Locality,Lorentz inv.gauge inv. spinor vector A interaction reprensatationperturbation S-matrix...
37
locality, Lorentz inv. 4 / F F gauge inv. spinor i i i i i m A g i ) ( tot L 2 / ) ( 2 A vector A 2 / A A i i i i m i ) ( 0 L i i i i A g I L 0 4 I ! 1 n n x d i T n S L interaction reprensatation n f f f T 2 1 ) ( j i j i f f T f f ) ( 2 2 2 1 1 2 1 k n l j i j i l l f f f f f f f N perturba tion I 0 tot L L L S- matrix provabi lity 2 in out S P Wick's theorem 0 )) ( ) ( ( 0 y x T i m k e i k d y x ik 2 2 ) ( 4 4 ) 2 ( 0 )) ( ) ( ( 0 y x T ) ( 2 2 4 4 ) ( ) 2 ( y x ik e i m k m k i k d 0 )) ( ) ( ( 0 y A x A T i k e i k d y x ik 2 ) ( 4 4 ) ( ) 2 (
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Transcript of Locality,Lorentz inv.gauge inv. spinor vector A interaction reprensatationperturbation S-matrix...
locality, Lorentz inv.
4/FF
gauge inv.spinor
i iiii mAgi )(totL 2/)( 2
A
vector A
2/ AA
i iii mi )(0L i iii Ag IL
0
4I!
1
n
nxdiT
nS L
interaction reprensatation
nfffT 21
)( jiji ffTff
)(
2221121 knljijill fffffffN
perturbationI0tot LLL
S-matrixprovability2
inout SP
Wick's theorem
0))()((0 yxT imk
ei
kd yxik
22
)(
4
4
)2(
0))()((0 yxT
)(
224
4 )(
)2(yxike
imkmk
ikd
0))()((0 yAxAT
ik
e
ikd yxik
2
)(
4
4 )(
)2(
0
4I!
1
n
nxdiT
nS L
2
inout SP i iii Ag IL
i iii Ag IL
0
4I!
1
n
nxdiT
nS LS-matrixprovability
2
inout SP
provability P?eqeq k
pk'
p'
0
4I!
1
n
nxdiT
nS L
2
inout SP
electron quark scattering
pk'p'k S
0)(q p†rb)(e k†sb
0
4
!
1
n
n
I xdiTn
L0 )'('q prb )'('e ksb
i iii Ag IL
provability amplitude (p'≠p,k'≠k)
Let us consider electron quark scattering.
We want to know the provability.
For this purpose, we calculate the provability amplitude.
provability amplitude
eqeq k
pk'
p'
electron quark scattering
pk'p'k S
provability P?
0
4I!
1
n
nxdiT
nS L
2
inout SP i iii Ag IL
†qb†
eb xdi I4L 2 4
2
1 xdiT IL=
0)(q p†rb)(e k†sb0 )'('q prb )'('e ksb
0
4
!
1
n
n
I xdiTn
Lbq
†be† 〉
0
1
p'≠p,k'≠k
)(2)2( 3
3ipxs
isipxss
ii evdeubE
dpp
p
p †
)*(2)2( 3
3ipxrripxrr eaea
Epd
A†
ppp
〈 bq'be' be†bq
† 〉rap 0
0
pk'p'k Sbq'
be'
〈
=abbreviations 〈 bq'be' be†bq
† 〉†rap 0
0'qb 'eb
'' eq bb iT 2
1 2 4xd ††
qebb i iii Ag 2
4 xd ††qebbqqqeee
AgAg xdAigxdAig 4qqq
4eee
e g i ee A xd 4
q g i qq A xd 4
∴∴
provability amplitude
eqeq k
pk'
p'
electron quark scattering
pk'p'k S
provability P? 'qb 'eb †qb†
eb xdi I4L 2 4
2
1 xdiT IL1
expand square
0
4I!
1
n
nxdiT
nS L
2
inout SP i iii Ag IL
0)(q p†rb)(e k†sb0 )'('q prb )'('e ksb
0
4
!
