Teaching

Detailed course plans, assignments, references, and occasionally some notes for the courses I have taught are available below.

Ashoka University

Quantitative Reasoning and Mathematical Thinking (January – April ‘26)

A (very) gentle introduction to some simple ideas in mathematics and their applications. We discussed what mathematics is, when it can be used to deepen our understanding, how one might go about doing so, and what the limitations of such analyses are. Through examples from many subfields within mathematics, I tried to cultivate an appreciation for the elegance, economy, and clarity of mathematical thinking.

Some of the topics I discussed were: the shapes of surfaces and knotted pieces of string; elementary results in the theory of numbers; what proofs are and where definitions come from; wordless or picture-perfect proofs; counting, games, and probabilities; the infinitely big and infinitesimally small in mathematics. For more details, please see the course information sheet.

Take-Home Assignment

The take-home assignment is available here.

Assignments

Lectures

25. Summing Up — 04/24

An overview of the topics discussed in this course.
24. Relativity — 04/22

The Michaelson-Morley null result and the death of luminiferous ether. Special relativity contrasted with Galilean relativity. The equivalence principle. Freely falling and accelerating elevators and the nature of gravitational forces. Early predictions regarding gravitational redshifts and the Pound-Rebka experiment. The curvature of spacetime induced by matter and energy. The Hulse-Taylor binary and early hints of gravitational waves. Direct detection and gravitational wave interferometry.
Suggested reading: I read Was Einstein Right? by Clifford M. Will as an undergraduate and it remains one of my favourite popular introductions to the general theory of relativity.
23. Quantum Mechanics — 04/17

Young’s double-slit experiment. The photoelectric effect and Einstein’s light quanta. Atomic spectra and Bohr’s model. Pauli’s aufbauprinzip and the periodic table. Fever dreams and the birth of quantum theory. Born’s rule and the probabalistic nature of quantum theory. The “Shut Up and Calculate!” philosophy and its utility in the development of the nascent quantum theory.
Suggested reading: A delightful account of the early days of quantum theory can be found in Ch. 1–6 of Quantum Drama by Jim Baggott and John L. Heilbron.
22. Class Test — 04/15

Based on take-home assignment.
21. Statistical Mechanics — 04/10

Reductionism and emergence. Entropy, energy, and their competition for dominance. Phase transitions and universality. Suggested reading: More is Different by P. W. Anderson.
20. Bayesian Inference — 04/08

Bayes’ theorem and hypothesis testing. Priors and posterior probabilities. Here’s the Python notebook from the lecture.
Suggested reading: Ch. 1–2 of Data Analysis: A Bayesian Tutorial by D. S. Sivia and J. Skilling; What is Bayesian Statistics? by Sean R. Eddy.
19. Hamiltonian Paths and Complexity — 04/01

Hamiltonian paths/cycles. Certificates. An informal discussion of complexity classes P and NP. Bipartite graphs. Dirac’s theorem.
18. Graphs — 03/28

The Bridges of Königsberg. Some elementary results on Euler cycles/paths in the theory of graphs.
17. Conditional Probability and Statistics — 03/27

Conditional probabilities and Bayes’ theorem. Random sampling of populations and Gaussian distributions.
Suggested reading: An Intuitive Explanation of Bayes’ Theorem by Eliezer S. Yudkowsky.
16. Probability — 03/25

Sample spaces, outcomes, and events. Elementary results in the theory of probability.
Suggested reading: Ch. 1–4 of A Philosophical Essay on Probabilities by Pierre-Simon Laplace.
15. Permutations and Combinations — 03/18

Some simple combinatorial problems. Binomial coefficients.
Suggested reading: Ch. 6 (titled Combinatorics) in Discrete Thoughts: Essays on Mathematics, Science, and Philosophy by Mark Kac, Gian-Carlo Rota, and Jacob T. Schwartz.
14. Kleiber Scaling and Fractal Networks — 03/13

The 3/4 scaling law. West-Brown-Enquist’s fractal model and a derivation of the 3/4 scaling law. Debates in the literature surrounding its validity.
Suggested reading: Life’s Universal Scaling Laws by Geoffrey B. West and James H. Brown.
13. Dimensional Analysis and Scaling Laws — 03/11

Simple dimensional analysis for equations in physics. Metabolic rate and Rubner’s surface law of metabolism.
Suggested reading: On Being The Right Size by J. B. S. Haldane.
12. Infinite Series and Primes — 02/27

Sequences and series. Exponential and logarithmic functions. The prime counting function and the prime number theorem. Eratosthenes’ sieve. Harmonic series and Euler’s product formula.
Suggested reading: A great popularisation of the Riemann Hypothesis can be found in Prime Obsession by John Derbyshire.
11. Primitive Ideas of Calculus — 02/25

On Archimedes’ Measurement of a Circle and Quadrature of the Parabola.
Suggested reading: Ch. 1 (titled Area, Number, and Limit Concepts in Antiquity) and Ch. 2 (titled Archimedes) of The Historical Development of the Calculus by C. H. Edwards, Jr.
10. Continuity and Calculus — 02/20

