前回のまとめ
Lagrangian を決める基準
対称性
局所性 簡単な形
変換 (Aq)I =D(A)IJ qJ 表現 D(AB)IK =D(A)IJD(B)JK
[Xm,Xn]=ifmnlXl
無限小変換 A=eiaX =1+iaX+O(a2),
D(A) =eiad(X) =1+iad(X)+O(a2), [d(Xm), d(Xn)]=ifmnld(Xl)
場の変換 (Af )i (Ax ) =D (A)ij fj (x)
時空の変換 x' =Ax
状態の変換 U (A) = e iau ( X ) = 1  iau(X )  O (a 2 )
[J i , J j ] = iijkJ k
回転群O(3)
generator J i (i = 1,2,3)
既約表現は半整数 j で指定される。
Lorentz群 SO(3,1) generator J i K i (i = 1,2,3)
既約表現は半整数 ( j (  ), j (  ) ) で指定される。
scalar field f (x ) ( j ( ) , j ( ) ) = (0,0)
1
1 2 2 1
2
Lagrangian密度 L = ( f )  m f  f 4
2
2
4!
自由scalar場の量子化
scalar 場 f
要請 (i) Lorentz不変性 (ii) f → fで不変 (iii) f の2次まで

1
1 2 2


Lagrangian 密度 L =  f f  m f   =  = ( 0 , )
2
2
x
2
2

  (f )
f
Lagrangian L =  Ldx
 = (1,  2 , 3 )
運動方程式  0 (L / ( 0f )) = L / f   (L / ( f )) = L / f
 2f で微分
これを
2

   f =  m f f   f = m 2f Klein Gordon方程式
正準共役運動量  (x ) = L / f(x )
 = f
pI = L / q I
[q I , pJ ] = i IJ
正準交換関係 =量子条件
[f (x ),  (y )] x =y = i (x  y ) (x = (x 0 , x ),y = (y 0 , y ))
Hamiltonian
dx  L
H
=

f

1 2

2
2 2
H =    (f )  m f dx
F = i[H , F ]
2
0
(
0
)

Klein Gordon 方程式    f = m f

  f =  m f
2
2
2


=

m
f
f  f
正準交換関係 =量子条件
[f (x ),  (y )] x =y = i (x  y )
Hamiltonian
1 2  2
H =    (f )  m 2f 2 dx
2
0
(
2
0
)
2
2
f   f = m f
Klein Gordon方程式
量子条件
[f (x ),  (y )] x =y = i (x  y )
Hamiltonian
1 2  2
2 2
H =    (f )  m f dx
2
0
(
0
)

Klein Gordon 方程式    f = m f
2
ikx
0
解 f (x ) = e とおく (kx = k0x  kx)
k0 2 k 2 = m 2 k 0 =  m 2  k 2
2
2
f   f = m f
xとxxのf 混じる
Normal modeで書く
負のenergy! 
f は状態でなく
演算子なのでOK
dkikx  ikx 2 † ikx2 
= m
k )
 a
一般解 f (x ) = e (2,)e3 2E (a E
ke
ke
0
 kx x=0Ex

で微分

kx
独立解


d
k

ikx
†
ikx
(ak e
 ak e )  E = m 2  k 2 
 (x ) = f(x ) = 
3


(
2

)
2
i
逆に解く
 kx = Ex 0  kx 

 E = m 2 k 2 


量子条件 
0

kx
=
Ex

kx
[f (x ),  (y )] x =y = i (x  y )
ikx
0
0
Hamiltonian
1 2  2
2 2
H =    (f )  m f dx
2
(
)

Klein Gordon 方程式    f = m f
2
2
2
f   f = m f
ipy
e
 dy = 2 ( p )
dk1
† ikx
ikx
ik'xdx
e
e
(
a

a
)
e
∫Ef
f (x ) ==
k
k
(2 )3 2E
eiEt (2)3 (kk' )
eiEt


dk 1
ikx
† ikx

ik'x
(
a
e

a
)eik'xd3x
(x)e) = fdx
(x ) = 
∫ i (x
k
ke
3
iEt
iEt
(
2

)
2
i
e
e
(2)3 (kk' )
逆に解く
ikx
iEt
a
=
e
dx
(
)
i

(x
)
 k e  Ef (x ) 
E = m 2 k 2 


量子条件 
0

kx
=
Ex

kx
[f (x ),  (y )] x =y = i (x  y )
(x )eik'xdx
0
0
Hamiltonian
1 2  2
H =    (f )  m 2f 2 dx
2
(
)

