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This is the Past Exam Paper of Mathematical Tripos which includes Solitons and Instantons, Smooth Function, Scalar Field Theory, Derrick Scaling Arguments, Bogomolny Equations, Topological Degree, Sigma Model Lumps etc. Key important points are: Quantum Field Theory, Dirac Equation, Gamma Matrices, Chiral Representation, Pauli Matrices, Clifford Algebra, Minkowski Metric, Klein-Gordon Equation, Plane-Wave Solutions, Negative Frequency Solutions
Typology: Exams
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Friday 1 June 2007 9.00 to 12.
Attempt THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 The Dirac equation is (i γμ∂μ − m) ψ = 0
where the gamma matrices are given in the chiral representation by,
γ^0 =
, γi^ =
0 σi −σi^0
Here σi^ are the Pauli matrices and 1 2 is the unit 2 × 2 matrix.
a) Show that these matrices satisfy the Clifford algebra
{γμ, γν^ } = 2 ημν^14
where ημν^ is the Minkowski metric.
b) Show that the each component of the spinor ψ(x) satisfies the Klein-Gordon equation.
c) Consider the ansatz for plane-wave solutions,
ψ(x) = u (~p) e−ip·x
where p^2 = m^2. Show that this ansatz solves the Dirac equation when
u (~p) =
√p^ ·^ σ ξ p · σ ξ¯
for any 2-component spinor ξ , with σμ^ = (1 2 , σi) and ¯σμ^ = (1 2 , −σi). Write down the ansatz for negative frequency solutions and solve it.
d) The action of a rotation ϕ~ on the Dirac spinor is given by the matrix
ei~ϕ·~σ/^2 0 ei~ϕ·~σ/^2
Write down the spinor u(~p) describing a stationary particle of mass m with
(i) Spin directed up along x^3.
(ii) Spin directed up along x^1.
For each of these cases, write down the spinor corresponding to a massless particle travelling in the positive x^3 direction.
Paper 50
3 The Lagrangian for a scalar field ϕ of mass m and charge e, interacting with the electromagnetic field is
Fμν F μν^ + Dμ ϕ?Dμϕ − m^2 |ϕ |^2
where Fμν = ∂μAν − ∂ν Aμ and Dμϕ = ∂μϕ + i eAμϕ.
a Show that this Lagrangian has a gauge symmetry.
b What is the physical difference between gauge symmetries and global symmetries? Justify your answer.
c The theory contains two interaction vertices with Feynman rules given by
p
q
and −i e (p + q)μ + 2i e^2 ημ ν
where ημν is the Minkowski metric. Identify the interaction terms in the Lagrangian corresponding to these two vertices.
d) When quantizing the theory in Coulomb gauge ∇ · A~ = 0 , the naive photon propagator is
Dμν (p) =
i p^2 + i
δij −
pipj | ~p |^2
μ = i 6 = 0 , ν = j 6 = 0 i | ~p |^2
μ, ν = 0
0 otherwise
Draw the leading order diagrams for ϕ ϕ¯ → ϕ ϕ¯ scattering and show that, when the external momenta are on-shell, the naive photon propagator may be replaced by the Lorentz invariant propagator
Dμν (p) = −i
ημν p^2
Paper 50
4 Write an essay on the role of anti-particles in quantum field theory. To obtain full credit the essay should include a discussion on the following topics: the difference between a real and a complex scalar field; the difference between a relativistic and non- relativistic field theory; a comparison of Dirac’s hole interpretation of anti-particles and the appearance of anti-particles in quantum field theory.
Paper 50