Plotting polynomial functions. How to define continuous functions rigorously. The trouble with infinitesimals and the notion of a derivative. The ubiquity of mathematics in the natural sciences. Quine-Putnam indispensability and our ontological commitments to mathematical objects. Field’s fictionalism.
Suggested reading: The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner. Also, here’s an interactive visualisation of the ε-δ definition of a limit.
9. Relations and Functions — 02/18

Relations on sets. Reflexivity, symmetry, and transitivity. Equivalence relations. Functions and their graphs.
Suggested reading: None for this class. However, you’d do well to learn how to plot simple functions using GeoGebra.
8. Countability and Correspondences — 02/13

Hilbert’s hotel. Countable and countably infinite sets. Injections, surjections, and bijections. The Cantor-Schröder-Bernstein theorem.
Suggested reading: Avatars of the Tortoise and The Library of Babel by Jorge Luis Borges.
7. The Infinite — 02/11

The one-to-one correspondence as the notion of “sameness of size” appropriate to infinite sets. Cantor’s diagonal argument.
Suggested reading: Introduction (titled Paradoxes of the Infinite) and Ch. 2 (titled Aristotle) and Ch. 8 (titled The Mathematics of the Infinite, and the Impact of Cantor) of The Infinite by A. W. Moore.
6. Induction and Primes — 02/06

More on mathematical induction. The fundamental theorem of arithmetic governing the existence and uniqueness of prime factors. Euclid’s theorem on the infinitude of primes. Arbitrarily long intervals of integers without primes.
Suggested reading: Why Isn’t the Fundamental Theorem of Arithmetic Obvious? and Proving The Fundamental Theorem of Arithmetic by Timothy Gowers.
5. Numbers — 02/04

Natural numbers, integers, and rationals. Irrational numbers and a proof of the irrationality of √2. Platonism and the Frege-Russell conception of cardinals. Simple statements and proofs in the theory of numbers.
Suggested reading: Enriching Divisibility: Multiple Proofs and Generalizations by Benjamin Dickman; Ch. 6 (titled Pythagoras and The Soul) and Ch. 18 (titled Philolaus and the Formal Cause) of The Presocratic Philosophers by Jonathan Barnes.
4. Knots — 01/30

Knots as embeddings of circles into three-dimensional space. Planar and ambient isotopy. Reidemeister moves. Tricolourability and its isotopy invariance.
Suggested reading: Chapters 1–3 of Knots: Mathematics with a Twist by Alexei Sossinsky.
3. The Poincaré-Hopf Index Theorem — 01/28

Winds on two-dimensional surfaces. Flowlines and their tangents. Isolated critical points and how to compute their index. Invariance of the index under “small disturbances.” The Euler characteristic as a sum of indices of critical points associated to a triangulation.
Suggested reading: Why Space Has Three Dimensions by Henri Poincaré.
2. The Euler Characteristic — 01/23

The topology of two-dimensional surfaces. Computing the Euler characteristic of a square using triangulations. An inductive proof that the square has unit Euler characteristic. How to construct a torus by identifying the edges of a square.
Suggested reading: Funes the Memorious by Jorge Luis Borges; Proofs and Refutations by Imre Lakatos.
1. Introduction — 01/21

Overview of course rules, attendance and grading policies. Rough outline of topics to be discussed during the course.
Suggested reading: A Mathematician’s Lament by Paul Lockhart.

Mathematical Physics I (January – April ‘26)

This was a first course in mathematical methods — both analytic and numerical — often used in physics. The course covered the following topics: linear algebra and vector spaces; simple ordinary differential equations; elementary Fourier analysis; vector calculus; and some computational techniques including numerical solutions to algebraic and differential equations. For more information, please see the course information sheet.

Take-Home Assignment

Linear Algebra and Vector Calculus

Assignments

Coding Challenges

Gram-Schmidt Orthogonalisation

Lectures

25. Summing Up — 04/24

Overview of topics covered in the course and how they’re related to each other.
24. Green’s Functions — 04/22

Green’s functions for second-order ordinary differential equations. The relationship between Green’s functions, eigenfunctions, and eigenvalues.
23. Parseval’s Identity — 04/17

Proof of Parseval’s identity. Using it to compute infinite sums.
22. Fourier Series — 04/15

Orthonormality of eigenfunctions. Extracting Fourier series coefficients.
21. Eigenfunctions of Differential Operators — 04/10

Eigenfunctions of the harmonic oscillator. Fourier series.
20. Harmonic Oscillator and Eigenvalue Equations — 04/08

The universality of the harmonic oscillator. Motivating eigenvalue equations in the study of differential equations.
19. More Second-Order Linear Differential Equations — 04/01

Solutions to homogeneous equations. Particular solutions. The method of undetermined coefficients.
18. Second-Order Linear Differential Equations — 03/28

Solutions to second order linear differential equations. The Wronskian and linear independence of solutions.
17. More First-Order Linear Differential Equations — 03/27

Some more first-order linear ODEs and how to solve them.
16. First-Order Linear Differential Equations — 03/25

Differential equations in mathematical physics. Some first-order linear ODEs and how to solve them.
15. Dirac Delta Functions — 03/18

Introducing distributions. The Dirac δ-function and its properties.
14. Curvilinear Coordinates and Stokes’ Theorems — 03/13

Proving the Stokes’ theorems in curvilinear coordinates.
13. Stokes’ Theorems — 03/11

Stokes’ theorems for gradients, divergences, and curls.
12. Taylor Expansions and Jacobians — 02/28

