Now add interactions:

For example, we can add

)()()(),(),( 43 xxxxx

to our Klein-Gordon or Dirac Lagrangian.

Interaction Hamiltonian:

These terms will add non-linear terms to the KG equation.

It means anything not quadratic in fields and derivatives.

Now we can study the time evolution.

Schrodinger Picture

Evolution Operator:

States evolve with time, but not the operators.

It is the default choice in wave mechanics.

狀態 波函數

測量 算子 O

x

ip ˆ

xxˆ

Heisenberg Picture

We move the time evolution to the operators:

Heisenberg Equation

For the same evolving expectation value, we can instead ask operators to evolve.

Now the states do not evolve.

Only the expectation values are observable.

The rate of change of operators equals their commutators with H.

Heisenberg Picture

Combine the two eq.

KG Equation

int0 HHH Interaction picture

S

States and Operators both evolve with time in interaction picture:

We can move just the free H0 to operators.

// 00)( tiHS

tiHI eOetO

Evolution of Operators

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!

For field operators in the interaction picture:

The Fourier expansion of a free field is still valid.

Evolution of States

S

States evolve like in the Schrodinger picture but with Hamiltonian replaced by HI(t).

HI(t) is just the interaction Hamiltonian Hint in interaction picture!

That means, the field operators in HI(t) are free.

)()( tHi

tdt

dIII

,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

)(),()( 00 tttUt II

Define time evolution operator U

All the problems can be answered if we are able to calculate this operator. It determines the evolution of states.

U operator, the evolution operator of states in the interaction picture

The transition amplitude for the scattering of A:

can be computed:

The transition amplitude for the decay of A:

can be computed: ASBCAUBC I ,

is the state in the future t which evolved from a state ψ in t0

is the state in long future which evolved from a state ψ from long past.

The amplitude for this state to appear as a state ϕ is their inner product:

Solve it by a perturbation expansion in small parameters in HI.

),(),(),( 0)1(

0)0(

0 ttUttUttU

To leading order:

Perturbation expansion

1),( 0)0( ttU

Define S matrix:

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

BC

ig

The remaining numerical factor is:

A

Momentum Conservation

For a toy ABC model

Three scalar particle with masses mA, mB ,mC

External Lines

Internal Lines

Vertex

ip

iq

-ig

1

imq

i

jj 22

321442 kkk

A B

C

1k

2k

3k

Lines for each kind of particle with appropriate masses.

The configuration of the vertex determine the interaction of the model.

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

In momentum space, the factor for a vertex is simply a constant.

The integration yields a momentum conservation.

A

BC

Interaction Lagrangian vertex

The Interaction Lagrangian is integrated over the whole spacetime.

Interaction could happen anywhere anytime.

The amplitudes at various spacetime need to be added up.

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!

Propagator

The integration of two identical interaction Hamiltonian HI. The first HI is always later than the second HI

Solve the evolution operator to the second order.

HI is first order.

The integration of two identical interaction Hamiltonian HI. The first HI is always later than the second HI

)()()()()()())()(( 1212212121 tAtBtttBtAtttBtAT

t’

t’’

We are integrating over the whole square but always keep the first H later in time than the second H.

t’ and t’’ are just dummy notations and can be exchanged.

)()()()()()())()(( 1212212121 tAtBtttBtAtttBtAT

This definition is Lorentz invariant! t’

t’’

This notation is so powerful, the whole series of operator U can be explicitly written:

The whole series can be summed into an exponential:

0)()(0 21)()(

24

14 142231 xCxCTeexdxd xppixppi

Amplitude for scattering BBAA

Propagator between x1 and x2 Fourier Transformation

p1-p3 pour into C at x2 p2-p4 pour into C at 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.

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)

Again every Interaction is integrated over the whole spacetime.

Interaction could happen anywhere anytime and amplitudes need superposition.

This construction ensures causality of the process. It is actually the sum of two possible but exclusive processes.

0)()(0 yxT

This propagator looks reasonable in coordinate space but difficult to calculate and the formula is cumbersome.

0)()(0 yxT

0)()(0 yxT

0)()(0 yxT

This doesn’t look explicitly Lorentz invariant.

But by definition it should be!

So an even more useful form is obtained by extending the integration to 4-momentum. And in the momentum space, it becomes extremely simple:

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)

The Fourier Transform of the propagator is simple.

0)()(0 yxT

00 yx

0)()(0 yxT

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)

For a toy ABC model

Internal Linesiq

imq

i

jj 22

Lines for each kind of particle with appropriate masses.

0)()(0 21)()(

24

14 142231 xCxCTeexdxd xppixppi

Amplitude for scattering BBAA

aa

aa

Every field either couple with another field to form a propagator or annihilate (create) external particles!

Otherwise the amplitude will vanish when a operators hit vacuum!

aa

For a toy ABC model

Three scalar particle with masses mA, mB ,mC

External Lines

Internal Lines

Vertex

ip

iq

-ig

1

imq

i

jj 22

321442 kkk

A B

C

1k

2k

3k

Lines for each kind of particle with appropriate masses.

The configuration of the vertex determine the interaction of the model.

Assuming that the field operator is a complex number field.

ipxp

ipxp ebea

pdx

2

1

)2()(

3

3

The creation operator b+ in a complex KG field can create a different particle!

Scalar Antiparticle

The particle b+ create has the same mass but opposite charge. b+ create an antiparticle.

ipxp

ipxp ebea

pdx

2

1

)2()(

3

3

ipxp

ipxp eaeb

pdx

2

1

)2()(

3

3 Complex KG field can either annihilate a particle or create an antiparticle!

Its conjugate either annihilate an antiparticle or create a particle!

So we can add an arrow of the charge flow to every leg that corresponds to a field operator in the vertex.

The charge difference a field operator generates is always the same!

charge non-conserving interaction

ipxp

ipxp ebea

pdx

2

1

)2()(

3

3 ipx

pipx

p eaebpd

x 2

1

)2()(

3

3

incoming particle or outgoing antiparticle

incoming antiparticle or outgoing particle

charge conserving interaction

incoming particle or outgoing antiparticle

incoming antiparticle or outgoing particle

can either annihilate a particle or create an antiparticle!

can either annihilate an antiparticle or create a particle!

incoming antiparticle or outgoing particle

incoming particle or outgoing antiparticle

U(1) Abelian Symmetry

)()( xex iQ

The Lagrangian is invariant under the field phase transformation

invariant

)(xee iQiQ

is not invariant

U(1) symmetric interactions correspond to charge conserving vertices.

A

BC

If A,B,C become complex, they all carry charges!

The interaction is invariant only if 0 CBA QQQ

The vertex is 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)

For a toy charged AAB model

Three scalar charged particle with masses mA, mB

External Lines

Internal Lines

Vertex

ip

iq

-iλ

1

imq

i

jj 22

321442 kkk

A A

B

1k

2k

3k

Lines for each kind of particle with appropriate masses.

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

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 )(