1
n
n
I xdiTn
L
'' eq bb iT 2
1 2 4xd ††
qebb 2 4 xd ††
qebbqqqeee
AgAg xdAigxdAig 4
qqq4
eee
Tbb2
1'' eq ††
qebb2
q(x ') A (x ')
xd 4 '4xd Tbb '' eq )( eee Ag )'''( qqq
Ag ††qebbi i
xdAig 4eee
xdAig 4qqq
xdAig 4eee
xdAig 4q
2 22
00
''
'' be†bq
† 〉
〈 b'qb'e
' '' ' 4qqq xdAig
xdAig 4eee
abbreviations
pk'p'k S
p'≠p ∴ k'≠k ∴
=
eqeq k
pk'
p'
electron quark scatteringprovability P?
0)(q p†rb)(e k†sb0 )'('q prb )'('e ksb
0
4
!
1
n
n
I xdiTn
L
'' eq bb iT 2
1 ††qebb i iii Ag
2 4 xd ††
qebbqqqeee
AgAg
Tbb2
1'' eq 2
xd 4 '4xd Tbb '' eq )( eee Ag )'''( qqq
Ag ††qebbi i
††qebb
xdAigxdAig 4qqq
4eee
xdAig 4eee
' '' ' 4qqq xdAig
nfffT 21
)( jiji ffTff
)(
2221121 knljijill fffffffN
Wick's theorem
pk'p'k S )(
2)2( 3
3ipxssipxss evdeub
Ed
ppp
p †
srrs Ebb )(2)2(},{ 3 qppqp †
srrs Edd )(2)2(},{ 3 qppqp †
b ipxseu
d ipxsev
†b ipxseu
†d ipxsev
pk'p'k S
'AA )'( AAT
iqe
iqd xxiq
2
)'(
4
4
)2(
) (A 'A 'eb e eg e †eb 'qb 'q ) ( qg 'q †
qb xid 4 '4xid
provability amplitude
provability amplitude
eqeq k
pk'
p'
electron quark scatteringprovability P?
0)(q p†rb)(e k†sb0 )'('q prb )'('e ksb
0
4
!
1
n
n
I xdiTn
L
'' eq bb iT 2
1 ††qebb i iii Ag
2 4 xd ††
qebbqqqeee
AgAg
Tbb2
1'' eq 2
xd 4 '4xd Tbb '' eq )( eee Ag )'''( qqq
Ag ††qebbi i
††qebb
xdAigxdAig 4qqq
4eee
xdAig 4eee
' '' ' 4qqq xdAig
pk'p'k S
) (A 'A xid 4 '4xid 'eb e eg e †eb 'qb 'q ) ( qg 'q †
qb
pk'p'k S )(
2)2( 3
3ipxssipxss evdeub
Ed
ppp
p †
srrs Ebb )(2)2(},{ 3 qppqp †
srrs Edd )(2)2(},{ 3 qppqp †
b ipxseu
d ipxsev
†b ipxseu
†d ipxsev
'44 xidxid
iqe
iqd xxiq
2
)'(
4
4
)2(xikeu '
e ' )( e g ikxeu e
''q ' xipeu )( q g '
qipxeu
'AA )'( AAT
iqe
iqd xxiq
2
)'(
4
4
)2(
pk'p'k S
pk'p'k S '44 xidxid
iqe
iqd xxiq
2
)'(
4
4
)2(xikeu '
e ' )( e g ikxeu e
''q ' xipeu )( q g '
qipxeu
provability amplitude
'44 xidxid
iqe
iqd xxiq
2
)'(
4
4
)2(xikeu '
e ' )( e g ikxeu e
''q ' xipeu )( q g '
qipxeu
xikeu 'e ' )( e g ikxeu
e''
q ' xipeu )( q g 'q
ipxeu
'44 xidxid
iqe
iqd xxiq
2
)'(
4
4
)2(pk'p'k S
4
4
)2( iqd
iq
2'eu )( e g eu 'qu )( q g qu
inout)4(4
inoutinout )''()2( TpkpkiS
i
2) ('eu )( e g eu 'qu )( q g qu
T : T-matrixpk'p'k T
()2( )4(4i kk ' )' pp