Taylor series. Jacobians of coordinate transformations.
11. Integrals in Two and Three Dimensions — 02/26

Surface and volume integrals.
10. Curls and Product Rules — 02/20

The curl of a vector field. Product rules. Scalar and vector potentials.
9. Multivariable Calculus, Gradients, and Divergences — 02/18

Functions of many variables and partial differentiation. Gradient of a function and its interpretation. Vector differential operators. The divergence of a vector field.
8. Review of Differential Calculus + Curves and Surfaces — 02/13

Outer products. Review of limits, continuity, and differentiability in single-variable calculus. Leibniz and chain rules. Curves and surfaces.
7. Eigenvalues and Eigenvectors — 02/11

Eigenvalues and eigenvectors of linear operators. Eigenspaces as subspaces. Characteristic polynomials and the eigenvalue equation.
6. Linear Functionals and Derivations — 02/06

Dual spaces and functionals. Derivations on algebras. The algebra of linear operators on a vector space. Structure constants.
5. Algebras and Linear Transformations — 02/04

Algebras and their examples. Linear transformations on vector spaces. Dual spaces and functionals. Derivations on algebras.
4. Inner Product Spaces — 01/30

Inner products on vector spaces. Orthogonality, normality, and orthonormality. The norm of a vector.
3. Vector Spaces II — 01/28

Linear independence. Subspaces. The span of a set of vectors. A basis for a vector space as a linear independent set of vectors that spans it. The dimension of a vector space.
2. Vector Spaces — 01/23

Abstracting away from vectors in three-dimensional space to define vector spaces in general.
1. Introduction — 01/21

Overview of course rules, attendance and grading policies. Rough outline of topics to be discussed during the course.

University of Delhi

Strategies for Scientific Dialogue in Research (August – December ‘25)

This is a skills-based course worth two credits. The course guidelines are available here.

Quantum Field Theory III (January – April ‘25)

This was a course on quantum field theory for PhD students. This course has as prerequisites PH-ET536 and PH-ET576, our QFT I and QFT II electives offered to MSc students.

Lectures

16. Monopoles — 04/25
15. Lattice Fermions — 04/24
14. Instantons — 04/17
13. Lattice Gauge Theory — 04/09
12. Gauge Anomalies and Anomaly Cancellation — 03/28
11. Background Field Method — 03/28
10. The Chiral Anomaly — 03/26
9. Soft Collinear Effective Field Theory — 03/21
8. 1-Loop β-Function for Non-Abelian Gauge Theories — 03/20
7. IR Divergences — 02/14
6. Basics of Thermal Field Theory — 02/13
5. BRST Quantisation — 02/07
4. Local Symmetries, Constraints, and Quantisation — 02/06
3. Weinberg’s Soft Photon Theorem — 01/31
2. Wigner’s Classification and Little Groups — 01/24
1. Coherent States in Quantum Mechanics — 01/23

Quantum Field Theory II (January – April ‘25)

This is the second of a two-semester sequence of courses on quantum field theory. Working chiefly in the path integral formalism, we studied more advanced aspects of quantum field theory including renormalisation, and effective field theories. The plan was to cover non-Abelian gauge theories, too, we didn’t have enough time to get to it.

Assignments

Lectures

26. Abelian Higgs Models, Superconductors, and Confinement — 04/25

The Higgs mechanism in Abelian gauge theory. Meissner effect. Nielsen-Oleson vortex strings. Type I and II superconductors. Electric-magnetic duality, dual superconductors, and charge confinement.
25. Effective Field Theories for Nambu-Goldstone Modes — 04/24

General discussion of effective field theories. Matching and using equations of motion.
24. Spontaneous Symmetry Breaking in Yukawa Theory — 04/17

Spontaneous symmetry breaking in Yukawa theory. The Adler zero principle.
23. Lie Algebras — 04/04

A lightning introduction to Lie algebras and their representations.
22. All Together Now — 04/03

The continuum and Wilsonian RG. Relevant, irrelevant, and marginal couplings revisited. Quantum triviality.
21. Wilsonian RG — 03/28

Integrating out high-energy degrees of freedom. The Wilsonian effective action.
20. RG Flows II — 03/27

Interpreting RG flows. The significance of fixed points.
19. RG Flows — 03/21

Anomalous dimensions and critical exponents. Renormalisation group flows. The Wilson-Fisher fixed point.
18. RG Equations — 03/20

1-loop β-function for QED. Landau pole. Asymptotically free and conformal theories. Anomalous dimensions and their interpretation.
17. Renormalisation of QED II — 03/07

Subtleties involving IR divergent integrals. Fixing counterterms.
16. Renormalisation of QED I — 03/06

Renormalised perturbation theory for quantum electrodynamics. Renormalisation conditions.
15. Renormalisation of Quartic Scalar Theory — 02/28

Renormalised perturbation theory for self-interacting scalar theories. Fixing counterterms. Renormalisation conditions.
14. Counting UV Divergences — 02/27

Superficial degree of divergence in theories with scalars and fermions. Superrenormalisable, renormalisable, and non-renormalisable theories. Listing divergent 1PI subgraphs.
13. Anomalous Magnetic Moment — 02/14

