ 0
%
& µ & ' 1# 4 µ ' 1# 4$.
0(e*e*e) ' e[D, e][D, e] =
$#MM$ 0
0 0
%
0((1# e)*(1# e)*(1# e)) ' (1# e)[D, 1# e][D, 1# e] =
$0 0
0 #M$M
%.
0(*2)
0(*2) = &0(e*e*e) + µ0((1# e)*(1# e)*(1# e)) = #(&+ µ)
$MM$ 0
0 M$M
%
(A,H, D) n
f 5 A
0(*f) ' [D, f ] = $µ(x)(µf = $(dMf) ,
$ : #(M,C(M)) #! B(H) $ dMM fj 5 A
0(f0*f1...*fp) ' f0[D, f1]...[D, fp] = $(f0dMf1 · ... · dMfp) ,
dMfj Cl1(M)
fj Cl0(M)
· C(M) = ,kCk(M)
M?
j fj0dMf j
1 f j0 , f
j1 5 A,
$1DA
!1(M)
$1DA + !1(M) .
$1DA B(H)
$ !1(M)
f 5 A,
2 =1
2(f*f # (*f)f) 7= 0 ,
*2 = *f*f .
0D(2) =1
2$µ(f(µf # ((µf)f) = 0 ,
0D(*2) = $µ$"(µf("f =1
2($µ$" + $"$µ)((µf("f) = #gµ"(µf("f I2n/2 7= 0
*2
A *2
n
(A,H, D) $DA = ,p $pDA
A
$1DA ) 5 $2
DA
) = dA+A2 .
A A =?
j aj [D, bj ] , aj , bj 5 AA A$ = A A
dA 5 $2DA
dA =?
j [D, aj ][D, bj ] )
A A2 dA
dA# (dA)$ =#
j
[D, aj ] [D, bj ]##
j
GD, a$j
H GD, b$j
H.
A$ = #?
j
ID, b$j
Ja$j = #
?j
ID, b$ja
$j
J+?
j b$j
ID, a$j
JA$ # A = 0
j2 ' dA # (dA)$ 7= 0
dA# (dA)$ = 0 )
U(A) A
U(A) A
A #! Au ' uAu$ + u[D, u$] , u 5 U(A) .
F
)u = dAu + (Au)2
= duAu$ + udAu$ # uAdu$ + du[D, u$] + uA2u$ +
+uA[D, u$] + u[D, u$]uAu$ + u[D, u$]u[D, u$]
= ...
= u(dA+A2)u$,
du = [D,u] udu$+(du)u$ = 0 d(u$u) = 0.
(), u) #! )u = u)u$,
SB(A) = (), ))2 = ,()2|D|!n) .
A 5 $1DA U(A)
SF (A, ') ' (', (D +A)')H , 6' 5 (D) = H, A 5 $1DA ,
(D +Au)u = ((D + u [D,u$] + uAu$)u
= Du+ u(Du$ # u$D)u+ uA
= uDu$u+ uA
= u(D +A)
('u, (D +Au)'u) = ('u$, (D +Au)u') = ('u$, u(D +A)') = (', (D +A)')
SF U(A)
D + A
J
u' = u'u† = uJ uJ' = U' .
U ' uJ uJ
uA = uAu† + u[D, u†] ,
D +A = $µ((µ + giAiµ) DA
D +A & DA ' D +A+ JAJ .
DA #! UDAU$ = UDU$ + UAU$ + UJAJU$
'DA'
A = C, CH = HL,HR < '$ = diag(#I H1 , I H2)
D =
$0 M
M$ 0
%
A A 5 $1DA
A = 0D(2) = &0D(e*e) + µ0D((1# e)*(1# e)) =
$0 &M
#µM$ 0
%
& = (%# 1) µ = (%$ # 1)
) (*2+ 22)
22 =
$0 &M
#µM$ 0
%$0 &M
#µM$ 0
%= #&µ
$MM$ 0
0 M$M
%
*2 = #(&+ µ)
$MM$ 0
0 M$M
%
)
) = #(&+ µ+ &µ)
$MM$ 0
0 M$M
%
SB(A) = ()2) = 2(|%| # ) (M$M)2 .
U(A) = U(1) " U(1) A
u =
$u1 0
0 u2
%, |u1|2 = 1 , |u22| = 1
A Au = uAu$+udu$
u$1u2 %
Au =
$0 (%u$1u2 # 1)M
(u$2u1%# 1)M$ 0
%
SB(A)
SB(A)
%
SF (A,') = (', (D +A)') 6' 5 (D) = H, A 5 $1DA .
A
D +A =
$0 %M
%$M$ 0
%
' =
$'L
'R
%
SF (A,') = '$L%M'R + '$
2%$M$'L .
