T2: Path integrals in QFT

Question 1

What spacetime dimensions do d = -1 and d = 0 correspond to?

Answer 1

d = -1 ordinary integrals
d = 0 quantum mechanics

Question 2

Generalise the transition amplitude to N particles

Answer 2

Add a [Dq_n] for each particle from 1,..,n

The action runs over n positions and n conjugate momenta.

2N boundary conditions q_n (t) = q_n and q_n (t’) = q_n’

Question 3

Generalise the transition amplitude to scalar fields ⟨ϕ’,t’|ϕ,t⟩

Answer 3

N INT [Dϕ] exp[iS]

S = INT_ti ^tf INT d^dx L(ϕ, ∂_μ ϕ)

Question 4

Define the generating functional Z[J] of the free scalar field theory

Answer 4

Z_0[J] = N exp(1/2 INT d^4x d^4y J(x)G_F^0(x,y)J(y) )

Note, the 4 can be any d+1 dimensions

Question 5

How do we relate free and interacting scalar field theories

Answer 5

Consider the Lagrangian of the interacting theory as a small perturbation from the free theory

S = S_0 + λS_I

Where |λ|«1

Question 6

Give the full expression for the generating function of the interacting field theory in terms of Z_0[J]

Answer 6

Z[J] = N exp(iλ INT d^4y L_I δ/δJ(y))Z_0[J]

Question 7

Draw the propagator G_F^0(x,y) in interacting field theory

Answer 7

X____Y

Question 8

Draw the loop G_F^0(0) in interacting field theory

Answer 8

0

Question 9

Draw an external source in interacting field theory

Answer 9

X____ = INT d^4x J(x)

Question 10

Draw Z_0[J]

Answer 10

Z_0[J] = N exp[1/2 X___X]

Question 11

Draw the first functional derivative of Z_0[J] wrt to J(y)

Answer 11

= N ∎____X exp(1/2 X____X)

Question 12

Describe how to take subsequent functional derivatives

Answer 12

Add a cross and connect to a dot or remove a cross from the end of a line and connect to a dot