Gordon’s identity. Vertex corrections and computation of electron g-2 in quantum electrodynamics.
12. On-Shell Subtraction — 02/13

1-particle irreducible graphs. Pole and renormalised mass. Further discussion of LSZ prescription.
11. Electron Self-Energy Graphs — 02/07

Self-energy graphs in QED. Mass and wavefunction renormalisation. Subtraction schemes.
10. Vacuum Polarisation in QED — 02/06

Vacuum polarisation graphs in scalar and spinor QED. 1-loop corrections to Coulomb’s law. Charge renormalisation.
9. Dimensional Regularisation — 01/31

How to do integrals commonly encountered in loop-level QFT using dimensional regularisation.
8. Vacuum Polarisation and Pauli-Villars Regularisation — 01/30

Pauli-Villars ghosts, and how to use them to regularise integrals. Example using vacuum polarisation graph in cubic scalar theory.
7. Renormalised Perturbation Theory and Schwinger Parameters — 01/24

Counterterms and their use. Schwinger and Feynman parametrisations for loop integrals.
6. Casimir Force, Regulators, and a Broad Overview of Renormalisation — 01/23

The Casimir effect. Cut-off and heat-kernel regulators. Wick rotation. On-shell perturbation theory and renormalisation conditions.
5. Lehmann-Symanzik-Zimmermann Reduction — 01/17

A derivation of the LSZ reduction formula, with examples of how it works. (I’m not writing up the notes for this lecture. Please see Sec. 6.1 of Schwartz instead!)
4. Dyson-Schwinger Revisited — 01/16

A derivation of the Dyson-Schwinger equation in the path integral formalism. Discussion of contact terms. The case of global symmetries and Ward identities.
3. Dyson-Schwinger, Canonically — 01/10

A derivation of the Dyson-Schwinger equation in the canonical formalism.
2. Path Integrals for Gauge Fields — 01/09

Derivation of the gauge-fixed path integral for an Abelian gauge theory. Proof that matrix elements of gauge invariant operators are independent of the choice of gauge.
1. Path Integrals for Scalars — 01/03

Review of path integral for scalar fields and how to compute the path integral for a free theory. Incorporating interactions and the reproducing Feynman rules. A first look at mass renormalisation due to interactions.

Quantum Field Theory I (August – November ‘24)

Between August–November 2024, I taught the first of a two-semester sequence of courses on quantum field theory. The course was divided into roughly four parts: (i) canonical and path integral quantisation of free scalar fields, (ii) spinors, Abelian gauge fields, and their quantisation, (iii) interactions and tree-level scattering amplitudes in various theories, and (iv) spontaneous symmetry breaking. It was planned initially that we would also discuss some aspects of the large-N approximation and lattice field theory, but this was not possible.

Assignments

Lectures

30. QED — 11/22

Feynman rules for quantum electrodynamics. Scattering amplitudes for elementary processes.
29. Scalar QED — 11/21

Feynman rules for scalar QED. Elementary processes.
28. Where Do Feynman Rules Come From? — 11/14

Tracking all the factors. Derivative couplings.
27. Scattering in Theories with Fermions — 11/08

Relating correlation functions computed in the free and interacting vacua. Feynman rules for theories with fermions and scalars. Computing correlation functions perturbatively.
26. More Amplitudes — 11/07

Streamlining Feynman rules. Mandelstam invariants.
25. Scattering Amplitudes in Scalar Theories — 10/25

Wick’s theorem relating time-ordered and normal-ordered operators. Feynman rules for theories with real and complex scalars.
24. The S-Matrix and Decay Amplitudes — 10/24

Decays amplitudes in a toy theory of scalars.
23. Spontaneous Symmetry Breaking II — 10/18

A special series of lectures by Prof. Debajyoti Choudhury.
22. Spontaneous Symmetry Breaking I — 10/17

A special series of lectures by Prof. Debajyoti Choudhury.
21. Interactions — 10/11

Dimensional analysis. Classically relevant, irrelevant, and marginal operators. The interaction picture and Dyson series.
20. Gupta-Bleuler — 10/10

Negative-norm states in Maxwell theory and the tension between Lorentz covariance and positive norm.
19. Maxwell Theory — 10/04

Gauge invariance and gauge fixing. Canonical quantisation of Maxwell theory. Polarisation vectors and mode expansions. Negative-norm states.
18. Path Integrals for Dirac Fermions — 10/03

Grassmann calculus. Fermionic path integrals. Re-deriving the Dirac propagator.
17. Propagators — 09/26

Propagators for free Dirac fermions. The Grassmann algebra.
16. Spin-Statistics Connection — 09/20

The relation between spin and statistics. Considerations of stability. Canonical quantisation of Dirac spinors. Causality and field commutators.
15. Dirac Lagrangian — 09/19

Plane wave solutions. Inner and outer products.
14. Dirac Lagrangian — 09/13

Constructing the Dirac Lagrangian. Field bilinears.
13. What is a Spinor? — 09/12

Clifford algebras. Properties of Dirac γ-matrices. Constructing representations of the Lorentz Lie algebra using γ-matrices.
12. The Lorentz Lie Algebra — 09/06

The relation between the complexified Lorentz Lie algebra and the su(2) Lie algebra. Scalar, vector, and tensor representations.
11. The Lorentz and Poincare Groups — 09/05

