T1: Interacting Relativistic Quantum Field Theories

Question 1

Define the contraction of the free fields ϕ_0(x) and ϕ_0(y)

Answer 1

For a string of fields containing ϕ_0(x) and ϕ_0(y) we pull them together and replace with Feynman propagator G(x,y)

Question 2

State Wick’s theorem

Answer 2

The time ordering of the free field at points x1 to xn is the sum of all ways the field can be contracted with any uncontracted fields being normal ordered.

Question 3

What is the vacuum expectation value of normal ordered fields?

Answer 3

Always zero!

Question 4

What is the vacuum expectation of odd n time ordered fields? Why?

Answer 4

Zero!

By Wick’s theorem we contract pairs and normal order remainders. Since contraction is pair-wise, for odd n there will always be a single normal-ordered term which annihilates.

Question 5

Define external points and their characteristics

Answer 5

• Points not integrated over
• Represent the initial space-time points (in front of exp in Feynman prop)
• Only have a single line

Question 6

Define internal points and their characteristics

Answer 6

• Points integrated over
• Represent the points induced by the interaction term in Hamiltonian
• Number of lines leaving internal point represent number of fields in interaction term

Question 7

Draw the Feynman diagram which represent spontaneous production and annihilation

Answer 7

Vacuum bubble; figure of 8

Question 8

How do we relate free a free and interacting field?

Answer 8

Assume that at some time t, the fields are equivalent. We evolve our free field backwards in time via H0 and then forward in time via H

Question 9

Give the relation between free and interacting fields

Answer 9

ϕ(t,x) = U†(t,t0)ϕ_0(t,x)U(t,t0)

Question 10

Give the differential equation relating U with H_int

Answer 10

i ∂U/∂t = H_int U(t,t0)

Question 11

Define H_I

Answer 11

The hamiltonian describing the interaction term

INT d^3x λ/4! ϕ_0 ^4

Question 12

How do we find Dyson’s formula?

Answer 12

Integrate the differential equation with initial condition t=t0 U= I

Question 13

State Dyson’s formula

Answer 13

U(t,t0) = T{exp(INT_t0 ^t d^3x H_int(t’) )}

Question 14

Give the composition property of U(t,t0)

Answer 14

U(t1,t2)U(t2,t3) = U(t1,t3)

Question 15

Describe the Feynman propagator for an interacting field

Answer 15

N: Dyson’s formula with ϕ_0 at each position x1,..xn in front of exponential

D: Dyson’s formula

N and D between ground states

Question 16

Give a brief description of evaluating the Feynman propagator analytically

Answer 16

Take the big exp formula and split into N and D. taylor expand around the small parameter and use Wicks theorem to evaluating the time-ordering

Question 17

Give the diagram for a Feynman propgator G(x,y)

Answer 17

x ______ y

Question 18

How do we find the coefficient of Feynmann diagram?

Answer 18

Permutation of the lines coming out an internal point/symmetries

Question 19

How many internal and external points does contribution N_n have? (for basic example)

Answer 19

Two external points and n internal points

Question 20

How many internal and external points does contribution D_n have?

Answer 20

No external points and n internal points. All vacuum bubbles

Question 21

Give the full Feynman propagator for interacting fields at n spacial points in diagrammatic form

Answer 21

Sum of connected diagrams with n external points

Question 22

In the Schrodinger picture, how do we evolve a state |a,ti⟩ in time?

Answer 22

Time evolution operator:

|a,ti⟩(tf) = E^-iH(tf - ti)|a,ti⟩

Question 23

What assumption motivates out defining of the S matrix

Answer 23

At very early and later times, we assume particles are well separated and we recover the free theory.

Question 24

Define the S matrix

Answer 24

The matrix which takes an initial state of the free theory at ti → -∞ to a final state at tf → ∞

Question 25

Write down in the S and H pictures, the probability of a state evolving at time limits

Answer 25

Check notes

Question 26

Define the operator a†_k ^in

Answer 26

The operator which prepares a particle of momentum k in the very far past; where ϕ and ϕ_0 coincide

Question 27

Define the four-momentum of a particle

Answer 27

K = (ω_k, k)

Energy and spacial momentum

Question 28

How do we define ingoing and outgoing states with definite momentum in the far past and future respectively

Answer 28

Act with a†_k ^in and a†_k ^out on the zero state

Question 29

How do we relate the s-matrix element to the time-ordered expectation valeus?

Answer 29

Prepare the incoming and outgoing states and take the ip. This gives a bunch of creation ops inside the vacuum state.

Question 30

State the Feynman rules for σϕ^2 theory

Answer 30

x _____ y G_ ϕ(x,y)
x ——–y G_ σ (x,y)

    |
    |  \_\_\_\_|\_\_\_\_\_ intersection at x'  -ig INT d^4 x' 

Divide by symmetry factor

Question 31

What constitutes particle scattering in Feynman diagrams?

Answer 31

External points connect to the same internal point

Question 32

Draw the s, t and u channel diagrams

Answer 32

Check notes

Question 33

Define connected for a Feynman diagram

Answer 33

Where all external points are connected by some combination of lines