U(A)
SU(2)L " SU(2)R ! U(1)" U(1)‡
gµ$ = (1, 1, 1, 1)
m("R"L + "L"R) " "†
A =C&(M)- (H,H) = (C&(M)-H)), (C&(M)-H) ,
M
H , H a 5 A qLqR H A
1(qL, qR) =
$qL 0
0 qR
%
H =(L2(M,S)- C2), (L2(M,S)- C2) ' HL ,HR ,
C2
SU(2)
SU(2)
' =
$'L
'R
%=
(
))))*
5L
$31
32
%
5R
$31
32
%
+
,,,,-
'L,R = 3 - 5L,R 3 5 L2(M,S) 5L,R 5 C2 qL 'L
qR 'R
D
D = /( - I+ $5 -DF ,
DF =
$0 M
M† 0
%.
D =
$/( - I2 $5M$5M† /( - I2
%
"
M =
$mI2 0
0 mI2
%
$5
cL,R L,R J
H SU(2)
M3(C) SU( , C U(1)
$1D = ($1
D(M)- $0D(Z2)), ($0
D(M)- $1D(Z2))
A $1D
) ) = dA+A2
A 5 $1D
A
A =#
i
ai [D, bi]
=
$aLi 0
0 aRi
%K$/( - I2 $5M$5M† /( - I2
%,
$bLi 0
0 bRi
%L
=
$aLi 0
0 aRi
%$(/(bLi ) $5(MbRi # bLi M)
$5(M†bLi # bRi M†) (/(bRi )
%
=
$aLi (/(b
Li ) $5aLi (MbRi # bLi M)
$5aRi (M†bLi # bRi M†) aRi (/(bRi )
%
$5
aL(R), bL(R)
G/(, bLi
H= /(bLi
bLi A
A = A†
A =
$/AL $5(%# %0)
$5(%† # %†0) /AR
%
/AL(R) ' $µAL(R)
µ %
AL(R)µ '
#
i
aL(R)i (µb
L(R)i , %# %0 ' aLi (MbRi # bLi M)
%0SU(2)
dA = dA†
dA = [D,A] =
$/( /AL + $5M$5(%† # %†0)# $5(%# %0)$5M† #$5/((%# %0) + $5M /AR # /AL$
5M$5M† /AL # /AR$
5M† # $5/((%† # %†0) /( /AR + $5M†$(%# %0)# $5(%† # %†0)$5M
%
$5M = #M$5 $5% = #%$5
dA =
$/( /AL #$5/((%# %0)
#$5/((%† # %†0) /( /AR
%
A2
A2 =
$/AL /AL + $5(%# %0)$5(%† # %†0) /Al$
5(%# %0) + $5(%# %0) /AR
$5(%† # %†0) /AL + /AR$5(%† # %†0) /AR /AR + $5(%† # %†0)$5(%# %0)
%
) =
$/( /AL + /AL /AL + $5(%# %0)$5(%† # %†0) #$5/((%# %0) + /AL$
5(%# %0) + $5(%# %0) /AR
#$5/((%† # %†0) + $5(%† # %†0) /AL + /AR$5(%† # %†0) /( /AR + /AR /AR + $5(%† # %†0)$5(%# %0)
%
Fµ" ' (µA" # ("Aµ + [Aµ, A" ]
$µ" ' 12 [$µ, $" ] /D% ' (/(+ /AL)%#% /AR
)
) =
$12$µ"F
µ"L + (%%† # %0%†0) #$5 /D%$5( /D%)† 1
2$µ"Fµ"R + (%†%# %†0%0)
%
)11
/( /AL + /AL /AL = $µ$"((µA" +AµA")
= ($µ" + gµ")((µA" +AµA")
=1
2($µ" # $"µ)((µA" +AµA") + gµ"((
µA" +AµA")
=1
2$µ"((
µA" # ("Aµ +AµA" #A"Aµ) + gµ"((µA" +AµA")
=1
2$µ"F
µ" + gµ"((µA" +AµA")
gµ"((µA" +AµA")
SB =1
N )2 =
ˆd4x [
1
4Fµ"L FL
µ" +1
4Fµ"R FR
µ" + (D%)†D%+ (%%† # %0%†0)2] .
V (%,%†) = (%%† # %0%†0)
2 |%min|2 = %20A u!1Au + u!1[D,u] u
SU(2) %
H SU(2)
% = aLi (MbRi # bLi M)
=
$x #y$
y x$
%K$m 0
0 m
%$x" #y"$
y" x"$
%#$
x"" #y""$
y"" x""$
%$m 0
0 m
%L
=
$m(xx" # xx"" + y$y"" # y$y") #m(xy"$ # xy""$ + y$x"$ # y$x""$)
m(yx" # yx"" # x$y"" + x$y") m(yy""$ # yy"$ + x$x"$ # x$x""$)
%
' m
$h1 #h$2h2 h$1
%' hM .
