Notes

Papers

• I. Path Integrals and the Quantum–Statistical Correspondence Imperial Year-2 essay — Feynman's path integral, Wick rotation, and the partition function. HTML ↗PDF ↓ Essay

Completed solutions to M. D. Schwartz’s Quantum Field Theory and the Standard Model Book website ↗

• 29. Chapter 29 — Weak interactions Completed Problems 29.1, 29.2, 29.3, 29.6, 29.7, 29.8, 29.9, collected in one chapter edition. HTML ↗PDF ↓ 7 completed problems

Solutions to David Tong’s The Standard Model example sheets Original course ↗

• I. Sheet 1: The Basics Weyl spinors, fermion masses, and discrete symmetries. Worked problems: 1, 2, 3, 4, 6, 7, 8. HTML ↗PDF ↓ Tong Standard Model · Sheet 1
• II. Sheet 2: Symmetry Breaking and QCD Goldstone modes, Higgs mechanisms, QCD running, and meson masses. Worked problems: 1, 2, 3, 4, 5, 6, 7, 8. HTML ↗PDF ↓ Tong Standard Model · Sheet 2
• III. Sheet 3: Anomalies Gauge anomalies, hypercharges, anomaly matching, and global symmetries. Worked problems: 1, 2, 3, 4, 5, 6. HTML ↗PDF ↓ Tong Standard Model · Sheet 3
• IV. Sheet 4: The Standard Model Running couplings, electric charges, flavour parameters, and two Higgs doublets. Worked problems: 1, 2, 4, 5, 6. HTML ↗PDF ↓ Tong Standard Model · Sheet 4

Solutions to Peskin & Schroeder’s An Introduction to Quantum Field Theory Publisher page ↗

• I. Equivalent Photon Approximation Peskin & Schroeder, Problem 6.2. HTML ↗PDF ↓ Peskin Ch. 6
• II. Alternative Regulators in QED Peskin & Schroeder, Chapter 7 — Pauli–Villars vs. dimensional regularization. HTML ↗PDF ↓ Peskin Ch. 7
• III. Final Project — Radiation of Gluon Jets Peskin & Schroeder, end-of-book project. HTML ↗PDF ↓ Peskin Final

Solutions to David Tong’s Quantum Field Theory problem sheets Original course ↗

• PS1. Lorentz, Symmetries & Currents \(\mathrm{SO}(3)\), infinitesimal Lorentz, energy-momentum tensor (incl. EM field), Proca, scale invariance. HTML ↗PDF ↓ Tong PS1
• PS2. Canonical Quantization & the Free Scalar String quantization, free scalar, normal ordering, Yukawa, Wick's theorem, Feynman propagator. HTML ↗PDF ↓ Tong PS2
• PS3. Dirac Field & Spin-Statistics Clifford algebra, Lorentz from \(\gamma\)-matrices, traces, plane-wave spinors, canonical quantisation. HTML ↗PDF ↓ Tong PS3
• PS4. Interactions & Tree-Level Amplitudes Wick's theorem at tree level — \(\phi^4\) 3-to-3, vacuum bubbles, Yukawa, Compton, \(e^-e^+\to\mu^-\mu^+\). HTML ↗PDF ↓ Tong PS4

Perimeter Scholars International · Quantum Field Theory II

• PS1. Correlation Functions in Quantum Mechanics Euclidean and real-time path-integral propagators of the harmonic oscillator at finite temperature. HTML ↗PDF ↓ PSI QFT II · PS1
• H1. Generating Functionals for \(\phi^4\) Theory Generating functionals, connected correlators, vacuum diagrams, and the 1PI effective action. HTML ↗PDF ↓ PSI QFT II · Homework 1
• H2. Renormalization in Massive \(\phi^4\) Theory Cutoff renormalization, running coupling and mass, and RG flow in the O(N) model. HTML ↗PDF ↓ PSI QFT II · Homework 2
• T3. Saddle-Point Approximation and Functional Determinants Saddle points, functional determinants, zeta regularization, and the Gel'fand-Yaglom formalism. HTML ↗PDF ↓ PSI QFT II · Tutorial 3
• T4. Exact Results in Quantum Field Theory The Källén-Lehmann representation, spectral functions, and Schwinger-Dyson equations. HTML ↗PDF ↓ PSI QFT II · Tutorial 4
• T5. Effective Action and Renormalization in \(\phi^3\) Theory The effective action, loop expansion, counterterms, and renormalization of \(\phi^3\) theory. HTML ↗PDF ↓ PSI QFT II · Tutorial 5
• T6. Renormalization-Group Flow in Three Dimensions Dimensionless couplings, beta functions, and the infrared fixed point of three-dimensional \(\phi^4\) theory. HTML ↗PDF ↓ PSI QFT II · Tutorial 6
• T7. Berezin Integration and the Gross-Neveu Model Grassmann Gaussian integrals, fermionic correlators, and large-N Gross-Neveu theory. HTML ↗PDF ↓ PSI QFT II · Tutorial 7
• T8. SU(N) Yang-Mills Theory and Scalar Fields Fundamental and adjoint representations, covariant derivatives, scalar couplings, and Yang-Mills zero modes. HTML ↗PDF ↓ PSI QFT II · Tutorial 8
• T9. The Faddeev-Popov Determinant Gauge-orbit volume, gauge slices, Jacobians, Gaussian gauge fixing, and ghost variables. HTML ↗PDF ↓ PSI QFT II · Tutorial 9