Computing Gaussian path integrals. The Poincare group and its various subgroups. The Poincare and Lorentz Lie algebras. Parity and time-reversal. The global structure of the Lorentz group.
10. Path Integral in Quantum Field Theory — 08/30

Gaussian fluctuations and effective actions. Operator insertions in the path integral and time-ordering. The path integral in quantum field theory. Green’s functions and the generating functional.
9. Path Integral in Quantum Mechanics — 08/29

Constructing the phase space and configuration space path integrals via time-slicing. Interpretation as sum over trajectories. Recovering the principle of least action.
8. Feynman Propagator and Complex Scalars — 08/23

Propagators, in particular the Feynman propagator as a Green’s function for the Klein-Gordon equation. Time-ordering symbol and the iε pole prescription.Note: Due to a power failure, we weren’t able to get very far into our discussion of the complex scalar. Please see the latest assignment for this material instead!
7. Causality and Propagation — 08/22

Lorentz-invariant measures and relativistic normalisation of one-particle states. Vanishing of field commutators at spacelike separations.
6. Mode Expansions and Fock Space — 08/16

Mode expansions of field operators, reality conditions, equal-time commutation relations. Hamiltonian and zero-point energy. Normal ordering. One-particle states. Quanta of real scalar field obey bosonic statistics. The number operator in free and interacting theories.
5. Canonical Quantisation — 08/10

Canonical quantisation of the free scalar field. Heisenberg picture operators and the Klein-Gordon equation.
4. Gauging Global Symmetries — 08/09

Wilson lines and covariant derivatives. Gauge invariance and charge conservation. Review of canonical quantisation for finite-dimensional dynamical systems.
3. Noether’s Theorem — 08/08

Locally conserved currents and associated charges. Noether’s theorem for internal and spacetime symmetries. Symmetry groups and the O(N) model.
2. Field Theory and Statistical Mechanics — 08/03

The Ising model, computing its partition function as a sum over functions of the average magnetisation of “blocks” of spins, the Landau-Ginzburg free energy functional, equivalence with the Hamiltonian of a scalar field in Minkowski spacetime, dominant contributions to the path integral as saddle points, Wick rotation (formally) relating QFTs and statistical mechanical systems.
1. Classical Scalar Fields — 08/02

The principle of least action for finite-dimensional dynamical systems, extension to scalar field theories, the (Euler-Lagrange) field equations for a scalar field, recovering Klein-Gordon as a field equation, plane-wave solutions and dispersion relations, classical field theory in phase space, the Hamiltonian for scalar fields.

Quantum Field Theory II (February – May ‘24)

This was the second of a two-semester sequence of courses on quantum field theory. Working chiefly in the path integral formalism, we studied more advanced aspects of quantum field theory including non-Abelian gauge theories, renormalisation, spontaneous symmetry breaking, and effective field theories. We were also able to discuss some aspects of quantum field theories in low dimensions.

Assignments

Lectures

38. Vertex Operators and Bosonisation — 05/10

Correlation functions of vertex operators. The bosonisation dictionary relating chiral fermions to vertex operators of chiral bosons. Duality between sine-Gordon theory and the massive Thirring model.
37. Two-Dimensional Fermions, Compact Bosons, and T-Duality — 05/09

Chiral fermions in two dimensions. Compact bosons. Momentum and winding modes. T-duality.
36. Abelian Higgs Model — 05/04

Spontaneous symmetry breaking with gauge fields. Landau-Ginzburg description of superconducting phase transition. Vortex strings.
35. Effective Field Theories — 05/03

Constructing an effective field theory of Nambu-Goldstone bosons via matching, and by explicitly eliminating heavy modes.
34. Scattering of Nambu-Goldstone Bosons — 05/02

Spontaneous symmetry breaking in a Yukawa theory. Nambu-Goldstone and Higgs modes. Scattering of Nambu-Goldstone bosons. The Adler zero principle.
33. Spontaneous Symmetry Breaking — 04/27

Spontaneous breaking of discrete symmetries. Tunneling in quantum mechanics vs. tunneling in field theory. Complex scalar fields in a quartic potential. Shift symmetry of the Goldstone mode.
32. Monopoles — 04/26

Another, extra special lecture on monopole solutions by Prof. Dileep Jatkar (Harish-Chandra Research Institute).
31. Symmetry and Renormalisation — 04/25

A special lecture by Prof. Dileep Jatkar (Harish-Chandra Research Institute).
30. Feynman Rules — 04/20

Feynman rules for non-Abelian gauge theories. Elementary processes.
29. BRST Symmetry — 04/19

A global symmetry of the gauge-fixed Yang-Mills Lagrangian. The BRST operator and its cohomology.
28. Faddeev-Popov for Non-Abelian Gauge Fields — 04/18

How to derive the gauge-fixed path integrals over non-Abelian gauge fields.
27. Effective Field Theory: Not Everything Goes! — 04/13

A special lecture on effective field theories by Dr. Subham Dutta Chowdhury (U. Chicago).
26. Yang-Mills Theory — 04/12

Non-Abelian gauge fields and their field strengths. The Yang-Mills action for non-Abelian gauge fields.
25. Lie Algebras and Wilson Loops — 04/06

A quick review of Lie groups, Lie algebras, and their representation theory. Wilson lines in Abelian gauge theory.
24. Wilsonian Renormalisation Group II — 04/05