%
'L(M+ %)'R
M+ % = m
$h1 + 1 #h$2h2 h$1 + 1
%
'$
H1 #H$2
H2 H$1
%
M+ % ' H
'L(M+ %)'R = 'LH'R
V (%,%†) =I(%+ %0)(%† # %†0)
J2
H
V (H) =I(H #M+ %0)(H
† +M† # %†0)J2
=I(H #M+ %0)(H
† +M† # %†0)J2
= (HH† #H0H†0)
2
H
$H1
H2
%
SU(2) H
LH = (D%)†D%+ (HH† #H0H†0)
2 + ('LH'R + 'RH†'L)
SF =M' | D +A | '
N
= 'L(/( + /AL)'L + 'L(/( + /AR)'L + ('LH'R + 'RH†'L) .
A
A =C&(M)-AF ,
C&(M) M AF
AF = M3(C),H, C
G = U(3)" SU(2)" U(1)
U(1) CSU(2) H ' (I2, i!pauli) U(3)
M3(C)AF
U(1)
H
H =L2(M ; S)-HF ,
HF
HL ,HR ,HcR ,Hc
L
AF HF
(c, q, b) 5 AF q, b, c 5 H, C, M3(C)1 AF HF
1(c, q, B) '$1w(q,B) 0
0 1s(c)
%
B = (b, b$)
1w(q,B) =
$q - IN - I3 0
0 B - IN - I3
%:
#######!
(
)))*
$uiLdiL
%
(uiR)
(diR)
+
,,,-"N
1s(c) =
$I2 - IN - c 0
0 I2 - IN - c
%:
#######!
(
)))*
$uiLdiL
%c
(uiR)c
(diR)c
+
,,,-"N
N = 1
1(c, q, B) '
(
)))*
q - I3 0
0 B - I30
0I2 - c 0
0 I2 - c
+
,,,-:
#######!
(
)))*
QL
QR
QcR
QcL
+
,,,-
Q u, d QcR ' (QR)c = (Qc)L
# = $5 , $F J =J , JF $5
J M $F JF
$F =
(
)))*
#I6NI6N
#I6N+I6N
+
,,,-, JF = J†
F =
$0 I12N
I12N 0
%C
C JF
C
D = /( - I+ $5 -DF
DF
DF =
(
)))*
0 M 0 0
M† 0 0 0
0 0 0 M$
0 0 MT 0
+
,,,-& D =
(
)))*
/( $5M 0 0
$5M† /( 0 0
0 0 /( $5M$
0 0 $5MT /(
+
,,,-
M
M =
$Mu - I3 0
0 Md - I3
%
Mu, Md
Mu = (mu,mc,mt)
Md = CKM (md,ms,mb)
Mu Md mu md
M 6" 6
A 24"24
A =#
i
2id!i =#
i
2i[D,!i]
2i,!i 5 A
2i '
(
)))*
q"i - I3 0
0 B"i - I3
0
0I2 - c"i 0
0 I2 - c"i
+
,,,-; !i '
(
)))*
qi - I3 0
0 Bi - I30
0I2 - ci 0
0 I2 - ci
+
,,,-
A(AL, B,%, G) =
(
)))*
/AL - I3 $5(%# %0)- I3 0 0
$5(%# %0)- I3 /B - I3 0 0
0 0 I2 - /G 0
0 0 0 I2 - /G
+
,,,-
%# %0 '?
iB"i(M†qi #BiM†) ; Bµ ' B"
i(µBi ; Gµ '
?i c
"i(
µci%# %0 '
?i q
"i(MBi # qiM) ; Aµ
L '?