1
kk '
xid 4 iqxe xike ' ikxe ''xipe 'ipxe '4xid 'iqxe xkkqi )'( ')'( xppqi
4
4
)2( iqd
) ()2( )4(4i
iq
2'eu )( e g eu 'qu )( q g qu
kkq '
)(2 XdqeiqX 2 ) ()2(
)4(4i 2 ppq 'k' k
k' k
qq q
q
kk'q
pk'p'k T
eg
qg
eu'eu
qu'qu
ikk
2)'(
Feynman rules
kk '
i
2) ('eu )( e g eu 'qu )( q g qu
kk '
inout)4(4
inoutinout )''()2( TpkpkiS
T : T-matrixpk'p'k T
i
2) ('eu )( e g eu 'qu )( q g qu
iq
2
内線
photon
q p
fermion
imp 1
vertex
ig
外線 incomingoutgoing粒子
反粒子
u u
v v
4
4
2 )( ipd
1loop fermion loop T matrix 1
internal line
external line
particle
ant-particle
''
' )'( rp
sp vgu s
krk ugv )('
g'g
sku
'spu
rkv '
''
rpv
ikk
2)'(
'' ffff
ffff
†')(
1)(''
sk
sk ug
imqkgu
ff
†')(
'
1)(''
sk
sk ug
imqkgu
iqg
imqkg
iqd
24
4
)(1
)()2(
loop diagram
imkg
imqkg
ikd
1)(
1)(Tr
)2( 4
4
impqkg
iqiqd
1
)()2( 24
4
loop diagram
)(1
)(
gimqk
g
ikk
2)'(sk
sk ugu )('' r
prp ugu )'(''
g
'g
sku'
'sku
rpu
''
rpu
ikk
2)'(
Feynman rules '' ffff
T
ikk
2)'(sk
sk ugu )('' r
prp ugu )'('' T
ikk
2)'(sk
sk ugu )('' r
prp ugu )'('' T
ikk
2)'(sk
sk ugu )('' r
prp ugu )'('' T
])'(/[))((' 2''
'' kkuuuugg r
prp
sk
sk
00 †
*'' )( s
ksk uu
*0
'' )( s
ksk uu †
''0
sk
sk uu
††
''0
sk
sk uu †
''
sk
sk uu
*''
'' )( s
ksk
sk
sk uuuu '
'''
sk
sk
sk
sk uuuu
0''
'' †s
ksk uu
0†s
ksk uu
)*( ''
''
rp
rp
rp
rp uuuu '
'''
rp
rp
rp
rp uuuu
*T ])'(/[*)(*)(' 2''
'' kkuuuugg r
prp
sk
sk
*''
'' )( s
ksk
sk
sk uuuu '
'''
sk
sk
sk
sk uuuu
)*( ''
''
rp
rp
rp
rp uuuu '
'''
rp
rp
rp
rp uuuu
'
''
''ss
sk
sk
sk
sk uuuu
'
*''
'' )(
ss
sk
sk
sk
sk uuuu
'
''
''ss
sk
sk
sk
sk uuuu
Trs'
''
'' s
sk
sk
sk
sk uuuu
'
''
''Tr
ss
sk
sk
sk
sk uuuu
mpuus
ss mpvv
s
ss
'Tr mkmk
41Tr 0Tr
4Tr 0Tr
Tr
(4
'Tr 2 mkk
)
])'(''[4 kkkkkk
24m*)(
'
''
''rr
rp
rp
rp
rp uuuu
])'(''[4 pppppp
2'4m
'
*''
'' )(
ss
sk
sk
sk
sk uuuu
])'(''[4 kkkkkk
24m*)(
'
''
''rr
rp
rp
rp
rp uuuu
])'(''[4 pppppp
2'4m
'
*''
'' )(
ss
sk
sk
sk
sk uuuu
])'(''[4 2 mkkkkkk
]')'(''[4 2 mpppppp
*)('
''
''rr
rp
rp
rp
rp uuuu '
*''
'' )(
ss
sk
sk
sk
sk uuuu
])'(''[4 2 mkkkkkk
]')'(''[4 2 mpppppp )'()')('()')('())(''[(16 2 ppmppkkkppkkppk
)'()')('())(''()')('( 2 ppmppkkkppkkppk )'(4)')('(4)')('()')('( 2 ppmkkppkkppkkpp