Solutions to Hugh Osborn’s Advanced Quantum Field Theory example sheets Course materials ↗

• O2. Functional Methods & Grassmann Integrals Osborn AQFT Example Sheet 2 — Legendre transform, 1PI effective action, Green's-function/vertex relations, Grassmann Gaussian integrals, Pfaffians, fermionic partition function, SUSY QM. (Q1–Q8) HTML ↗PDF ↓ Osborn AQFT · ES2
• O3. Feynman Graphs & RG Calculations Osborn AQFT Example Sheet 3 — superficial divergence, Feynman parameters, position-space loop integrals, \(\phi^4\) counter-terms, two-loop β functions, RG equations, \(\phi^3\) in \(d=6\). (Q1–Q10) HTML ↗PDF ↓ Osborn AQFT · ES3

Solutions to Mark Srednicki’s Quantum Field Theory Author’s page ↗

• S1. Nonabelian Symmetries Srednicki Ch. 24 — antisymmetry of the generator, structure constants from \(\mathrm{SO}(N)\), Noether current and charge algebra, generators of \(\mathrm{Sp}(2N)\). (Q24.1–24.4) HTML ↗PDF ↓ Srednicki Ch. 24
• S2. Renormalization Group for \(\phi\text{-}\chi\) Theory Srednicki Ch. 28 — one-loop Z-factors in MS-bar, \(\beta\)-functions for the \(\phi^3\) + \(\phi\chi^2\) couplings, and asymptotic freedom condition on \(h/g\). (Q28.3) HTML ↗PDF ↓ Srednicki Ch. 28

Independent computation

• I. Computation for \(\phi^3\) Theory Counter-terms, partition functional, and one-loop diagrams in \(\phi^3\) scalar theory; follows Srednicki and Peskin & Schroeder. HTML ↗PDF ↓ \(\phi^3\)

Solutions to David Tong’s General Relativity problem sheets Original course ↗

• PS1. Differential Geometry Manifolds, tensors, Lie and exterior derivatives, Maurer–Cartan, Poincaré lemma. HTML ↗PDF ↓ Tong GR PS1
• PS2. Connections & Curvature Christoffel transformation, torsion, Bianchi, parallel transport, Riemann / Ricci / Weyl, geodesics, Reissner–Nordström. HTML ↗PDF ↓ Tong GR PS2
• PS3. Geodesics, Killing Vectors & Energy Conditions Timelike geodesics, Brans–Dicke scalar-tensor gravity, 11-d supergravity, Killing vectors, conformal compactification, energy conditions. HTML ↗PDF ↓ Tong GR PS3
• PS4. Linearised Gravity & Gravitational Waves Linearised gravity from a point mass, cosmic strings, Lense–Thirring, Fierz–Pauli, transverse-traceless gauge, binary gravitational-wave emission. HTML ↗PDF ↓ Tong GR PS4

Worked exercises from Robert M. Wald’s General Relativity Book website ↗

• 3. Chapter 3: Curvature Torsion, two-dimensional geometry, Riemann symmetries, curvature, affine parameters, and geodesics. Selected worked problems. Includes links to related Tong GR solutions. HTML ↗PDF ↓ Wald GR Ch. 3
• 4. Chapter 4: Einstein's Equation Maxwell theory, frame dragging, linearised gravity, and gravitational radiation. Selected worked problems. Includes links to related Tong GR solutions. HTML ↗PDF ↓ Wald GR Ch. 4
• 5. Chapter 5: Homogeneous, Isotropic Cosmology Robertson-Walker coordinates, Friedmann equations, de Sitter spacetime, redshift, and null rays. Selected worked problems. HTML ↗PDF ↓ Wald GR Ch. 5
• 6. Chapter 6: The Schwarzschild Solution Isotropic coordinates, Reissner-Nordström, stationary observers, a draft time-delay calculation, and horizon crossing. Selected worked problems. HTML ↗PDF ↓ Wald GR Ch. 6

Solutions to Cambridge Quantum Mechanics examples accompanying David Tong’s notes Original exercises ↗

• I. Non-Degeneracy & Reality of Bound States in 1D Cambridge Quantum Mechanics, Example Sheet 1, Q4; accompanies David Tong’s notes. HTML ↗PDF ↓ Cambridge QM · ES1
• II. Absence of Odd-Parity Bound States in a Shallow Square Well Cambridge Quantum Mechanics, Example Sheet 1, Q7; accompanies David Tong’s notes. HTML ↗PDF ↓ Cambridge QM · ES1
• III. Factorisation Method for the Sech-Squared Potential Cambridge Quantum Mechanics, Example Sheet 1, Q8; accompanies David Tong’s notes. HTML ↗PDF ↓ Cambridge QM · ES1

