T1: Free Relativistic Quantum Field Theories

Question 1

How do we move from Lagrangian to Hamiltonian field theories?

Answer 1

Define some generalised coordinate ϕ(x,t)

Define some conjugate π = ∂L/∂ϕ ̇

Question 2

Define normal ordering

Answer 2

Field ordering such that all energy raising operators appear to the left of of energy lowering operators.

Question 3

Give the relation between ω k and m

Answer 3

ω_k = sqrt(k^2 +m^2)

Question 4

Define the Hamiltonian density (and hence Hamiltonian)

Answer 4

H(ϕ,π) = -L + SUM_a πϕ ̇

Question 5

State the particle commutators (in H and S pictures)

Answer 5

[qa, qb] = 0
[pa, pb] = 0
[qa, pb] = δab

(all at t1=t2 for H)

Question 6

How do the quantum commutators change for QFT?

Answer 6

[ϕ_a (x,t), π_b (y,t)] = i δ_ab δ(x-y)

Question 7

How can we show the free complex scalar field behaves as a quantised SHO?

Answer 7

Fourier transform the solution in terms of some new label k. Rewrite the Klein-Gordon in separate terms and act on the state.

Question 8

Give the commutator relations for the creation annihilation operators with eachother (Q)

Answer 8

[a, a†] = 1
[a, a] = 0
[a†, a†] = 0

Question 9

Give the commutator relations for the creation annihilation operators with the Hamiltonian (Q)

Answer 9

[H, a†] = ωa†
[H, a] = -ωa

Question 10

State the Hamiltonian for the SHO (Q)

Answer 10

H = ω/2 (a†a +a a†) = ω(a†a + 1/2)

Question 11

Give the excited state and energy spectrum for SHO (Q)

Answer 11

(a†)^n |0⟩

H|n⟩ = ω(n +1/2)

Question 12

State the general solution of ϕ~ for SHO (QFT)

Answer 12

ϕ(k,t) = 1/sqrt(2ω_k) [a_k e^-iωt + b*_k e^iωt]

Question 13

State the general solution ϕ for the Klein gordon SHO (QFT)

Answer 13

Integrate ϕ~ over d^3k/(2π)^3 and add an e^-k⋅x with each exponential.

Make change of integration variables in t2 sending k to -k

Question 14

Give the commutator relations for the creation annihilation operators with eachother (QFT)

Answer 14

[a_k, a_k’†] = (2π)^3 δ(k-k’)

Question 15

State the Hamiltonian for the SHO with frequency ω_k (QFT)

Answer 15

H = ω(a†a + bb†)

ω, a and b have _k

Integrate over d^3k/(2π)^3

Question 16

Where does the QFT SHO hamiltonian fail?

Answer 16

When we try to determine the energy spectrum by commuting annihilation ops to the right we get an infinity.

When we promote from fields to operators we gain an infinite number of non-commutators.

Question 17

How do we resolve the QFT SHO hamiltonian fail?

Answer 17

We employ normal ordering: place all annihilation ops to the right. Denoted by ::

Question 18

What is the energy of the vacuum state in QFT?

Answer 18

0!

Question 19

Define normal-ordered charge :Q: (QFT)

Answer 19

Hamiltonian with a negative sign in front of b operators

Question 20

How do our QFT SHO ops change for a real scalar field

Answer 20

A particle is equal to its anti-particle a = b and a† = b†

A particle has charge Q = 0

Question 21

How can we create a particle at definite position

Answer 21

Denote ϕ in terms of raising and lowering ops and act on the vacuum.