]'4)'('4)'(')'(' 22222 mmkkmkkmkkm )')('(2))(''(2[16 kppkkppk
]'4)'('2)'(2 2222 mmkkmppm
*)('
''
''rr
rp
rp
rp
rp uuuu '
*''
'' )(
ss
sk
sk
sk
sk uuuu
)')('(2))(''(2[16 kppkkppk ]'4)'('2)'(2 2222 mmkkmppm
*)('
''
''rr
rp
rp
rp
rp uuuu '
*''
'' )(
ss
sk
sk
sk
sk uuuu
)')('(2))(''(2[16 kppkkppk
]'4)'('2)'(2 2222 mmkkmppm
)')('(2))(''(2[16 kppkkppk
]'4)'('2)'(2 2222 mmkkmppm
2/)'(2/)'[(16 222222 mmumms ]'4)2(')'2( 222222 mmmtmmtm
2222222 )'()')((2/)[(16 mmmmusus ]'8)'( 2222 mmmmt
]'8)'(2/)[(16 2222222 mmmmus
)'12'22(8 224422 mmmmus
2
'' Trrss 2
22442222 '12'22'8
tmmmmus
gg
*)('
''
''rr
rp
rp
rp
rp uuuu '
*''
'' )(
ss
sk
sk
sk
sk uuuu
scattering cross section target cross section
A area
N2 # of targetsN1 # of projectiles
Ns # of scatterings
ANNN/
/
1
2s
P 2s / NNP
AN /1
scattering provability
flux
reaction a1+a2→ a3+a4
VEdE
EN
23
2
32222
22
)0()2(2
x
pp
22iii mE pmomentumipmassim energy
volume of space:V
ANNN /21s
2
21434
34
3
33
33
s 2)2(2)2(pppp
ppS
Ed
Ed
N
TEAN ||2/ 1211 vv
VTEE
SE
dE
d
||22
2)2(2)2(
1221
2
21434
34
3
33
33
vv
pppppp
total time:T
VTEE
SE
dE
d
||22
2)2(2)2(
1221
2
21434
34
3
33
33
vv
pppppp
VTEE
SE
dE
d
||22
2)2(2)2(
1221
2
21434
34
3
33
33
vv
pppppp
VTEE
SE
dE
d
||22
2)2(2)2(
1221
2
21434
34
3
33
33
vv
pppppp
2143 pppp S 21432144
43 })()2(1{ pppp Tppppi
2
21431221
432144
43
43
33
33
||22
)()2(
2)2(2)2(pppp
vvpp
TEE
ppppE
dE
d
)(22)2(||22
)2(4321
2
214343
63
3
1221
4
EEEETEE
dEE
pppp
pvv
243
221 )()( pppps
242
231 )()( ppppt
232
241 )()( ppppu
24
23
22
21 uts
),0,0,(1 LL pEp )0,0,0,(2 Mp
)cos,0,sin,( 3333 ppEp )cos,0,sin,( 4444 ppEp
),0,0,( 11 pEp ),0,0,( 22 pEp
)cos',0,sin',( 33 ppEp
)cos',0,sin',( 44 ppEp
invariant variables 不変変数
center-of-mass frame 重心系laboratory frame 実験室系
time1a
2a
3a4a
s
t
s-channel
t-channels-channelt-channelu-channel
reaction 反応4321 aaaa
4231 aaaa 2341 aaaa
sEE 21 sEE
22
21
21
ss
E2
22
21
1
s
sE
2
21
22
2
ss
E2
24
23
3
s
sE
2
23
24
4
sss
p4
))()()(( 221
2212
sss
p4
))()()(('
243
2432
'4
))(()2(cos
24
23
22
21
22
spptss ii
'2cos
ppdt
d
1
22
21lab
1 2
s
E
21
221
2212ab
4
))()()(()(
ss
pl
center-of-mass frame