Solutions to the radiation exercises accompanying David Tong’s Electromagnetism Original exercises ↗

• I. Retarded Potentials and Far-Field Radiation Electromagnetism, Chapter 7 exercises, Q4. HTML ↗PDF ↓ Cambridge EM · Ch. 7
• II. Liénard–Wiechert Potential and the Field Tensor Electromagnetism, Chapter 7 exercises, Q6. HTML ↗PDF ↓ Cambridge EM · Ch. 7

Solutions to Electromagnetism Problem Sheet 3 — Waves and Relativity Original course ↗

• I. Relativistic Motion in a Uniform Electric Field D. Tong, Electromagnetism, PS3 Q9. HTML ↗PDF ↓ Tong EM · PS3
• II. Gauge Transformation of a Plane EM Wave D. Tong, Electromagnetism, PS3 Q5. HTML ↗PDF ↓ Tong EM · PS3
• III. Reflection of an EM Wave from a Moving Mirror D. Tong, Electromagnetism, PS3 Q7. HTML ↗PDF ↓ Tong EM · PS3
• IV. Covariant Form of Ohm's Law D. Tong, Electromagnetism, PS3 Q10. HTML ↗PDF ↓ Tong EM · PS3

Solutions to the electromagnetism-in-matter exercises accompanying David Tong’s Electromagnetism Original exercises ↗

• I. Dielectric Sphere in a Uniform Electric Field Electromagnetism, Chapter 8 exercises, Q1. HTML ↗PDF ↓ Cambridge EM · Ch. 8
• II. Frustrated Total Internal Reflection & Evanescent Waves Electromagnetism, Chapter 8 exercises, Q3. HTML ↗PDF ↓ Cambridge EM · Ch. 8

Standalone reference note

• I. Jordan's Lemma Standard statement, proof, and applications. HTML ↗PDF ↓ Complex Analysis

Solutions to J. E. Santos’s Cambridge Part IB Complex Methods example sheets Cambridge examples ↗

• I. Convolution and Fourier Representation of \(e^{-\lvert x\rvert}\) J. E. Santos, Cambridge Part IB Complex Methods, Example Sheet 3, Q2. HTML ↗PDF ↓ Santos CM
• II. Residues at Simple and Higher-Order Poles J. E. Santos, Cambridge Part IB Complex Methods, Example Sheet 2, Q7. HTML ↗PDF ↓ Santos CM
• III. Contour Integration via Residues and Cauchy J. E. Santos, Cambridge Part IB Complex Methods, Example Sheet 2, Q9. HTML ↗PDF ↓ Santos CM
• IV. Laplace Transform of \(t^{-1/2}\) and the Bromwich Contour J. E. Santos, Cambridge Part IB Complex Methods, Example Sheet 3, Q12. HTML ↗PDF ↓ Santos CM

Cambridge Mathematical Tripos · Variational Principles Original sheet ↗

• I. Variational Derivation of Euler's Equation for Inviscid Flow Variational Principles, Example Sheet 2 (2023), Q9. HTML ↗PDF ↓ VarPrin

Unofficial solution to a public Cambridge Mathematical Tripos past paper Original paper ↗

• I. Green's Identity & Method of Images for the Half-Space Cambridge Mathematical Tripos Part IB 2025, Paper 3, 14D (Methods). Original question linked, not reproduced. HTML ↗PDF ↓ Cambridge IB

Solutions to David Tong’s Statistical Physics problem sheets Original course ↗

• I. The Joule–Thomson Process D. Tong, Statistical Physics, PS4 Q4. HTML ↗PDF ↓ Tong SP
• II. Partition Function of a Spin-\(\tfrac{1}{2}\) System D. Tong, Statistical Physics, PS1 Q3. HTML ↗PDF ↓ Tong SP
• III. Heat Capacity of an Interacting Spin System D. Tong, Statistical Physics, PS1 Q4. HTML ↗PDF ↓ Tong SP

Solutions to David Tong’s Statistical Field Theory example sheets Original course ↗

• I. Sheet 1: Landau Theory Transfer matrices, mean-field and tricritical transitions, XY order, and scaling. Worked problems: 1–8. HTML ↗PDF ↓ Tong SFT · Sheet 1
• II. Sheet 2: Correlations and Scaling Correlations, Lifshitz scaling, gauge coupling, anisotropy, and Wick's identity. Worked problems: 1, 2, 5, 6, 7, 8; Problems 3 and 4 remain pending. HTML ↗PDF ↓ Tong SFT · Sheet 2
• III. Sheet 3: Renormalisation Group Critical exponents, fixed-point flows, sine-Gordon theory, and membranes. Worked problems: 1–6. HTML ↗PDF ↓ Tong SFT · Sheet 3