前回のSummary Path Integral Quantization
D    d x
path Integral D    dx
D   d x
x
x
x
generating functional
L  L0  L1
Z [ , , ]   DD D e
1
1 2 2
2

L0   (  )     (i    m )
2
2
1 4
L1 ( , , )     g 
4
0 T ( (x1 ) (x j ) (xk ))) 0
(i )n
 n Z [ , , ]

Z (0)  (x1 ) (x j )  (x k )
  0 ,  0,  0
i  d 4 x L      
   
i  d x L1 
,
,
i

i

i

4
Z [ , , ]  e



1

1
1


i  d x  2 2  

 2  
i    m 

4
e
a
Gauge Theory
for gauge field G  & spinor field 
Assume Lorentz inv., locality, superficial renormalizability &
local gauge symmetry with Lie group G. Lie algebra g.
 i :parameter, depends on x
G i  gf ijk  jG k     i   ig iT i
j
j
j
i
i

X

X : generator of G X  g
e
G
gauge transformation
[ X i , X j ]  if ijk X k
f
ijk:
structure constants of G
T i: representation of Xi on 
[T i ,T j ]  if ijk T k
1 i 2
Lagrangian L   (G  )  (i  D   m )
4
i
field strength G 
  Gi   G i  gf ijk G jGk
covariant derivative
D  (   igT G  )
i
i
example SU(3)group of complex 3×3 matrices U
withUU †=1 (unitary) & det U = 1 (special)
generator X i  i / 2 (i  1,2,,8)
i Gell-mann matrices
0
0 1 0
1   1 0 0 , 2   i



0
0
0
0


0 0 1
0
4   0 0 0 , 5   0



1
0
0


i
i
0
0
0
1
0 , 3   0


0
0
0 i  6 0

0 0 ,    0

0 0
0
f
commutators [i ,  j ]  2if ijk k
f 123  1, f
458
f
678
0 0
1
1
0
 1 0 , 8 

3 0
0 0

0 0
0 0
0 1 , 7   0 0


1 0
0 i
0 0 
1 0 

0  2
0
i 

0
ijkは完全反対称
147
165
246
257
345
376
f

f

f

f

f

f
 1/ 2
 3 / 2,
irreducible representations
are specified by two integers
0
1
2
3
0
1
3
6
10
1
3*
8
15
2
6*
15*
27
3
10*
......
Path intdegral quantization of gauge theories
L  L0  L1 (G a , , ) としてみる
1 i  j
1
i
i 2
L0   ( G   G  )   G  K G
K       2     
2
4
∂K ∂∂2∂2∂ 0
generating functional

Z [J a , , ]   DGDD exp i  d 4 x L  G i J i     
e
 
 
4
i  d xL1 
,
,
 iJ a i i






1 a

1
1   b

i  d x J  ( K ) J 


2
i   m 

4
e
∂
is inappropriate ∵ (K 1 )  does not exist. ∂ (K  K 1    )
need gauge fixing
we choose the gauge with  G i  B i
0
矛盾
∂
(K1)
does not exist.
gauge fixing  G i  B i
need gauge fixing
we choose the gauge with  G i  B i
gauge fixing  G i  B i
Z [J ]   DG i e iS iJ G
J  G   J iG i d 4 x
xi
  DG i e iS iJ G D  ( G i(  )  B i) det K
yj
 i ( )
  G  (x )
i ( )
i
ijk
j k
i
G  G  gf  G     
K xi ,yj 
j
   (y )

4
ij 2
ijk 
k

G  ) (x  y )
 (    gf
 y j
 i dx i (y j (xk )) det xk

    dyi  (y j )  1
i

gauge fixing  G i  B i
Z [J ]   DG i e iS iJ G
J  G   J iG i d 4 x
  DG i e iS iJ G D  ( G i(  )  B i ) det K

Gi (gauge不変性より)
i
ijk
j k
i
 G   gf  G     
i ( )
  G  (x )
i ( )
G
K xi ,yj 
j
 (y )
4
ij 2
ijk 
k

G  ) (x  y )
 (    gf
Z [J ]   DG  e
i
iS iJ G

D ( G   B ) det K
i
i
無限大の定数
  DG i e iS iJ G  ( G i  B i ) det K
  DG  DB e
i
  DG e
i
i
i
iS iJ G

e
1
(B i )2 d 4x
2
i

物理はBi
によらない
無限大の定数
e
iS iJ G
1
(  G i ) 2 d 4 x
2

 ( G   B ) det K
det K
i
i

Z [J ]   DG i e
iS iJ G i

1  i 2 4
( G ) d x
2
det K
K xi ,yj  (   gf
ij
2
ijk

 G  ) (x  y )
K xi ,yj 
 (    gf
ij
2

ijk
 G k ) 4 (x  y )
Z [J ] 
  DG e
i
iS iJ G
e
i

1
(  G i ) 2 d 4 x
2
det K
k
4
Z [J ]   DG i e
iS iJ G i

1  i 2 4
( G ) d x
2
det K
K xi ,yj  (   gf
ij
 i , ~ i Grassman number
2
ijk

 G  ) (x  y )
k
4
Faddeev Popov ghost
i  d 4 xd 4y~ i ( x ) K xi ,y j  (y ) j
~
det K   DDe
i  d 4 xd 4y~i ( x )(  ij  2 gf ijk  G k ) 4 ( x y )  (y ) j
~
 DDe

~
  DDe
  DD~e
~
i  d 4 x~i ( x )(  ij  2 gf ijk  G k )  (y ) j
i  d 4 x   ~ i ( ij    gf ijk G k )
Z [J ,  ,  ]   D G  
i
DD~ e
e
j
iS iJ G i

1
(  G i ) 2 d 4 x
2
i  d 4 x  ~ i ( ij    gf
~
G k )  j i   i~ 
ijk
~
Z [J ,  ,   , ]   DGDD~DD


fermionも加える
~i i
i i
~
exp i  d x L  G  J           
Lagrangian
4
i
i
L  LG  LGF  LFP  LF
1
i
i
ijk
j
k 2
LG   ( G   G   gf G G )
4
1  i 2
LGF  
( G  )
2
 ~i
ij
ikj
k
j
LFP    (    gf G  ) 
~
Z [J ,  ,  ]   D G  
i
DD~ e
e
iS iJ G i

1
(  G i ) 2 d 4 x
2
i  d 4 x  ~ i ( ij    gf
~
G k )  j i   i~ 
ijk

~
Z [J ,  ,   , ]   DGDD~DD


fermionも加える
~i i
i i
~
exp i  d x L  G  J           
Lagrangian
4
i
i
L  LG  LGF  LFP  LF
1
i
i
ijk
j
k 2
LG   ( G   G   gf G G )
4
1  i 2
LGF  
( G  )
2
 ~i
ij
ikj
k
j
LFP    (    gf G  ) 
LF   (i     g T iG i  m )
T i  i / 2


Chiral Primordial Gravitational Waves from a

arXiv:1502.00322v1 [quant