Scaling and renormalisation group flows revisited. Comparison between the continuum and Wilsonian renormalisation group. Radiative corrections to scalar masses. Fine-tuning, naturalness, and the hierarchy problem.
23. Wilsonian Renormalisation Group I — 04/04

Momentum shell renormalisation group. The Wilsonian low-energy effective action.
22. Running Masses and Renormalisation Group Flows — 03/21

Quantum corrections to the Yukawa potential. Critical exponents. The renormalisation group flow. Gaussian and Wilson-Fisher fixed points.
21. Renormalisation Group Equations — 03/16b

The beta function. Anomalous dimensions. The Callan-Symanzik equation.
20. 1-Loop Renormalisation of Quantum Electrodynamics — 03/16a

Consolidating earlier computations in QED and doing some more. Counterterms, relations between them, and implications thereof.
19. Renormalised Perturbation Theory II — 03/14

Setting up renormalised perturbation theory for QED.
18. 1-Loop Renormalisation of 𝜙⁴ Theory — 03/09

Actually computing counterterms in scalar and Yukawa theories.
17. Renormalised Perturbation Theory I — 03/08

Setting up renormalised perturbation theory. Counterterms and renormalisation conditions.
16. Counting UV Divergences — 03/07

Superficial degree of divergence. Super-renormalisable, renormalisable, and non-renormalisable theories. Examples.
15. Anomalous Magnetic Moment — 03/02

Computing 1-loop corrections to the magnetic moment of the electron.
14. Pole Mass and Subtraction Schemes — 03/01

1-particle irreducible graphs. Pole mass. Minimal vs. on-shell subtraction. Amputation via LSZ. Subtraction points.
13. Electron Self-Energy — 02/29

Computing 1-loop corrections to the electron propagator. Mass and wavefunction renormalisation. Counterterms and subtraction schemes.
12. Vacuum Polarisation in Quantum Electrodynamics — 02/24

Computing quantum corrections to the Coulomb potential. The Lamb shift and effective charge. The Landau pole in QED.
11. Dimensional Regularisation — 02/23

How to do integrals in 4-ε dimensions.
10. Vacuum Polarisation in Scalar Theories — 02/22

Vacuum polarisation in scalar field theories. Feynman and Schwinger parameters. Pauli-Villars ghosts. Wick rotation.
9. A Bird’s Eye View of Renormalisation — 02/17

How renormalisation works. Bare and physical couplings. Counterterms and renormalised perturbation theory.
8. Infinities and Regulators— 02/16

Using cut-off and heat kernel regularisation schemes to regulate divergent sums. Computation of Casimir force.
7. Takahashi-Ward Identities — 02/15

A proof of the Takahashi-Ward and Ward identities.
6. LSZ Reduction — 02/10

Lehmann-Symanzik-Zimmermann (LSZ) reduction, or how to relate correlation functions to scattering amplitudes.
5. Dyson-Schwinger, Path Integrally — 02/09

Derivation of the Dyson-Schwinger equations in the path integral formalism. Dyson-Schwinger equations for global symmetries.
4. Dyson-Schwinger, Canonically — 02/08

Derivation of the Dyson-Schwinger equations in the canonical formalism. Recovering position space Feynman diagrams using the Dyson-Schwinger equations.
3. Path Integrals for Fermions and Abelian Gauge Fields — 02/02

Review of path integral for fermions and Abelian gauge fields. Field redefinitions. Gauge invariance. Stueckelberg’s trick.
2. Path Integrals for Scalars — 02/01

Review of path integral for scalar fields and how to compute the path integral for a free theory. Incorporating interactions and the reproducing Feynman rules. A first look at mass renormalisation due to interactions.
1. Introduction — 01/29

Broad introduction to the topics to be covered in this course, discussion of course plan.

Critical Phenomena (February – May ‘24)

These lectures covered critical phenomena and the renormalisation group.

Assignments

Sextic Free Energies and Multi-Critical Points

Lectures

11. Gaussian Fixed Points — 05/10

The Gaussian fixed point. Relevant and irrelevant operators.
10. Momentum-Shell Renormalisation Group — 05/09

What renormalisation group transformations look like in momentum space. Flows in the space of theories. Fixed points.
9. Ginzburg Criterion — 05/04

When is mean field theory reliable?
8. Correlation Length — 05/03

Computing the 2-point function in a Gaussian theory explicitly. The correlation length.
7. Green’s Functions — 05/02

Connected correlation functions. Gaussian path integrals with sources. Green’s functions.
6. Path Integrals — 04/27

How to do Gaussian path integrals.
5. Landau-Ginzburg Theory II — 04/26

Saddle points. Kink solutions and their proliferation in one dimension.
4. Landau-Ginzburg Theory — 04/25

Local order parameters. Coarse-graining. The Landau-Ginzburg functional.
3. Landau’s Theory of Phase Transitions — 04/20

Continuous and first-order phase transitions. Critical exponents.
2. Mean Field Theory, Again — 04/19

Global order parameters. Landau free energy and its critical points.
1. Ising Model and Mean Field Theory — 04/18

Ising model and its solution via mean field theory.