Klein Gordon 方程式    f = m f
2
ikx
0
解 f (x ) = e とおく (kx = k0x  kx)
k0 2 k 2 = m 2 k 0 =  m 2  k 2
2
2
f   f = m f
xとxxのf 混じる
Normal modeで書く
負のenergy! 
dk1
† ikx
ikx
ik'x
Ef (x )e f (x
(
a

a
) f は状態でなく
dx) ==
e
∫一般解
k iEt
ke
3
(2 ) 2E
e
演算子なのでOK

x0で微分

dk 1
ikx
† ikx

ik'x
(
a
e

a
) E = m 2 k 2 
(x)e) = fdx
(x ) = 
i (x
∫
k
ke
3
iEt


(
2

)
2
i
e
逆に解く
a k = e
iEt
 (Ef (x ) i (x )) e
ikx
 kx = Ex 0  kx 

dx
 E = m 2 k 2 


量子条件 
0

kx
=
Ex

kx
[f (x ),  (y )] 
= i (x  y )
x 0 =y 0
Hamiltonian
1 2  2
2 2
H =    (f )  m f dx
2
(
)

Klein Gordon 方程式    f = m f
2
ikx
0
解 f (x ) = e とおく (kx = k0x  kx)
k0 2 k 2 = m 2 k 0 =  m 2  k 2
2
2
f   f = m f
xとxxのf 混じる
Normal modeで書く
負のenergy! 
dk
† ikx
ikx
(
a

a
) f は状態でなく
e
f
(x
)
=
一般解
ke
 (2 )3 2E k
演算子なのでOK
x0で微分
dk
ikx
† ikx

=
(
a
e

a
) E = m 2 k 2 

(
x
)
=
f
(
x
)

ke
 (2 )3 2i k


逆に解く
0


ikx
iEt
a
e
dx
=
(
)
i

(x
)
 k e  Ef (x ) 
 [ak ,a ] = (2 )3 2E  (k  k ' )
= [ eiEt ∫ (Ef(x)i(x))eikxdx ,
eiE't ∫ (E'f(y)i(y))eik'ydy ]
= ∫ ( i E i(xy)  i E' i(xy) )
eiEt eiE't eikx eik'y dx dy
= 2 EE '(2 )3 (k  k ' )
†
k'
 kx = Ex  kx 
量子条件
[f (x ),  (y )] x =y = i (x  y )
Hamiltonian
1 2  2
2 2
H =    (f )  m f dx
2
0
(
ipy
e
 dy = 2 ( p )
0
)

Klein Gordon 方程式    f = m f
2
ikx
0
解 f (x ) = e とおく (kx = k0x  kx)
k0 2 k 2 = m 2 k 0 =  m 2  k 2
2
2
f   f = m f
xとxxのf 混じる
Normal modeで書く
負のenergy! 
dk
† ikx
ikx
(
a

a
) f は状態でなく
e
f
(x
)
=
一般解
ke
 (2 )3 2E k
演算子なのでOK
x0で微分
dk
ikx
† ikx

=
(
a
e

a
) E = m 2 k 2 

(
x
)
=
f
(
x
)

ke
 (2 )3 2i k


逆に解く
0


ikx
iEt
a
e
dx
=
(
)
i

(x
)
 k e  Ef (x ) 
 [ak ,a ] = (2 )3 2E  (k  k ' )
ak 0 = 0
真空状態 0
Fock space ak1† ak2† akn† 0
†
ak 生成演算子 ak 消滅演算子
†
k'