i q"i(
µqi
A
(%# %0) = (%# %0)†
AL, B,G = A†L, B
†, G†
AµL SU(2) Bµ U(1)
Gµ U(3)
AµL =
3#
i=1
AµiL
+i
2
Bµ = BµI2
Gµ =8#
a=1
Gaµ&a
2+G0
µI3
+i SU(2) I3 I2U(1) &a SU(3)
3 " 3 U(3)
SU(3) - U(1) Gµ
SU(3) G0µ U(1)
DF M3(C) A
) ) ' dA + A2
L = LB + LF = 1N )2 + 'DA'
' A
) dA A2
dA =
$[dA]1 0
0 [dA]2
%
[dA]1 =
$(/( /AL +M$5(%† # %†0)# $5(%# %0)M†)- I3 (#$5/((%# %0) +M /B # /ALM)- I3
(M† /AL # /BM† # $5/((%† # %†0))- I3 (/( /B +M†$5(%# %0)# $5(%† # %†0)M)- I3
%
=
$(/( /AL (#$5/((%# %0))- I3
($5/((%† # %†0))- I3 (/( /B
%
[dA]2 =
$I2 - /( /G I2 - (M$ /G# /GM$)
I2 - (MT /G# /GMT ) I2 - (/( /G)
%
=
$I2 - /( /G 0
0 I2 - /( /G
%
A2 =
(
)))*
( /AL /AL + $5(%# %0)$5(%† # %†0))- I3 ( /AL$5(%# %0) + $5(%# %0) /B)- I3 0 0
($5(%† # %†0) /AL + /B$5(%† # %†0))- I3 ( /B /B + $5(%† # %†0)$5(%# %0))- I3 0 0
0 0 /G/G
0 0 /G/G
+
,,,-
)
) =
(
)))*
12$µ"F
µ"L + (%%† # %0%†0) #$5 /D%$5( /D%)† 1
2$µ"Bµ" + (%†%# %†0%0)
0
012$µ"G
µ" 0
0 12$µ"G
µ"
+
,,,-
/D% ' (/( + /AL)%# % /B
AF U(3)" SU(2)" U(1) U(3) = SU(3)" U(1)
U(1)
unimodularita A (A+
JAJ) = 0
A+JAJ =
(
)))*
/AL - I3 + I2 - /G $5(%# %0)- I3 0 0
$5(%† # %†0)- I3 /B - I3 + I2 - /G 0 0
0 0 I2 - /G+ /B - I3 $5(%† # %†0)- I30 0 $5(%# %0)- I3 I2 - /G+ /AL - I3
+
,,,-
(A+ JAJ) = 2 /AL - I3 + I2 - 4/G+ 2/B - I3)= (4 /G0 /G0 - I3 + 2/B - I2)= 12 /G0 + 4/B = 0
U(1) G0µ
U(1)
F 2
SB =1
N )2 =
ˆd4x [
1
4Gµ"Gµ" +
1
4Fµ"Fµ" +
1
4Bµ"Bµ" + (D%)†D%+ (%%† # %0%†0)
2] .
% SU(2)L
V (%) = (%%† # %0%†0)2 .
SU(2)
%
% = q"i(MBi # qiM)
=
$x" #y"$
y" x"$
%$muI3 0
0 mdI3
%$b 0
0 b$
%#$
x" #y"$
y" x"$
%$x #y$
y x$
%$muI3 0
0 mdI3
%
=
$mu(bx" + x"x# y"y) #md(y"$b$ + x"y$ + y"x$)
mu(y"b+ y"x+ x"$y) md(x"$b$ # y"y$ + x"$x$)
%- I3
'$41mu #4$
2md
42mu 4$1md
%- I3
=
$41 #4$
2
42 4$1
%$mu 0
0 md
%- I3
(41,42) x, x", y, y", b,
AF
SU(2)
V (4) = 3(m4u +m4
d)|4|4 # 2(%0%†0)3(m
2u +m2
d)|4|2 + (%0%†0)
2
4! 4" = 4/%0
V (4) = (%0%†0)
2G3(m4
u +m4d)|4|4 # 6(m2
u +m2d)|4|2 + 1
H
' K
&|4|4
16L2# |4|2
2L+ 1
'
K L
&
K L
SF
SF = (' | D +A+ JAJ | ') = (' | D | ')+ (' | A+ JAJ | ')
= '
(
)))*
/( $5M 0 0
$5M† /( 0 0
0 0 /( $5M$
0 0 $5MT /(
+
,,,-'
+'
(
)))*
/AL + /G $5(%# %0)- I3 0 0
$5(%† # %†0)- I3 /B + /G 0 0
0 0 /G+ /B $5(%† # %†0)0 0 $5(%# %0) /G+ /AL
+
,,,-'
' '
' = (QL QR QcR Qc
L) , ' =
(
)))*
QL
QR
QcR
QcL
+
,,,-.
SF = QL(/( + /AL + /G)QL +QcL(/( + /AL + /G)Qc
L +
+QR(/( + /B + /G)QR +QcR(/( + /B + /G)Qc
R +
+[QL(M+ %)QR +QcL(M$ + %)Qc
R + h.c. ] .
HHF
L2(M ; S)
hF 5 HF
hF = hL + hR + hcR + hcL
HL HR HcR, Hc
L
x 5M '
'(x) = 'L + 'R + 'cR + 'c
L .
