The Quantum Electrodynamics - Outline | PHYS 544, Study notes of Physics

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Quantum Electrodynamics

Phys 544

Eric Thrane

Outline

• Foch Space: The playground of QFT
• The Lagrangian Formulation of QM
• Feynman Diagrams: A tool for doing

perturbation theory

• Gauge Invariance and Photons
• The “running” of the fine structure

constant:

Fig 1: The running

of α.

e

2

4 #h c

e

2

Foch Space

• Relativistic QM (Foch Space)
  • States are still described by kets:
  • In addition to having different energies,

states can have different numbers of

particles in them.

• We can define operators that add or remove

particles at a given momentum:

!

" = c 1

E 1

• c 2

E 1

, E 1

• c 3

E 1

, E 2

• ...

a ˆ

( k )

, a ˆ

( k )

†

Foch Space

a ˆ

(

r

k )

†

Vacuum =

r

k

• We’ve expanded our vector space to

accommodate antimatter.

• What kind of Hamiltonian can we write

down? It has to be Lorentz invariant.

• Enter the quantum field.

Writing a Hamiltonian

• Now with our fields handy, we can write

down a Hamiltonian that might look

something like this:

• The Hamiltonian looks everywhere in space

and tries to remove particles. When it finds

one, it hits it with E

2 = (p

2

• m

2 ), and then

puts back the particle. (Good thing we

divided by E

1/ in each of the fields to cancel

one of the E’s.)

!

H = d

3 x "

$

( x )

(%&

2

• m

2 ) $ ( x )

• ...

The Lagrangian Formulation

• Indeed, we could write down a Hamiltonian

like that, but it turns out to be a bit of pain to

do calculations.

• In the 1940’s Feynman came up with a new

way of thinking about quantum mechanics,

which makes things easier.

• Fortunately, our discussion of fields will be

directly applicable to Feynman’s Lagrangian

formulation of Quantum Mechanics.

Perturbation Theory

• Doing the sum over all paths is hard. There

are many paths, and exact solutions are rare.

• We can, however, do the

sum over all paths for a free

particle, which we can then

do perturbation theory about.

2

U "

1

= e

i d

4 x # L $

all _ paths

%

Field Lagrangian

• Before we get into perturbation theory,

let’s write down an example of a

Lagrangian. Recall our Hamiltonian:

• If we transform it into a Lagrangian, we

get:

!

H = d

3 x "

$

( x )

(%&

2

• m

2 ) $ ( x )

• ...

L =

1

2

" μ

( )

2

$

1

2

m

2

( x )

2

• L int

Feynman Diagrams

• There is lots of fun math involved in

calculating these expansions, but

Feynman devised a nice bookkeeping

method for doing them.

• Once you know the “Feynman Rules”

for a given Lagrangian, you just have to

plug and chug.

• The first thing you do is draw a cartoon.

For the Lagrangian I wrote down

before––(called “φ

4 theory”)––the

rules are pretty simple.

• Every order of perturbation gets

a vertex.

• Every leg corresponds to a φ−

particle. (Here two φ‘s are

scattering off each other.)

• The amplitude for this process is

λ.

L =

μ

2

m

2

( x )

2

4

The QED Lagrangian

• Here it is. Everything we know about light

and its interactions with matter resides in this

Lagrangian.

• What type of scattering can we achieve with

this Lagrangian?

L = " ( i

$ m ) " $

( F

μ%

2

$ e " &

μ

" A μ

!

F μ"

$

μ

A "

% $ "

A μ

" = ( i # 0

")

†

γ -> e

e

Can we get a photon to turn into an e+ e- pair?

L = " ( i

$ m ) " $

( F

μ%

2

$ e " &

μ

" A μ

!

F μ"

$

μ

A "

% $ "

A μ

" = ( i # 0

")

†

CPT Theorem

• Also interesting is the CPT theorem.
• Almost all Lagrangians that we use in particle

physics are invariant under the combined

operations of C, P, and T.

• This means that antiparticles moving forward

in time are mathematically equivalent to

particles moving backward

in time.

The QED Lagrangian

• Let’s take another look at the QED

Lagrangian. What symmetry properties does

it have?

• Noether’s theorem tells us that each

symmetry of the Lagrangian corresponds to

a conserved quantity.

L = " ( i

$ m ) " $

( F

μ%

2

$ e " &

μ

" A μ

!

" = ( i # 0

")

†

!

F μ"

$

μ

A "

% $ "

A μ