one particle states: wave packets states. heisenberg picture
TRANSCRIPT
One particle states:
Wave Packets States
Heisenberg Picture
Combine the two eq.
KG Equation
Dirac field and Lagrangian
The Dirac wavefunction is actually a field, though unobservable!Dirac eq. can be derived from the following Lagrangian.
mimi
LLL
00 mimi
Negative energy!
00 mimi
Anti-commutator!
A creation operator!
bbbb~
,~
b annihilate an antiparticle!
pppppp aaaaaa 0,
0ppaa
0 pap
Exclusion Principle
Now add interactions:
For example, we can add
)()()(),(),( 43 xxxxx
to our Klein-Gordon or Dirac Lagrangian.
Interaction Hamiltonian:
Schrodinger Picture
Heisenberg Picture
We can move the time evolution t the operators:
Heisenberg Equation
int0 HHH Interaction picture
// 00)( tiHS
tiHI eOetO
S
States and Operators both evolve with time in interaction picture:
// 00)( tiHS
tiHI eOetO
Evolution of Operators
// 00)( tiHS
tiHI eet
,0H
i
dt
d
Operators evolve just like operators in the Heisenberg picture but with the full Hamiltonian replaced by the free Hamiltonian
II OH
i
dt
dO,0
Field operators are free, as if there is no interaction!
Evolution of States
S
States evolve like in the Schrodinger picture but with Hamiltonian replaced by V(t).
V(t) is just the interaction Hamiltonian HI in interaction picture!That means, the field operators in V(t) are free.
,0H
i
dt
d
Operators evolve just like in the Heisenberg picture but with the full Hamiltonian replaced by the free Hamiltonian
States evolve like in the Schrodinger picture but with the full Hamiltonian replaced by the interaction Hamiltonian.
)()( tHi
tdt
dIII
Interaction Picture
)()( tHi
tdt
dIII
)(),()( 00 tttUt II
Define time evolution operator U
All the problems can be answered if we are able to calculate this operator. It’s determined by the evolution of states.
)(),()(),()( 0000 tttUHi
tttUdt
dt
dt
dIIII
),(),( 00 ttUHi
ttUdt
dI
U operator
),(),( 00 ttUHi
ttUdt
dI
Solve it by a perturbation expansion in small parameters in HI.
),(),(),( 0)1(
0)0(
0 ttUttUttU
II Hi
ttUHi
ttUdt
d
),(),( 0
)0(0
)1(
To leading order:
t
t
I tHdti
ttU0
)''('),( 0)1(
Perturbation expansion
1),( 0)0( ttU
Define S matrix:
)(),()( 43 xxditxxddtitHdtiS II LH
It is Lorentz invariant if the interaction Lagrangian is invariant.
Vertex
Add an interaction term in the Lagrangian:
The transition amplitude for the decay of A:
can be computed:
ASBCAUBC I ,
To leading order:
In ABC model, every particle corresponds to a field:
)()( xAxA A
aa
A
BC
ig
Numerical factors remain
Momentum Conservation
A
BC
Every field operator in the interaction corresponds to one leg in the vertex.Every field is a linear combination of a and a+
aa
interaction Lagrangian
vertex
Every leg of a vertex can either annihilate or create a particle!
This diagram is actually the combination of 8 diagrams!
aa
aa
aa
This is in momentum space.
The integration yields a momentum conservation.
A
BC
interaction Lagrangian
vertex
There is a spacetime integration.
Interaction could happen anytime anywhere and their amplitudes are superposed.
Every field operator in the interaction corresponds to one leg in the vertex.
aa
interaction Lagrangian
vertex
Every leg of a vertex can either annihilate or create a particle!
4IL
Every field operator in the interaction corresponds to one leg in the vertex.
aa
interaction Lagrangian
vertex
Every leg of a vertex can either annihilate or create a particle?
gI L ba ab
aa
interaction Lagrangian
vertex
Every leg of a vertex can either annihilate or create a particle?
gI L ba ab
can either annihilate a particle or create an antiparticle!
ba
can either annihilate an antiparticle or create a particle!
ab
The charge flow is consistent! So we can add an arrow for the charge flow.
p
ipxipx evbeuax
)( )( 1pe
p
ipxipx euaevbx
)(
01pu
Feynman Rules for an incoming particle
gI L ba ab
External lineWhen Dirac operators annihilate states, they leave behind a u or v !