# a# ('L - hL + 'R - hR + 'L - hcR + 'R - hcL)
#b# ('cL - hcL + 'c
R - hcR + 'cL - hR + 'c
R - hL)
#c# ('L + 'cR)- (hR + hcL) + ('R + 'c
L)- (hL + hcR)
(c) 'L - hR'L
hR (a)
(b)
(a) (b) QL,R ' QcL,R
J (c)
P
P =1# $5
2- PL +
1 + $5
2- PR ,
PL PR PL =
(1, 0, 1, 0) PR = (0, 1, 0, 1)
P =1
2
(
)))*
1# $5 0 0 0
0 1 + $5 0 0
0 0 1# $5 0
0 0 0 1 + $5
+
,,,-.
SF = (' | PDAP | ') ,
qL,R 3L,R - q'L,R
SF = qL(/( + /AL + /G)qL + qR(/( + /B + /G)qR +
+[ qL(M+ %)qR + qcR(M$ + %)qcL + h.c. ] .
A =C&(M)-AF ,
C&(M) M AF
AF = H, C
G = SU(2) " U(1)
H
H =L2(M ; S)-HF ,
HF HL,HR,Hc
R ,HcL
AF HF
(q, b) 5 AF q, b,5 H, C 1 AF
HF
1(q,B) '$12(q,B) 0
0 11(b)
%
B = (b, b$)
12(q,B) =
$q - IN 0
0 B - IN
%:
#######!
(
)))*
$"LeL
%"N
("R)
(eR)"N
+
,,,-
11(b) =
$b$I2 - IN 0
0 b$I2 - IN
%:
#######!
(
)))*
$"LeL
%c
"N
("R)c
(eR)c"N
+
,,,-
N = 1
1(q,B) '
(
)))*
q 0
0 B0
0b$I2 0
0 b$I2
+
,,,-:
#######!
(
)))*
'L
'R
'cR
'cL
+
,,,-
' (", e) 'cR ' ('R)c = ('c)L
# = $5 , $F J =J , JF $F JF
$F =
(
)))*
#I2NI2N
#I2NI2N
+
,,,-, JF = J†
F =
$0 I4N
I4N 0
%C
C
D = /( - I+ $5 -DF
DF
DF =
(
)))*
0 M 0 0
M† 0 0 0
0 0 0 M$
0 0 MT 0
+
,,,-& D =
(
)))*
/( $5M 0 0
$5M† /( 0 0
0 0 /( $5M$
0 0 $5MT /(
+
,,,-
M
M =
$M" 0
0 Ml
%
M" , Me
M" = (m"e,m"µ,m"! )
Ml = CKM (me,mµ,m! )
M" Ml m" me
M 6" 6
SB =
ˆd4x [
1
4Fµ"Fµ" +
1
4Bµ"Bµ" + (D%)†D%+ (%%† # %0%†0)
2] .
SF = 'L(/( + /AL)'L + 'cL(/( + /AL)'
cL +
+'R(/( + /B)'R + 'cR(/( + /B)'c
R +
+['L(M+ %)'R + 'cL(M
$ + %)'cR + h.c. ] .
'L,R ' 'cL,R P
SF = 'L(/( + /AL)'L + 'cL(/( + /B)'c
L +
+['L(M+ %)'R + h.c. ] .
'DF' =4'L 'R '
cR '
cL
5
(
)))*
0 M 0 ML
M† 0 0 0
0 0 0 M$
† 0 MT 0
+
,,,-
(
)))*
'L
'R
'cR
'cL
+
,,,-
LM = ('LML'cL + '
cLML'L) .
MR
ML =1
2
$mL 0
0 0
%
mL
LM =1
2mL("L"
cL + "cL"L) .
G[D, a] , b0
H= 0 ,
b0 = Jb$J ML
[D, a] =
(
)))*
/(q $5(MB # qM) 0 (MLb$ # qML)
$5(M†q #BM†) /(B 0 0
0 0 /(b$I2 0
(M †Lq # b$M †
L) 0 0 /(b$I2
+
,,,-
b0 =
(
)))*
b!I2 0 0 0
0 b!I2 0 0
0 0 B!$ 0
0 0 0 q!$
+
,,,-
G[D, a] , b0
H=
(
)))*
0 0 0 (MLb$ # qML)q!$ # b
!I2(MLb$ # qML)
0 0 0 0
0 0 0 0
(M †Lq # b$M †
L)b!I2 # q
!$(M †Lq # b$M †
L) 0 0 0
+
,,,-
DF
DF =
$M TT† M$
%
M =
$0 M
M† 0
%T
T =
$0 ML
0 0
%
T
T =
$ML 0
0 0
%
'cR %! 'c
L
' #! '
' =
(
)))*
'L
'R
'cR
'cL
+
,,,-! ' =
(
)))*
'L
'R
'cL
'cR
+
,,,-
DF
O' | DF '
P=
4'L 'R 'c
L 'cR
5
(
)))*
0 M ML 0
M† 0 0 0
M †L 0 0 M$
0 0 MT 0
+
,,,-
(
)))*
'L
'R
'cL
'cR
+
,,,-
= 'LM'R + 'RM†'L + 'cLM$'c
R + 'cRMT'c
L +
+'LML'cL + 'c
LM†L'L .
