Quantum field theory
Tutorials for the Oxford Master of Physics (MPhys) course on theoretical physics (C6)
Hilary and Trinity terms 2026
Administrative information:
Tutorials for Hilary term are on Mondays of weeks 4, 6, and 8.
Please hand in your solutions before Sunday 12.30pm: either via email or to DWB653; during the final weekend please slide it into the mailbox at the front entrance of DWB level 4.
The lecturer's course webpage is https://sites.google.com/view/ehardy/teaching/c6-qft.
Course content:
Quantum field theory (QFT) concerns the quantum physics of relativistic particles and their interactions. Although it has deep and far-reaching consequences, this course can only scratch the surface of its extremely wide range of applications. In particular, the main functionality that this course focuses on is to build a scaffolding for calculating decay widths and cross sections for simple processes.
From first year and third year materials on special relativity (CP1) and particle physics (B4), particle interactions at the relativistic scale differ from the those at the non-relativistic scale (for example the electromagnetic attraction between the nucleus and the electrons in atomic physics) in the sense that, at relativistic energies, the number and/or type of particles is not conserved. To give some concrete examples, the neutron decays into three particles (proton, electron, and anti-electron neutrino) in its decay, and $e^+e^-$ collisions can give rise to single resonances (for example $c\bar c$ or $b\bar b$ bound states) and subsequently decay to more particles. Complications arise quickly for a theory that is so general which takes into account so many intricacies.
As a starting point hopefully you would realise that, since a theory with varying particle numbers is essential, making field theory you studied in Michaelmas term relativistic is a much better starting point than making single-particle quantum mechanics relativistic: as it already include tools specifically designed for describing many particles collectively. Therefore it would be beneficial to go through the course with these tools sharp, in particular
- generating functionals, correlation functions, Wick's theorem, Feynman diagrams, perturbation theory;
- path integrals for single-particle quantum mechanics, propagators, imaginary time, Euclidean action, the partition function and Helmholtz free energy as path integrals;
- Landau free energy in terms of one or many order parameters, Fourier transforms of order parameters on a lattice, spontaneous symmetry breaking, Goldstone's theorem;
- the occupation number representation, creation and annihilation operators, quantisation of operators, basis transformations.
Although in many cases the exact approaches in the Michaelmas and Hilary term material are not identical, the underlying logic is roughly the same; additionally keeping these tools in mind would also, of course, help you to tackle the C6 exam in Trinity term.
Problem sheet 1: symmetries
Monday week 4, Hilary term
Link to problem sheet 1 via the lecturer's webpage
Lagrangians, Hamiltonians, and Euler-Lagrange equations of fields. Mass dimensions and dimensional analysis. Real and complex scalar fields, the Klein-Gordon equation. Vector fields, Maxwell's equations.
Nöther's theorem.
External symmetries. Generators of the Lorentz group. Irreducible representations of the Lorentz group characterised by $(j_+, j_-)$. Nöther current and Nöther charges corresponding to Lorentz and Poincaré invariance.
Global internal symmetries. Discrete and continuous symmetries, e. g. $\mathbb{Z}_2$, $U(1)$, $SO(3)$, $SU(2)$. Fundamental and adjoint representations. Structure functions. Global $U(1)$ symmetry and associated Nöther current and Nöther charges. Spontaneous symmetry breaking in e. g. $\mathbb{Z}_2$ and $U(1)$ with Sombrero-shaped potentials. Goldstone's theorem.
Local internal symmetries and gauge theories. Gauge theories of e. g. $U(1)$, $SU(2)\times U(1)$. Abelian and non-Abelian Higgs Mechanism. Analysis of spontaneous symmetry breaking pattern and Higgs mechanism in cases with non-diagonal mass matrices, un-broken subgroups and/or un-eaten Goldstone bosons.
Problem sheet 2: quantisation
Monday week 6, Hilary term
Quantisation: promoting Poisson brackets to commutators. Equal-time correlation relations.
Quantisation and mode expansion(s) of one or many real and/or complex scalar field(s). Annihilation and creation operators. Normal ordering.