laboratory frame
)(22)2(||22
)2(4321
2
214343
63
3
1221
4
EEEETEE
dEE
ppppp
vv
)(22
''
)2(4
14321
2
214343
2
2EEEET
EEddpp
ps
pppp
)()'/)((22
)('
)2(4
14321
2
2143432143
43212
2EEEET
dpEEEEdEEdEEEEdp
ps
pppp
2
21434343
2
2 ))/'/'(22
'
)2(4
1pppp T
EpEpEEdp
ps
2
2143
2
2 '4
cos2'
)2(4
1pppp T
psdp
ps
2
2143cos32
'pppp Td
spp
2
2143264
1pppp Tdt
sp
2
434
34
3
33
33
2)2(2)2(ppp
ppS
Ed
Ed
EV2
T
2
434
34
3
33
33
2)2(2)2(2
1ppp
ppS
Ed
Ed
EVT
)()()(
)(43
42
434
34
3
33
334
22222
2pppT
Ed
Ed
E
ppppp
)()(
)(43
2
4343
63
34
2222
2EEET
EEd
E
pppp
decay width =decay provability/total time
= # of decays/ # of particles /total time
# of decays
# of particles
total time
2
434
34
3
33
33
2)2(2)2(2
1ppp
ppS
Ed
Ed
EVT
)()()(
)(43
42
434
34
3
33
334
22222
2pppT
Ed
Ed
E
ppppp
)()(
)(43
2
4343
63
34
2222
2EEET
EEd
E
pppp
)(22)2(2
)2(43
2
4343
63
34
EEMTEE
dM
ppp
p )(22)2(2
)2(43
2
4343
63
34
EEMTEE
dM
ppp
p
)(22)2(2
)2(43
2
4343
63
34
EEMTEE
dM
ppp
p
)(''
)( 43
2
4343
2
2 2222
1EEMT
EEddpp
M
ppp
)()'/)((
)(')( 43
2
434343
432
2 2222
1EEMT
dpEEMdEEdEEMdp
M
ppp
2
434343
2
2 2222
1ppp T
EpEpEEdp
M
))/'/'('
)(
2
43
2
2 4
2
22
1ppp T
Mpdp
M '
cos')(
2
432cos
16
'ppp Td
Mp
2
4328ppp T
Mp
'
center-of-mass frame
'' ffff k
pk'
p'
fermion fermion scattering
ikk
2)'(sk
sk ugu )('' r
prp ugu )'(''
g
'g
sku'
'sku
rpu
''
rpu
ikk
2)'(
T
2
'' Trrss 2
22442222 '12'22'8
tmmmmus
gg
2
'' Trrss 2
22442222 '12'22'8
tmmmmus
gg
2
'' Trrss 2
22442222 '12'22'8
tmmmmus
gg
2
'' Trrss 2
22442222 '12'22'8
tmmmmus
gg
tsmmu 22 '22
)cos1(2 2 kts
mmsmms4
))'()()'(( 222 k
''
2
2 4
1
64
1rrss
Tdtsk
''
2
4
1cos
32
1rrss
Tds
''
2
2 4
1
64
1rrss
Tsdt
dk
''
2
4
1
32
1
cos rrssT
sdd
cos2 2ddt k
散乱断面積scattering cross section
微分断面積differential scattering cross section
2
264
1Tdt
sp
2cos
32
'Td
spp
2
264
1T
spdtd
2
32
'
cosT
spp
dd
differential scattering cross section 微分散乱断面積
scattering cross section 散乱断面積
''
2
212
1
64
1rrss
Tnn
dtsp
''
2
21
1cos
32
1rrss
Tnn
ds
2
12 ii
jn
ni :# of sipn states of particle ai
unpolarized scattering cross section 非偏極散乱断面積
2143 pppp TT
ji :spin quantum #(massive particle)
(massless particle)