O' | DF '
P=
:'LM'R + 'c
LM$'cR + h.c.
;+:'LML'
cL + h.c.
;.
A '?
i ai [D, bi] ,
A =
(
)))))*
qi/(q!i $5qi
4MB
!i # q
!iM5
$5qi4MLb
!$i # q
!iML
50
$54M†q
!i #B
!iM†
5Bi/(B
!0 0
$5b$i
4M †
Lq!i # b
!$i M
†L
50 b$i /(b
!$ 0
0 0 0 b$i /(b!$
+
,,,,,-
=
(
))))*
/AL $5 (%# %0) $5 (3# 30) 0
$54%† # %†0
5/B 0 0
$543† # 3†
0
50 /b 0
0 0 0 /b
+
,,,,-,
3
(3# 30) = qi4MLb
!$i # q
!iML
5
=
$31 0
32 0
%ML
31, 32 b, x, y, x!, y
!
qi biI2
A+ JAJ =
(
))))*
/AL + /b $5 (%# %0) $5 (3# 30) 0
$54%† # %†0
5/B + /b 0 $5
43† # 3†
0
5
$543† # 3†
0
50 /b + /B $5
4%† # %†0
5
0 $5 (3# 30) $5 [(%# %0)] /b + /AL
+
,,,,-
S0F
SF = S0F + 'L
GML + $5 (3# 30)
H'cL + h.c. ,
$5 3 $5 'cL
SF = S0F + 'L [ML + (3# 30)]'
cL + h.c. ,
3 SU(2)
% 3
DA #! DuA = UDU$ + UAU$ + U(JAJ)U$
' #! 'u = U'
U = uJuJ$ U
U =
(
)))*
qu - I2b$uBu - I2b$u
I2b$u - qu
I2b$u -Bu
+
,,,-
'
(
)))*
'L
'R
'cL
'cR
+
,,,-#!
(
)))*
qu - I2b$u 'L
Bu - I2b$u 'R
I2b$u - qu 'cL
I2b$u -Bu 'cR
+
,,,-
'L 'cL SU(2)
U(1) 'R 'cR SU(2) A
UAU$
Au =
(
))))*
qub$u /ALq$ubu qub$u (%# %0)B$ubu qub$u (3# 30) q$ubu
Bub$u4%† # %†0
5q$ubu Bub$u /BB$ubu
q$ubu43† # 3†
0
5qub$u b$uqu/bbuq$u
b$uBu/bbuB$u
+
,,,,-,
% SU(2) U(1)
3
SU(2) 'L3'cL
3
ML + (3# 30) ' 3ML
[ML + 3# 30] = ML +
$31 0
32 0
%ML =
KI+
$31 0
32 0
%LML =
$1 + 31 0
32 1
%ML =
= mL
$1 + 31 0
32 0
%'$
mL31 0
mL32 0
%' 3ML .
SF = S0F + 'L3ML'
cL + h.c. ,
(31, 32)
32 = 0
'L3ML'cL = mL"L31"
cL
3 =
$31
0
%#! 1#
2
$v + H
0
%
LMmass =
mLv*2"L"
cL + h.c. .
3
) = dA+A2
# =
!
"""#
12%µ$F
µ$L + (--† ! -0-
†0) + (..† ! .0.
†0) !%5 /D- !%5( /D.) %5.M†
%5( /D-)† 12%µ$B
µ$ + (-†-! -†0-0) !-†. 0
%5( /D.)† !.†- 12%µ$b
µ$ ++(..† ! .0.†0) 0
M$%5.† 0 0 12%µ$bµ$
$
%%%&
/D3 ' #(/( + /A)3+ 3/b
SB =1
N )2
=
ˆd4x [
1
4Fµ"Fµ" +
1
4Bµ"Bµ" + (D%)†D%+ (%%† # %0%†0)
2 +
+(D3)†D3+ (33† # 303†0)
2 + %†33†%+ 3MTM$3†] .
D
DA = D+A+JAJ$, A J
DA
D DA
D ! DA = D +A+ JAJ$
A
A$ = A
A =#
j
aj [D, bj ] , aj , bj 5 A .
DA
SB(D,A) = H(3D2
A
!2)
H H !
