T1: Interacting Relativistic Quantum Field Theories Flashcards
Define the contraction of the free fields ϕ_0(x) and ϕ_0(y)
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)
State Wick’s theorem
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.
What is the vacuum expectation value of normal ordered fields?
Always zero!
What is the vacuum expectation of odd n time ordered fields? Why?
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.
Define external points and their characteristics
- Points not integrated over
- Represent the initial space-time points (in front of exp in Feynman prop)
- Only have a single line
Define internal points and their characteristics
- 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
Draw the Feynman diagram which represent spontaneous production and annihilation
Vacuum bubble; figure of 8
How do we relate free a free and interacting field?
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
Give the relation between free and interacting fields
ϕ(t,x) = U†(t,t0)ϕ_0(t,x)U(t,t0)
Give the differential equation relating U with H_int
i ∂U/∂t = H_int U(t,t0)
Define H_I
The hamiltonian describing the interaction term
INT d^3x λ/4! ϕ_0 ^4
How do we find Dyson’s formula?
Integrate the differential equation with initial condition t=t0 U= I
State Dyson’s formula
U(t,t0) = T{exp(INT_t0 ^t d^3x H_int(t’) )}
Give the composition property of U(t,t0)
U(t1,t2)U(t2,t3) = U(t1,t3)
Describe the Feynman propagator for an interacting field
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
Give a brief description of evaluating the Feynman propagator analytically
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
Give the diagram for a Feynman propgator G(x,y)
x ______ y
How do we find the coefficient of Feynmann diagram?
Permutation of the lines coming out an internal point/symmetries
How many internal and external points does contribution N_n have? (for basic example)
Two external points and n internal points
How many internal and external points does contribution D_n have?
No external points and n internal points. All vacuum bubbles
Give the full Feynman propagator for interacting fields at n spacial points in diagrammatic form
Sum of connected diagrams with n external points
In the Schrodinger picture, how do we evolve a state |a,ti⟩ in time?
Time evolution operator:
|a,ti⟩(tf) = E^-iH(tf - ti)|a,ti⟩
What assumption motivates out defining of the S matrix
At very early and later times, we assume particles are well separated and we recover the free theory.
Define the S matrix
The matrix which takes an initial state of the free theory at ti → -∞ to a final state at tf → ∞
Write down in the S and H pictures, the probability of a state evolving at time limits
Check notes
Define the operator a†_k ^in
The operator which prepares a particle of momentum k in the very far past; where ϕ and ϕ_0 coincide
Define the four-momentum of a particle
K = (ω_k, k)
Energy and spacial momentum
How do we define ingoing and outgoing states with definite momentum in the far past and future respectively
Act with a†_k ^in and a†_k ^out on the zero state
How do we relate the s-matrix element to the time-ordered expectation valeus?
Prepare the incoming and outgoing states and take the ip. This gives a bunch of creation ops inside the vacuum state.
State the Feynman rules for σϕ^2 theory
x _____ y G_ ϕ(x,y)
x ——–y G_ σ (x,y)
|
| \_\_\_\_|\_\_\_\_\_ intersection at x' -ig INT d^4 x'
Divide by symmetry factor
What constitutes particle scattering in Feynman diagrams?
External points connect to the same internal point
Draw the s, t and u channel diagrams
Check notes
Define connected for a Feynman diagram
Where all external points are connected by some combination of lines