 kx = Ex  kx 
量子条件
[f (x ),  (y )] x =y = i (x  y )
Hamiltonian
1 2  2
2 2
H =    (f )  m f dx
2
0
(
dk
†
†
E (ak ak  akak ) / 2
Hamiltonian H = 
2
(2 ) 2E
0
)
Normal mode!
Lagrangian をLorentz不変に書くため既約表現の場を使う
前回のスライドより
Lorentz群の既約表現は ( j ( ) , j ( ) ) で指定される。
( j ( ) , j ( ) ) = ( 0 , 0 ) scalar field f (x )
d (J i ) = 0
d (K i ) = 0
d(J i() ) = 0
(1/2, 0 ) right-handed Weyl spinor field  R (x )
d(J i() ) =  i / 2
d(J i() ) = 0
d (J i ) =  i / 2 d(Ki ) = i i / 2
( 0 ,1/2) left-handed Weyl spinor field  L (x )
d(J i() ) = 0
(1/2, 0 )  ( 0 ,1/2)
(1/2,1/2)
d(J i( ) ) = J i
d(J i() ) =  i / 2
Dirac spinor field
vecrtor field
d(K i ) = i i / 2
 L (x ) 

 (x ) = 
 R (x ) 
d (J i ) =  i / 2
V (x )
d(J i ) = J i
d (K i ) = K i
(1/2, 0 ) right-handed Weyl spinor field  R (x )
d(J i() ) =  i / 2
d(J i() ) = 0
d (J i ) =  i / 2 d(Ki ) = i i / 2
( 0 ,1/2) left-handed Weyl spinor field  L (x )
d(J i() ) = 0
d(J i() ) =  i / 2
d (J i ) =  i / 2
d(K i ) = i i / 2
(1/2, 0 ) right-handed Weyl spinor field  R (x )
d(J i() ) =  i / 2
d(J i() ) = 0
d (J i ) =  i / 2 d(Ki ) = i i / 2
( 0 ,1/2) left-handed Weyl spinor field  L (x )
d(J i() ) = 0
d(J i() ) =  i / 2
d (J i ) =  i / 2
d(K i ) = i i / 2
(1/2, 0 ) right-handed Weyl spinor field  R (x )
d(J i() ) =  i / 2
d (J i ) =  i / 2 d(Ki ) = i i / 2
d(J i() ) = 0
( 0 ,1/2) left-handed Weyl spinor field  L (x )
d(J i() ) = 0
d(J i() ) =  i / 2
d (J i ) =  i / 2
d(K i ) = i i / 2
状態空間上の無限小変換演算子u(X)
(Xf )i = [u(X ),fi ] = d(X )ij f j  X  x  fi
(
)
] = (d(K )  i (x   x  ))
[J i , ( ) ] = d(J i )  iijkx j k  ( )
[Ki , ()
Weyl spinor 場
i
i
 ( )
0
0
(1/ 2,0) 表現
i
( )
 ( )
(0,1/ 2) 表現
 i
j k 
†
† i
†
[J i , (  ) ] =   i ijk x   (  )
[J i , (  ) ] =  (  )
 i ijk x j k (  )
 2

2
 i

i
0
[K i , (  ) ] =    i  i (x  0  x i )  (  ) [K i , (  )† ] =  i  (  )†  i  i (x i  0  x 0i ) (  )†
 2

2
Weyl spinor 場
 ( )
(1/ 2,0) 表現
 ( )
(0,1/ 2) 表現
 i
j k 
†
† i
†
[J i , (  ) ] =   i ijk x   (  )
[J i , (  ) ] =  (  )
 i ijk x j k (  )
 2

2
i
 i

[K i , (  ) ] =    i  i (x i  0  x 0i )  (  ) [K i , (  )† ] =   (  )†  i  i (x i  0  x 0i ) (  )†
2
 2

Weyl spinor 場
 ( )
(1/ 2,0) 表現
 ( )
(0,1/ 2) 表現
 i
j k 
†
† i
†
[J i , (  ) ] =   i ijk x   (  )
[J i , (  ) ] =  (  )
 i ijk x j k (  )
 2

2
 i

i
0
[K i , (  ) ] =    i  i (x  0  x i )  (  ) [K i , (  )† ] =  i  (  )†  i  i (x i  0  x 0i ) (  )†
 2