3 D2A !2
!
d m#M g O ' D2
A/!2
(O!s) =1
#(s)
&
0
dt e!tOt!s , (s) 4 0 ,
e!tO
e!tO =#
n(0
tn"m
d
ˆM
*gdx an(x; O) .
an(x; O)
an(x; D2/!2) 3
(3O) =#
n(0
fnan(x; O) ,
fn
f0 =
&
0
du3(u)u
f2 =
&
0
du3(u)
f2(n+2) = (#1)n3(n)(0) , n 4 0 ,
3(n) n 3 3
[0, 1] f2 = f4 = 2f0 = 1
D2A !2
D2 = 2S +1
4R ,
R gµ" 2S
"s = #gµ"("sµ"s
" # #)µ""s
)) .
O
piu
SB =
ˆ
M
*gdx
2I1!
2 + I2 + I3!!2 + o(
1
!4)
3
I1 =45
802,
I2 =1
1602(#15R# 8K1|%|2) ,
R K1 = (3M †uMu + 3M †
dMd + M †eMe) M
!!2
I3 =1
1602
2240Fµ"Fµ" + 12Gµ"Gµ" + 4K1|Dµ
A%|#2
3K1R|%|2 +K2|%|4 #
9
4C2
3+
+ .
F G K2 = (3(M †uMu)2+
3(M †dMd)2 + 3(MMe)2) C2
Cµ"() = Rµ"() # gµ"[)Rµ(] +1
6(gµ)g"( # gµ(g"))R .
GeometriaRiemanniana #! Gravita
@
Geometria non commutativa #! Gravita + Y ang #Mills
G
g1, g2 5 G G
g1, g2 5 G g3 = g1g2 5 G
g1, g2, g3 5 G g1(g2g3) = (g1g2)g3 5 G
A I : 6g 5 G Ig = gI = g
6g 5 G A g!1 | gg!1 = g!1g = I
G G
g1, g2 5 G g1g2 = g2g1
N La a = 1, ..., N
[La, Lb] = iCcabLc .
Ccab
G
g = ei,aLa , 6g 5 G
/a Ccab
!Ccab = #Cc
ba
CnabC
dnc + Cn
bcCdna + Cn
caCdnb = 0
G
N U(1) RN
SU(2)
G V 0
G G GL(V )
V
R C0 G V 0 : G #! GL(V )
0(I) = 1 GL(V )
6g1, g2 5 G, 0(g1)0(g2) = 0(g1g2)
0(g!1) = 0(g)!1 g
g
0
N 0 0"
C N " N 0(g) ! 0"(g) = C0(g)C!1 g 5 G
0(g1)0(g2) = 0(g1g2)& 0"(g1)0"(g2) = C0(g1)C
!1C0(g2)C = C0(g1g2)C!1 = 0"(g1g2).
W = V
G
0(g) W W 0(g)W =
W, 6g 5 G
(
))*
01(g) · · ·
02(g)
+
,,-
0i(g) i
0c.r.(g)
Di(g)
0c.r.(g) = ,ni=10i(g)
Zn n Zn {0, 1, ..., n#1} n 0
{1, e2/i/n, e4/i/n, ..., e2/(n!1)/n} = {e2/ik/n}k=0,1,...,n!1
1 k
GL(n,C) n " n
In 2n2
GL(n,C). SL(n,C)2n2 # 1
O(n) n" n ggT = I, 6g 5 O(n)
SO(n) O(n)
Rn n(n# 1)/2
O(1, 3) g 4"4 g-gT = - - = (1,#1,#1,#1).U(n) n " n gg† = g†g = I n2
SU(n)
n2 # 1
R2n2
U(1)
R2 U(1) SO(2)
U(1) SO(3)
vc ' ! 5 [0, 1[
0(g)
0† = 0!1
01 02 G H1 H2
01 - 02 H1 -H2 < '1 - '2
(01 - 02)(g)('1 - '2) ' 01(g)'1 - 02(g)'2
An = SU(n+ 1)
Bn = SO(2n+ 1)
Cn = Sp(2n)
Dn = SO(2n)
E6, E7, E8, F4, G2.
SU(N) N " N
U 5 SU(N)
U †U = 1 (unitaria)
U = 1 (speciale)
SU(N)
N2 # 1
H H† = H
U = eiH .
N2 # 1 N "N
U = exp
(
*iN2!1#
i=1
)a#a
+
-
#a SU(N)
[#a, # b] = ifabc# c.