Quantum Field Theory I (September – December ‘23)

Between September–December 2023, I taught the first of a two-semester sequence of courses on quantum field theory. We discussed the canonical and path integral quantisation of scalars, spinors, and Abelian gauge fields, building up to a study of elementary processes in quantum electrodynamics.

Assignments

Lectures

53. Soft Photons — 12/22b

A proof of the Weinberg’s soft photon theorem, charge conservation.
52. Klein-Nishina and Thomson — 12/22a

The Klein-Nishina formula, recovering the Thomson scattering cross-section.
51. Compton Scattering — 12/21

The kinematics for Compton scattering.
50. Photon Polarisation Sums — 12/20

Towards differential cross-sections for Compton scattering, summing over polarisations.
49. Elementary Processes w/ Spin — 12/15b

Differential cross-sections for electron-muon and electron-proton scattering, crossing relations, the Mott formula and recovering Rutherford scattering.
48. Unpolarised Scattering — 12/15a

Using trace identities to do spin sums.
47. Signs and Traces — 12/14

Tracking factors of (-1) due to statistics, and computing traces of products of Dirac matrices.
46. Light-Matter Interactions — 12/13

The Feynman rules for quantum electrodynamics.
45. Gauge Invariance — 12/08b

A sketch of a diagrammatic proof of gauge invariance.
44. Elementary Processes w/o Spin — 12/08a

Elementary processes in scalar electrodynamics.
43. Scalar Electrodynamics — 12/07

The Lagrangian for scalar electrodynamics, Feynman rules.
42. Where Do Feynman Rules Come From? — 12/06

How to derive Feynman rules, symmetry factors, derivative couplings.
41. Amplitudes for Fermions — 12/01b

More scattering amplitudes in theories with interacting scalars and fermions.
40. Yukawa Theory — 12/01a

Feynman rules for theories with Yukawa couplings.
39. Green’s Functions — 11/30

Computing time-ordered products of operators in the interacting vacuum, connected diagrams, vacuum bubbles.
38. Cross Sections and Decay Rates — 11/29

Interpreting the “square of a delta function” as the volume of spacetime, relating scattering amplitudes to cross sections and decay rates.
37. Amplitudes for Scalars — 11/24b

More scattering amplitudes in scalar theories.
36. Feynman Rules — 11/24a

Reproducing the results using Feynman diagrams.
35. Scattering Amplitudes — 11/23

Computing scattering amplitudes in cubic scalar theories using Wick’s theorem.
34. The S-Matrix and Decay Amplitudes — 11/22

Asymptotic states, the S-matrix, the amplitude for decay in cubic scalar theory.
33. Interactions: Dyson’s Formula — 11/09

The interaction picture, Dyson’s formula and time-ordered exponentials.
32. Path Integrals for Photons and Introducing Interactions — 11/08

The Feynman propagator for free Maxwell theory and its derivation via the path integral for Abelian gauge fields, covariant gauges. Dimensional analysis, relevant, irrelevant, and marginal perturbations.
31. Faddeev-Popov — 11/03b

Gauge fixing in the path integral for Abelian gauge field.
30. Canonical Quantisation of Maxwell Theory — 11/03a

Gauge symmetry and the Gauss law constraint, canonical quantisation and the Gupta-Bleuler condition.
29. Free Maxwell Theory and Gauge Symmetry — 11/02

Free Maxwell theory, equations of motion and solution, polarisation vectors, gauge symmetry, different choices of gauge and their properties.
28. Path Integrals for Fermions — 10/27b

Grassmann numbers, finite-dimensional Gaussian integrals using Gaussians, generalisation to field theory, generating functional for free massive Dirac fermions and recovering the Feynman propagator.
27. Dirac Propagator — 10/27a

Feynman propagator for Dirac fields, time ordering for fermionic fields.
26. Canonical Quantisation of Dirac Fields — 10/26

Canonical anticommutation relations for fermionic fields, the Fock space, multiparticle states and Fermi-Dirac statistics, the commutator of Dirac fields, causality,
25. Spin-Statistics Theorem — 10/25

What goes wrong when you impose commutation relations on fermionic oscillators? Anticommutation relations and Fermi-Dirac statistics, issues of stability.
24. Inner and Outer Products — 10/20b

Inner products between plane wave solutions, spin sums, inner and outer products.
23. Plane Wave Solutions to Dirac Equation — 10/20a

Solving the free Dirac equation, particles and antiparticles, spin and helicity.
22. Weyl and Majorana Spinors — 10/19

Reducibility of the Dirac spinor, Weyl spinors, parity, complex conjugation, Majorana spinors.
21. The Dirac Lagrangian — 10/18

The Dirac adjoint, constructing Lorentz covariant field bilinears, the Dirac equation, vector and axial vector currents.
20. Dirac Spinors — 10/13b

Introducing the Dirac spinor, its transformation properties under rotations and boosts.
19. Spinor Representations — 10/13a

Using the Clifford algebra to construct (spinor) representations of the Lorentz group.
18. Tensor Representations — 10/12

Fields as representations, transformation rules for scalars, vectors, and tensors.
17. Groups and Algebras: Lorentz and Poincaré— 10/11

The Lorentz and Poincaré groups and their subgroups, the algebra of generators, parity and time-reversal, global structure of the Lorentz group, factorisation of the complexified algebra into two copies of su(2).
16. Green’s Functions— 10/06b