2
Weyl spinor 場
 ( )
(1/ 2,0) 表現
 ( )
(0,1/ 2) 表現
 i
j k 
†
† i
†
[J i , (  ) ] =   i ijk x   (  )
[J i , (  ) ] =  (  )
 i ijk x j k (  )
 2

2
i
 i

[K i , (  ) ] =    i  i (x i  0  x 0i )  (  ) [K i , (  )† ] =   (  )†  i  i (x i  0  x 0i ) (  )†
2
 2

 ( )† (0,1 / 2) 表現
 ( )† (1 / 2,0) 表現
Lorentz不変な演算子 †( ) ( ) , †( ) ( ) , †( ) (0   i i ) ( )
†
†
†
[
J
,


]
例えば
i
( ) ( ) = [J i , (  ) ] ( )  (  ) [J i , ( ) ] = 0
[Ki ,†() () ] = [Ki ,†() ] () †() [Ki , () ] = 0
Lorentz不変なhermite演算子
†( )i (0   i i ) ()
†()()  †()()
Lagrangian density
L =†( )i (0   i i ) ( ) †( )i (0   i i ) ( )  m(†() () †() ( ) )
Lagrangian density
†
i
†
i
†
†
=

i
(




)



i
(




)


m
(




L
( )
0
i
( )
( )
0
i
( )
(  ) ( )
( ) (  ) )
Lagrangian density
L =†( )i (0   i i ) ( ) †( )i (0   i i ) ( )  m(†() () †() ( ) )
Lagrangian density
†
i
†
i
†
†
=

i
(




)



i
(




)


m
(




L
( )
0
i
( )
( )
0
i
( )
(  ) ( )
( ) (  ) )
†() () †( ) ( ) = (†()
  () 
†
†
†




= () ()
 () )

  () 
†
g0
(

 0 1  (  ) 
) 1 0  (  )  =  


 i (0  i ) ()  i (0  i ) ()
i
†
( )
†
( )
i
†
†
0
 0  i i  () 
= i ( ()  () ) 


  0  i i




0
(
)



= i †( ) †( )   0 1 
 0
 i     (  ) 
 i

  0  
 
i

0    (  ) 
 1 0


(
Dirac行列
= i g  
g0
 (  ) 
Dirac spinor =  
 () 
gi
 =  †g 0
g
∂
0 1

g = 
1 0
i
 0


i
g = 

 i 0 


{g  , g  } = 2 
0
)





 = g   
Cliford algebra
Lagrangian density L =  (ig     m) =  (i  m )
Lagrangian density
L = (ig     m) = (i
  m)
Lagrangian density L = (ig     m) =  (i  m )
L = (ig     m) = (i
  m)
Lagrangian density

equation of motion (ig    m) = 0
Dirac equation
1 2 3

   1 0
i

 
0
5
0 1 2 3
 = 

g = ig g g g = 
1 2 3
 i     0 1 
 0

i  
= [g , g ]
2

M

J

i0
 i
= 
 0
1 
i  
=  = [g , g ]
2
4
i


1
1
J i =  ijkM jk = 
2
2 0
i
()
i
0 

i 
 i 
0

 i 
1 i
1 0 0 
i

= (J  iK ) = 
i 
2
2 0  
 12
K =M
i
J
i
( )
i0
 3 0 

= 
3
 0  
1  i i
= 
2 0
0 

i 
 i 
i


1 i
1
= (J  iK i ) = 
2
2 0
0

0 
Lagrangian density
L = (ig     m) = (i
  m)
0 = (ig     m)(ig     m) = (2  m2 )
 = ueipx
(p
  m )u = 0
 = ve ipx
(p
  m )v = 0
u u = 2m
s
r
u
s
s
sr
u = p
 m
s
2