0'22 3' pppa pp
)( 1pe 01pv
Feynman Rules for an incoming antiparticle
1pu
gAI L
ba ab
2pu
g
aaxA )(
Propagator
BBAA
),(),( 00 ttUHttUdt
di I
t
t
III tHdttHttUtHdt
ttdUi
0
)'(')(),()(),(
0)1(0
)2(
t
t
t
t
II tHdttHdtttU0 0
''
0)2( )'(')''(''),(
The integration of two identical interaction Hamiltonian HI. The first HI is always later than the second HI
t
t
II
t
t
tHtHTdtdtttU0 0
)'()''('''2
1),( 0
)2(
)()()()()()())()(( 1212212121 tAtBtttBtAtttBtAT
This definition is Lorentz invariant!
)()(2
1),( 212
41
4)2()2( xxTxdxdUS II LL
)()()()( 2143 pApASpBpB
)()()()()()()()()()( 2122211124
14
43 pApAxCxBxgAxCxBxAgTxdxdpBpB
0)()(0 21)()(
24
14 142231 xCxCTeexdxd xppixppi
Amplitude for scattering
BBAA
Propagator between x1 and x2
Fourier Transformation
p1-p3 pour into x2 p2-p4 pour into x1
000)()(00)()(0212121 xx aaaaxCxCxCxCT
Cx2
x1
A(p1) A(p2)
B(p3)
C(p1-p3)
B(p4)
A(p1) A(p2)
B(p3) B(p4)
21 tt
A particle is created at x2 and later annihilated at x1.
000)()(00)()(0121221 xx aaaaxCxCxCxCT
C
x2
x1 Cx2
x1
A(p1) A(p2)
B(p3)
C(p1-p3)
B(p4)
A(p1) A(p1)A(p2) A(p2)
B(p3) B(p3) B(p4)B(p4)
21 tt
A particle is created at x1 and later annihilated at x2.
)(0)()(0 422212
41
4 21 qpmq
ixCxCTeexdxd
C
iqxipx
C
x2
x1 Cx2
x1
A(p1) A(p2)
B(p3)
C(p1-p3)
B(p4)
A(p1) A(p1)A(p2) A(p2)
B(p3) B(p3) B(p4)B(p4)
0)()(0 yxT
0)()(0 yxT
0)()(0 yxT
0)()(0 yxT
This doesn’t look explicitly Lorentz invariant. But it is!
0)()(0 yxT
00 yx
aa
aa
Every field either couple with another field to form a propagator or annihilate (create) external particles! Otherwise it will vanish!
Antiparticles can be introduced easily by assuming that the field operator is a complex number field.
ipxp
ipxp ebea
pdx
2
1
)2()(
3
3
ipxp
ipxp eaeb
pdx
2
1
)2()(
3
3
20 m
L
Complex KG field can either annihilate a particle or create an antiparticle!
Its conjugate either annihilate an antiparticle or create a particle!
The charge flow is consistent! So we can add an arrow for the charge flow.
Scalar Antiparticle
33 ggIL
vertex
Charge non-conserving
2 IL
vertex
Charge conserving
000)()(00)()(0212121 xx baabxxxxT
Cx2
x1
A(p1) A(p2)
B(p3)
C(p1-p3)
B(p4)
A(p1) A(p2)
B(p3) B(p4)
21 tt
An antiparticle is created at x2 and later annihilated at x1.
0)()(0 21 xxT
Propagator:
000)()(00)()(0121221 xx abbaxxxxT
C
x2
x1 Cx2
x1
A(p1) A(p2)
B(p3)
C(p1-p3)
B(p4)
A(p1) A(p1)A(p2) A(p2)
B(p3) B(p3) B(p4)B(p4)
21 tt
A particle is created at x1 and later annihilated at x2.
)(0)()(0 422212
41
4 21 qpmq
ixxTeexdxd
C
iqxipx
C
x2
x1 Cx2
x1
A(p1) A(p2)
B(p3)
C(p1-p3)
B(p4)
A(p1) A(p1)A(p2) A(p2)
B(p3) B(p3) B(p4)B(p4)
C
x2
x1 Cx2
x1
A(p1) A(p2)
B(p3)
C(p1-p3)
B(p4)
A(p1) A(p1)A(p2) A(p2)
B(p3) B(p3) B(p4)B(p4)
BI L
B B
20 m
L
U(1) Abelian Symmetry
)()( xex iQ
The Lagrangian is invariant under the phase transformation of the field operator:
2 IL invariant
)(xee iQiQ
A
BC
If A,B,C become complex, they carry charges!
