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!

Hermann Weyl, 1885-1955

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