2
p
=
p
p
=
m


v sv r = 2m sr
s s
v
 v = p  m
s
L = (ig     m) = (i
  m)
†

=
i

canonical conjugate momentum
Lagrangian density
quantization
quantization condition { a (x ), i b (y )† } |x
d3p
solution  (x ) = 
(2 )3 2E p
s
0
=y
0
= i (x  y ) ab
s ipx
s†
ipx
(
b
u
e

d
v
e
)
 p
p
s
s
s
r†
{bp ,bq } = (2 )3 2E p ( p  q ) sr
s
r†
{d p , dq } = (2 )3 2E p ( p  q ) sr
dp 0 = 0
bp 0 = 0
vacuum state 0
Fock space
s
s
†
†
†
†
†
†
bp1 bp2 bpn dq1 dq 2 dqn 0
particle
antiparticle
†
creation operator
bp
†
dp
annihilation operator
bp
dp
discrete symmetry
P, T, C
space inversion
Pb P = b
s
p
s
p
P (t, x )P = g  (t,x )
0
Pdps P = dsp
time reversal
T bpsT = bps
T (t, x )T =  * g 1g 3 (t, x )
T d T = d
TcT = c *
s
p
s
p
charge conjugation
CbpsC = dps
C (t, x )C = i ( *  (t, x )g 0g 2 )T
CdpsC = bps
CcC = c *

i g 
5

g g 

g 
=0
5
  
 1  = 0
 =  = 0   1, = 0or
  1 or,  1  = 1,  0
P
1
1
1
1
1
1
T
1
1
1
1
1
1
C
1
1
CPT
1
1
1
1
1
1
1
1
1
1
1
1
Lorentzian invariant Lagrangian density
L =  i  m 
 f  f
 ig g 5f '
5
2

G
(

g

)
 F (  )
 F ' ( g  )2  G ' ( g g 5 ) 2
2
Electromagnetic field
E electric field

Maxwell equation E = 

(c =  = 1)
B = 0
4-dimensional description
F
0
= 
E

F

E

B 

  B   0E = j

  E   0B = 0

  = ( 0 , )
j  = ( , j )
  F 
  F  = j 
   F  = 0

 = (0 ,)
 0 B

= 2
B E 



E
 = ( E ,   0E    B ) = (  , j ) = j 
B 
  0 B


= ( 0 ,) 
=0
 B  E  = (B,0B    E )


 0
= ( 0 ,  ) 
E
1
    F 
2
v
B magnetic field
 0
 3
= v
 v 2

v 3
0
v1
v2 

1
v 
0 
  : totally anti-symmetric tensor
( 0123 = 1)
Electromagnetic field
E electric field
B magnetic field

Maxwell equation E = 

(c =  = 1)
B = 0

  F  = j 
  B   0E = j

 


F =0
  E   0B = 0




  = ( 0 , )
4-dimensional description
 = (0 ,)

 0 E   0
   0 A    0  B

 =  j = ( , j )    F = (= 2A   A )
F = 
     A  (  A ) 
E

B
B E 

 
0

A  = ( , A)
scalar potential , vector potential A



 


F
=

A


A
E =   0 A B =   A
gauge transformation A  A ' = A    
E, B : invariant under gauge transformation
v
 0
 3
= v
 v 2

v 3
0
v1
v2 

1
v 
0 
  : totally anti-symmetric tensor
( 0123 = 1)
require (i) vector field A  (dynamical variable)
Lorentzian invariance, locality
Maxwell(ii)
equation
(iii) gauge invariance
(iv) simple interaction with the currentj 
1 
Lagrangian densityL =  F F  A j  F 
4
Maxwell eq.
  F  = j 
   F  = 0
=   A   A 
F  =   A   A 
gauge transformation A  A ' = A    
E, B : invariant under gauge transformation
require (i) vector field A  (dynamical variable)
(ii) Lorentzian invariance, locality
(iii) gauge invariance
(iv) simple interaction with the currentj 
1 
Lagrangian densityL =  F F  A j  F 
4
Maxwell eq.
  F  = j 
   F  = 0
=   A   A 
1 
1  

L =  F F  A j =  (  A   A  )(  A   A  )  A j 
4
4
 


F
L
L

=
Euler equation  
(  A ) A
=    (  A   A  )




   ( A   A )
F
Maxwell equation
 j

  F  = j 
=      A
      A
  F  = 0
0
0
Quantization of free electromagnetic field A
free-field Lagrangian L = (  A   A    A  A ) / 2
canonical conjugate momentum  = A   A  = 0

quantization condition


 2
0
[A (x ), (x )] |x 0 =y 0 = i (x  y ) 
???
0
gauge fixing
0
0
positive frequency part
add LGF = (  A ) / 2 to L
and impose the subsidiary condition   A()  physical = 0
physical states
L  LFG =  A   A / 2
canonical conjugate momentum   = A 
 0  0 good!
quantization condition [A (x ), (x )] |x =y = i (x  y )
0
0
eq. of motion     A = 0 solution A =  e ipx
2
2
2
p
=
(
p
)