The interaction is invariant only if
0 CBA QQQ
U(1) symmetry is related to charge conservation!
mi L
The Dirac Fermion Lagrangian is also invariant under U(1)
LL miee iQiQ
)()( xex iQ
SU(N) Non-Abelian Symmetry
n
3
2
1
Assume there are N kinds of fields
If they are similar, we have a SU(N) symmetry!
)()()( xexUxiiTi
20 m
L 2 IL
are invariant under SU(N)!
u-d 互換對稱
量子力學容許量子態的疊加
a + b
c + d
1**
**
db
ca
dc
baUU
u
u
u
d
d
dd
u
量子力學下互換群卻變得更大！
0
1
1
**
22
22
bdac
dc
ba
dc
baU
d
uU
d
u,
古典量子
2
2
20
m
UUmUU
m
L
222 UUIL
)()(
)()(
xUx
xUx
They are invariant under SU(N)!
Gauge symmetry
)()( )( xex xiQ
Gauge (Local) symmetry
)()( )( xex xiQ )()( xex iQ
)()()( xee xiQxiQ )(xee iQiQ
)(xee iQiQ
)()()(
)()()(
)(
xeiQxe
xexiQxiQ
xiQ
)(
)(
xe
xeiQ
iQ
Kinetic energy is not invariant under gauge transformation!
Global Symmetry
)(xe iQ
)()( )( xDexD xiQ
Could we find a new “derivative” that works as if the transformation is global?
)()()(
)()()(
)(
xeiQxe
xexiQxiQ
xiQ
To get rid of the extra term, we introduce a new vector field:
)()()( xxAxA
AiQD
)()()()()( )()()( xeiQxeiQxDeiQAD xiQxiQxiQ
)()( xDe xiQ
Gauge (Local) symmetry
)()( )( xex xiQ )()( xex iQ
)(xee iQiQ
is invariant under gauge transformation!
Global Symmetry
)(xe iQ
)()( )( xDexD xiQ
DD
xDeeD
DDiQiQ )(
DDReplacing derivative
with covariant derivative,
AAQAiQAiQDD 2
The scalar photon interaction vertices
mi L
To force it to be gauge invariant,
)()( )( xex xiQ
you only need to replace derivative with coariant derivative.
D
mDi L is gauge invariant!
AQmi
mAQimDi
L
This gauge invariant Lagrangian gives a definite interaction between fermions and photons
gAI L g
1pu
gAI L
ba ab
2pu
g
aaxA )(
This form is forced upon us by gauge symmetry!
It is really a Fearful Symmetry! Tony Zee
Tyger! Tyger! burning brightIn the forests of the nightWhat immortal hand or eyeCould frame thy fearful symmetry!
William Blake
gAI L
Let there be light!
In the name of gauge symmetry!
Yang and Mills
SU(N) Non-Abelian Symmetry
n
3
2
1
Assume there are N kinds of fields
If they are similar, we have a SU(N) symmetry!
)()()( xexUxiiTi
20 m
L 2 IL
are invariant under SU(N)!
Non-Abelian Gauge Symmetry
)()()()( )( xexxUxii Txi
ii AigTD
We need one gauge field for each generator.
UUg
iUTAUTA iiii
1
Gauge fields transform as:
DUD
DUUUUAUTigUU
UUUg
iUAUTigUAigTD
ii
iiii
1
11
is invariant under gauge transformation!
DD
mDi L
ii
ii
ATQmi
mATQimDi
L
iiI TgAL iTg
3
1iiiW
e
We
ee ?
2 × 2 matrices
e
Weg ei
iie
2,
22121 iWW
WiWW
W
3
3
321
213
2
2
WW
WW
WiWW
iWWWW i
ii
10
01
0
0
01
10321
i
i
2
1
22 mL
22 mV
22 mV
22
vm
Vacua happen at:
v
0
Choose:
0000 iiTU
For infinitesimal transformation:
0)(,0)(,0 03
03
0 YTYTT
SU(2)χU(1)Y is broken into U(1)EM
033
033
00 ''
YBgWgTWgTYBgWgTWgTDDDD
BggWBggWTTWWTTg '' 33
03
03
002
W become massive
Z become massive
BggWgg
Z ''
1 322
Photon is massless.
gBWggg
A
322
''
1