Concrete calculations using commutation relations and annihilation/creation operators: the Pauli-Jordan function $\mathit{\Delta}\left(x-y\right) = [\phi\left(x\right), \phi\left(y\right)]$, the Wightman function $D_+\left(x-y\right) = \left\langle 0 | \phi\left(x\right) \phi\left(y\right) | 0 \right\rangle$, and the Feynman propagator $D_F\left(x-y\right) = \left\langle 0 | T \phi\left(x\right) \phi\left(y\right) | 0 \right\rangle$. Microcausality.
The Fock space: one-particle and many-particle states.
Quantisation of vector fields. Using the gauge degrees of freedom to remove unphysical states in the Fock space. The Feynman propagator for vector fields $D_F^{\mu\nu}\left(x-y\right) = \left\langle 0 | T A^\mu\left(x\right) A^\nu\left(y\right) | 0 \right\rangle$.
Problem sheet 3: interactions
Monday week 8, Hilary term
The interaction picture, Dyson's formula.
Time-ordered correlation functions in the interaction picture, perturbative expansion of $\left\langle \mathit{\Omega} | T \phi\left(x_1\right)\cdots\phi\left(x_n\right) | \mathit{\Omega} \right\rangle$ in terms of $\left\langle 0 \Big| T \phi_{\mathrm{ip}}\!\left(x_1\right)\cdots\phi_{\mathrm{ip}}\!\left(x_n\right)\prod_j \int \mathrm{d}^4 y_j \mathcal{H}_{\mathrm{int, ip}}\!\left(y_j\right) \Big| 0 \right\rangle$.
Evaluation of $\left\langle 0 \big| \phi_{\mathrm{ip}}\!\left(x_1\right)\cdots\phi_{\mathrm{ip}}\!\left(x_n\right) \big| 0 \right\rangle$. Contraction. Wick's theorem, Feynman diagrams, Feynman rules. Symmetry factors.
Scattering theory. $S$ and $T$ matrices. $\left\langle \mathbf{p}_1 \cdots \mathbf{p}_n | \mathrm{i}T | \mathbf{k}_1 \cdots \mathbf{k}_m \right\rangle$, decay widths and cross sections in terms of the scattering amplitude $\mathcal{A}_{fi}$.
Relations between $\left\langle \mathbf{p}_1 \cdots \mathbf{p}_n | \mathrm{i}T | \mathbf{k}_1 \cdots \mathbf{k}_m \right\rangle$, $\left\langle \mathit{\Omega} | T \phi\left(x_1\right)\cdots\phi\left(x_n\right) | \mathit{\Omega} \right\rangle$, and $\left\langle 0 \big| \phi_{\mathrm{ip}}\!\left(x_1\right)\cdots\phi_{\mathrm{ip}}\!\left(x_n\right) \big| 0 \right\rangle$. Removal of particles from initial and final states, the LSZ formula.
Concrete calculations of decay widths and cross sections for simple models e. g. $\phi^4$, scalar Yukawa theory, scalar quantum electrodynamics.
Problem sheet 4: integrals
Trinity term
Path integral quantisation for relativistic fields. Path integral representation of $\left\langle \mathit{\Omega} | T \phi\left(x_1\right)\cdots\phi\left(x_n\right) | \mathit{\Omega} \right\rangle$. Schwinger-Dyson equations. Symmetries in the path integral formalism: derivation of Nöther current and Nöther charges.
Evaluation of $\left\langle \mathit{\Omega} | T \phi\left(x_1\right)\cdots\phi\left(x_n\right) | \mathit{\Omega} \right\rangle$ using generating functionals. Wick's theorem and Feynman diagrams. Removal of vacuum bubbles using $Z[J]=-\mathrm{i} \log W[J]$.
Perturbation theory. Loop counting.
Effective action. Classical fields. The effective action $\mathit{\Gamma}\left[\phi_c\right]$ as a Legendre transform of $Z[J]$. One particle irreducible (1PI) Green's functions. Concrete calculations of 1PI Green's functions for an interacting theory.