(
p
)
=0
p = (k 0 0 k )
0

 1
 2
 3
0
= (1
= (0
= (0
= (0
0
1
0
0
0
0
1
0
0)
0)
0)
1)
polarization vectors
r s

  rs = 
r
general solution 3
d p
r
r
ipx
r
r † ipx
A (x ) = 
  ( p )a pe
  (p) * ap e
3
(2 ) 2E p
r
s†
[a p , a q ] = 2E p (2 ) 3 rs  ( p  q )
(
vacuum state 0
a 0 =0
r
k
r1 †
k1
Fock space a a
r2 †
k2
a
(a p0  a p3 ) physical = 0
subsidiary condition
s†
a q creation operator
aqs annihilation operator
rn †
kn
)
0
gauge invariant Lagrangian density
  m)
L =|  f  igQf Af |  (i  gQ A
2
complex scalar f = 1  i 2
gauge transformation for matter field
igQ 
igQf 
  ' = e

f f' = e
f
covariant derivative
Df =  f  igQf Af
D =    igQ A
まとめ
自由scalar場の量子化
1
1 2 2

Lagrangian 密度 L =  f f  m f
2
2
2

運動方程式    f =  m f Klein Gordon方程式
正準共役運動量 = f 量子条件 [f (x ),  (y )] x
1 2  2
Hamiltonian H =    (f )  m 2f 2 d 3x
2
(
)
0
=y 0
= i (x  y )
dk
ikx
† ikx
f
=
(
a
e

a
)
ke
一般解  (2 )3 2E k
[ak ,ak ' ] = (2 )3 2E (k  k ' )
†

†
†
†
Fock space ak1 ak2 akn 0
ak 0 = 0
真空状態 0
†
ak 生成演算子 ak 消滅演算子
dk
†
†
E (ak ak  akak ) / 2
Hamiltonian H = 
2
(2 ) 2E

 (  ) 
0 1 i  0
0
 Dirac行列 g = 
 g = 
Dirac spinor = 

 i

1
0
(

)





 = †g 0
i 

0 
 = g   
Lagrangian密度 L =  (i  m ) 正準共役運動量  = i†

Dirac equation (ig    m) = 0
{g  , g  } = 2 
a
b
†
quantization condition { (x ), i (y ) } |x
d3p
solution  (x ) = 
(2 )3 2E p
0
=y
0
= i (x  y ) ab
s ipx
s†
ipx
(
b
u
e

d
v
e
)
 p
p
s
s
s
r†
s
3
sr
{bp , bq } = (2 ) 2E p ( p  q ) , {d p , dq } = (2 ) 2E p ( p  q )
s
r†
sr
3
particle
antiparticle
dp 0 = 0
bp 0 = 0
vacuum state 0
Fock space
s
s
†
†
†
†
†
†
bp1 bp2 bpn dq1 dq 2 dqn 0
creation operator b p†
†
d
creation operator p
annihilation operatorbp
annihilation operator d p
1 
=

F F
Lagrangian密度L
4
Electromagnetic field
1
 2
( ) 
L
=

(

A
)
gauge固定 GF
physical = 0
補助条件   A

2
canonical conjugate momentum   = A 
quantization condition [A (x ), (x )] |x 0 =y 0 = i (x  y )
r s

  polarization vectors

  rs = 
r
general solution 3
d p
r
r
ipx
r
r † ipx
A (x ) = 
  ( p )a pe
  (p) * ap e
3
(2 ) 2E p
r
s†
[a p , a q ] = 2E p (2 ) 3 rs  ( p  q )
)
(
vacuum state 0
r1 †
k1
Fock space a a
a 0 =0
r
k
r2 †
k2
a
(a p0  a p3 ) physical = 0
補助条件
s†
a q creation operator
aqs annihilation operator
rn †
kn
0