SU(N) N
%i 'i
%i ! U ij%
j
'$i ! '$
j (U†)ji
'$i %
i SU(N)
SU(N)
'$i %
i %i
T i1i2...iMj1j2...jN
! (U i1k1U i2k2...U iM
kM)(U †l1
j1U †l2
j2...U †lN
jN)T k1k2...kM
l1l2...lN
SU(3) " SU(2) " U(1)
SU(5)
= ,
$ #morfismo A B 0 : A! B
0(ab) = 0(a)0(b)
0(a$) = 0(a)$, 6a 5 A
rappresentazione 0 C$#algebraA H $ morfismo
A H
0 : A!B (H)
A
rappresentazion fedele $ # morfismo0
0
ker(0) = {a 5 A |0(a) = elemento neutro diB}irriducibile H
0 H0(a) B (H)
0(a)
I A
A6a 5 A, b 5 I & ab 5 I ( ba 5 I )
II[a, b]
A $ # ideale
C$ # algebraA semplice
I essenziale A AI C$ # algebraA primitivo
60 diA , I = (0)
I AA
insieme risolvente r(a) a 5 A C
r(a) = {& 5 C| (a# &I) A}
& (a# &I)!1 a &
r(a) C +(a) a
+(a) = {& 5 C | (a# &I) A}
A C$#algebra a
C. 1(a) a 5 A
1(a) = {|&|, & 5 +(a)}
C$# algebra
||a||2 = 1(a$a), 6a 5 A.$#algebraA a normale a$a = a a$. autoaggiunto
a = a$. C$ # algebra
+(a) D [#||a|| , ||a|| ] +(a2) D [0, ||a||2 ] proiezione
A p = p$ = p2 isometria parziale v 5 A v$v
a unitario a$a = a a$ = I $#algebra
M3(C)U(3) C U(1)
|a| ' (a$a)1/2 a 5 A.
A a = u|a|,u ' a|a|!1
T 5 B (H) HT = U |T | U T
stato C$ # algebra A
% : A!C,
%(I) = 1
%(a$a) 4 0, 6a 5 A||%|| ' sup
a'A{|%(a) : ||a|| : 1|} = 1.
%
S(A) A S(A)
6%1,%2 5 S(A) 0 : & : 1
&%1 + (1# &)%2 5 S(A)
% puro
PS(A)
C$
B(H)
% 5 S(A) (H-,0-)
A
N- A
N- = {a 5 A |%(a$a) = 0}.
%(a$b$b a) : ||b||2%(a$a) N- ApreHilbertianoA/N-, a
[a]
A/N- ' { [a] = a+ c , 5 A, 5 N-}
A/N- "A/N- ! C , (a+N-, b+N-)! %(a$b)
A/N- a 5 A 0(a) 5 B(A/N-)
0
0(a)(b+N-) ' ab+N-
||0(a)(b+N-)82 = %(b$a$ab) : ||a||2%(b$b) = ||b+N-||2 ,
||0(a)|| : ||a|| 0(a) 5 B(A/N-). 0(a)
0-(a) 5 B(H-) 0-(a1a2) = 0-(a1)0-(a2)
0-(a$) = 0-(a)$ $#
0- : A #! B(H-), a! 0-(a).
A = C0(M)
M * *x0 = a(x0)
*x0
N0x0= {a | a(x0) = 0}
A = M2(C)
%1
$Ka11 a12a21 a22
L%= a11 , %2
$Ka11 a12a21 a22
L%= a22
%1 %2
N1 =
!K0 a120 a22
LQ, N2 =
!Ka11 0
a21 0
LQ
H1 =
!Kx1 0
x2 0
LQ+ C2 =
!X =
$x1x2
%Q, < X,X
!>= x$1x
!1 + x$2x
!2.
H2 =
!K0 y10 y2
LQ+ C2 =
!Y =
$y1y2
%Q, < Y, Y
!>= y$1y
!1 + y$2y
!2.
H1 H2$0
0
%
a 5M2(C)
01(a)
Kx1 0
x2 0
L=
Ka11x1 + a12x2 0
a21x1 + a22x2 0
L' a
$x1x2
%
02(a)
K0 y10 y2
L=
Ka11y1 + a12y2 0
a21y1 + a22y2 0
L' a
$y1y2
% .
gµ" = (1,#1,#1,#1)
(i$µ(µ #m)' = 0
' =
$'L
'R
%
$0 =
$0 II 0
%, $i =
$0 +i
#+i 0
%, i = 1, 2, 3
'L 'R $5
$5 ' i$0$1$2$3 =
$#I 0
0 I
%
'
{$µ, $"} = 2gµ"
x0 ! x4 = ix0
#!x ! #!x
gµ" = (#1,#1,#1,#1)
{'(x),'(y)} = 0R'(x),'(y)
S= 0
R'(x),'(y)
S= 0
'
'
' '
{$µ, $"} = #2*µ"
$4 ' i$0 = i
$0 II 0
%, $i =
$0 +i
#+i 0
%, i = 1, 2, 3 .
O(4)
' '
$5 ' $1$2$3$4 =$#I 0
0 I
%
'' '$5' '$µ'
S = #ˆ
d4x (i$µ(µ # im)' .
'
' '
(/p+ im)/(p2 +m2) i
C$
$
$