Generating functionals, computing the n-point Green’s function in the free field theory, Wick’s theorem and diagrammatics, recovering the pole prescription.
15. Path Integrals in Field Theory — 10/06a

The path integral for scalar fields, vacuum persistence amplitudes, time-ordering, and recovering the Feynman propagator.
14. Functional Determinants II: Gel’fand-Yaglom — 10/05

How to compute functional determinants using a method due to Gel’fand and Yaglom.
13. Functional Determinants I: Gaussian Fluctuations— 10/04

Small perturbations around classical trajectories, writing the Gaussian fluctuations about a saddle point as a determinant, analytic continuation to imaginary time and relation to partition function revisited.
12. Path Integrals in Quantum Mechanics— 09/29b

Time-slicing, propagator in quantum mechanics as a path integral, recovering the principle of least action in the classical limit.
11. Complex Scalar Field — 09/29a

Conserved charges in quantum theories, mode expansion for free complex scalar field, the charge operator, 1-particle states organised as representations of the global U(1) symmetry.
10. Propagators — 09/27

Definition of propagator, relation between propagator and commutator, time ordering, Feynman propagator, momentum-space representation, pole prescriptions, Green’s functions.
9. Lorentz Invariance and Causality — 09/22b

Lorentz-invariant measures, the relativistic normalisation of states, commutator of field operators vanishing outside the lightcone and implications for causality.
8. Zero-Point Energy and Fock Space — 09/22a

The Hamiltonian in terms of mode operators, UV and IR divergences in zero-point energy, normal-ordered operators, Fock space and multiparticle states, bosonic statistics.
7. Mode Expansions — 09/21

Mode expansions and commutation relations.
6. Rules for Canonical Quantisation — 09/20

Review of canonical quantisation, free scalar fields again, equal-time commutation relations, Schrodinger and Heisenberg pictures, Klein-Gordon equation for the free field operator.
5. How To Gauge a Global Symmetry — 09/15b

Global and local symmetry transformations, covariant derivatives, gauge potentials and gauge symmetries, the field strength associated to a gauge field, the QED Lagrangian.
4. Noether’s Theorem II: Spacetime Symmetries — 09/15a

Noether’s theorem applied to spacetime symmetries, translations and the energy-momentum tensor, rotations and angular momentum.
3. Noether’s Theorem I: Internal Symmetries — 09/14

Noether’s theorem on the existence of locally conserved currents associated to continuous global symmetries, internal symmetries, complex scalars and U(1) global symmetry, symmetry groups and their representations on fields, the O(N) model.
2. Field Theory and Statistical Mechanics — 09/11

The Ising model, computing its partition function as a sum over functions of the average magnetisation of “blocks” of spins, the Landau-Ginzburg free energy functional, equivalence with the Hamiltonian of a scalar field in Minkowski spacetime, dominant contributions to the path integral as saddle points, Wick rotation (formally) relating QFTs and statistical mechanical systems.
1. Classical Scalar Fields — 09/04

The principle of least action for finite-dimensional dynamical systems, extension to scalar field theories, the (Euler-Lagrange) field equations for a scalar field, recovering Klein-Gordon as a field equation, plane-wave solutions and dispersion relations, classical field theory in phase space, the Hamiltonian for scalar fields.

Quantum Mechanics II (April – July ‘23)

Assignments

Notes

See here for notes discussing Wigner matrices, Clebsch-Gordan coefficients, the addition of angular momentum, spherical tensors, and the Wigner-Eckart theorem.

Bosonic String Theory (January – April ‘23)

A series of ten lectures on bosonic string theory delivered to an audience of PhD scholars and faculty working in particle theory and cosmology.

Lectures

Free Point Particles Revisited
The Nambu-Goto and Polyakov Actions
Gauge Fixing, Mode Expansions, and Issues with Quantisation
Lightcone Quantisation and the Closed String Spectrum
Open Strings and D-Branes
Background Fields I: Einstein’s Equations
Background Fields II: Frames and Solutions
Brane Dynamics and Tension
T-Duality I: Kaluza-Klein Compactifications
T-Duality II: Open Strings and D-Branes

Notes

See here.

Madhusudhan Raman

Ashoka University

Quantitative Reasoning and Mathematical Thinking (January – April ‘26)

Take-Home Assignment

Assignments

Lectures

Suggested Reading

Mathematical Physics I (January – April ‘26)

Take-Home Assignment

Assignments

Coding Challenges

Lectures

Suggested Reading

University of Delhi

Strategies for Scientific Dialogue in Research (August – December ‘25)

Suggested Reading

Quantum Field Theory III (January – April ‘25)

Lectures

Suggested Reading

Quantum Field Theory II (January – April ‘25)

Assignments

Lectures

Suggested Reading

Quantum Field Theory I (August – November ‘24)

Assignments

Lectures

Suggested Reading

Quantum Field Theory II (February – May ‘24)

Assignments

Lectures

Suggested Reading

Critical Phenomena (February – May ‘24)

Assignments

Lectures

Suggested Reading

Quantum Field Theory I (September – December ‘23)

Assignments

Lectures

Suggested Reading

Quantum Mechanics II (April – July ‘23)

Assignments

Notes

Bosonic String Theory (January – April ‘23)

Lectures

Notes