## Curriculum

The coursework is divided into four phases and a short research project, the essay. The full course will advance:

- Research Skills: problem formulation and solving, presentation skills and necessary background in numerical methods and mathematics (Four weeks, full time).
- Core Topics: foundational subjects, such as quantum mechanics, relativity, field theory, statistical physics, dynamical systems, data analysis, and scientific computation. (Three three-week sessions, each with two courses running in parallel).
- Reviews: subdisciplinary subjects, such as particle physics, cosmology, quantum information, quantum foundations and condensed matter physics. Students will choose from a selection of courses (Three three-week sessions, each with three courses running in parallel. Students are required to take at least four review courses in order to meet course requirements).
- Explorations: short, in-depth courses on specialized fields which are currently "hot." Students will choose from a selection of courses (Three three-week sessions, each with three courses running in parallel. Students are required to take at least two exploration courses in order to meet course requirements).

Essay: Each student undertakes a short research project supervised by a local or outside Faculty member, and produces an essay which is publicly presented and defended.

The program also includes remedial English courses, training in scientific writing, and presentation workshops.

## ASSESSMENT

Although all course grades are either "credit" or "no credit," PSI's approach to evaluation involves assessment throughout the year conducted by the PSI Fellows. This assessment is in the form of oral interviews (for core courses), homework assignments, and tutorial participation. The goal is to encourage all students to achieve their potential and to avoid grade-chasing competition.

## FRONT-END COURSES

The Front-End begins the second week of August and runs for four weeks. The Front-End courses are short courses intended to provide mathematics and physics background useful for the rest of the PSI year. Recent courses and topics include:

**Complex Analysis**

Complex numbers, holomorphic functions, analyticity, Riemann surfaces and applications of Cauchy’s residue theorem.

**Special Topics in Quantum Theory**

Density matrices, non-relativistic path integral, non-relativistic Feynman diagrams.

**Lie Groups and Lie Algebras**

Definitions and examples of groups, Lie groups and Lie algebras, representation theory, SU(2) irreducible representations, SU(N) and Young tableaux, classification of semi-simple Lie algebras.

**Theoretical Mechanics**

Introduction to differential and symplectic geometry and its applications to mechanics, principle of least action, Noether’s theorem, Poisson brackets, canonical transformations, Liouville’s theorem, integrable systems, constraints, Hamilton-Jacobi theory.

**Functions, “Functions,” etc.**

Distributions, Green’s functions, asymptotic series, saddle-point approximation and Stokes’ phenomenon, summation of divergent series.

**Introduction to Mathematical Computing**

Mathematica: plots, movies, advanced features, numerical solution of partial differential equations; Julia: linear algebra, numerical methods.

## CORE COURSES

September to December. The core courses cover foundational graduate-level subjects, and each course is three weeks long. Students take all six core courses.

*Sample Core Course Schedule(Click to enlarge)*

Topics covered in previous years include:

**Quantum Theory**

Operational approach to quantum theory, postulates of quantum theory, generalized preparations (density operators), generalized evolutions (CPTP maps, Krauss operators), generalized measurement (POVMs, weak measurements), foundational origins of classical and quantum mechanics.

**Relativity**

Special relativity, foundations of general relativity, Riemannian geometry, Einstein's equations, black holes, gravitational waves.

**Statistical Mechanics**

Review of thermodynamics, statistical ensembles, Gibbs distribution, Ising model, phase transitions, critical exponents, mean field theory, Wilsonian renormalization, systems with continuous symmetry, vortices, Monte Carlo methods for studying phase transitions.

**Quantum Field Theory I**

Canonical quantization of scalar, spinor, and electromagnetic fields, Feynman diagrams, applications to particle physics, quantum electrodynamics.

**Quantum Field Theory II**

Path integral quantization, renormalization group, quantization of non-abelian gauge theories, gauge fixing and ghosts, spontaneous symmetry breaking.

**Condensed Matter**

Crystal lattices, reciprocal lattices, band structure, phonons, quantum Hall effect, superconductivity and BCS theory.

## REVIEW COURSES

January to March. Review courses introduce students to modern topics and research subfields. Students take at least four out of nine courses. Each course is three weeks. Recent courses and topics include:

**Standard Model**

Higgs Mechanism and spontaneous symmetry breaking, Yukawa interaction, deep inelastic scattering, Higgs decay and production in LHC, hadron spectroscopy, flavour physics, open questions.

**Cosmology**

Homogenous and isotropic Universe, FRW metric, thermodynamics in expanding Universe, big bang nucleosynthesis, cosmic microwave background, dark matter and dark energy, inflation.

**Quantum Foundations**

Operational and realistic approaches to the interpretation of quantum mechanics, local realism and the EPR argument, Bell's theorem and non-locality, contextuality and the Kochen-Specker theorem, deBroglie-Bohm interpretation, many worlds interpretation.

**Gravitational Physics Review**

Differential forms, Cartan's structure equations, black holes in four and higher dimensions, cosmic strings and domain walls, Einstein-Hilbert action, Gibbons-Hawking term, black hole thermodynamics, hypersurfaces, Gauss-Codazzi formalism, Kaluza-Klein theory, Gregory-Laflamme instability, gravitational instantons, Randall-Sundrum model.

**Quantum Information Review**

Qubits, quantum gates, quantum circuits, entanglement, quantum algorithms and complexity, information theory and implementations, quantum error correction, quantum cryptography, quantum information theory.

**String Theory**

Nambu-Goto action, Polyakov action, the bosonic string spectrum, BRST quantization and the Hilbert space, tree amplitudes, effective action, D-branes and T-duality, Type II superstring spectrum.

**Quantum Field Theory III**

Conformal field theory in D dimensions, the conformal anomaly, conformal bootstrap, conformal field theory in D=2, Virasoro algebra, unitary minimal models, anomalies, instantons.

**Condensed Matter Theory**

Quantum phase transitions, entanglement and entanglement entropy, Lieb-Robinson bounds, lattice gauge theories, topological order.

**Beyond the Standard Model**

Evidence for physics beyond the Standard Model: neutrinos, baryogenesis, dark matter, scale hierarchies, electroweak precision experiments; BSM Physics: supersymmetry, technicolor, extra-dimensions.

## EXPLORATIONS COURSES

March to April. Explorations courses cover cutting-edge research topics from various specialized subfields. Students take at least two out of five or six courses offered. Courses are three weeks long. Recent courses and topics include:

**Explorations of Quantum Integrable Models**

Topics in 1+1 dimensional theories: spin chains, algebraic Bethe ansatz, Yang-Baxter equation, anti-ferromagnetic vacuum, spinons, magnons; R-matrices in 4d topological field theories, Yangian algebra, RTT relations, Wilson lines, Thirring model, defect and beta functions.

**Explorations in Relativistic Quantum Information**

Introduction to relativistic quantum information, phenomenology of particle detectors and the relativistic light-matter interaction, entanglement in quantum field theory, entanglement harvesting and farming. relativistic quantum communication, relativistic quantum information view of the Unruh effect and Hawking radiation.

**Explorations in Cosmology**

Cosmological perturbation theory, inflationary perturbations, ADM formalism, quantum field theory in curved spacetime, Fisher matrix for cosmic microwave background.

**Explorations in Quantum Gravity**

Actions for gravity and first order formulation, canonical formulations of constrained systems and gauge symmetries, Dirac program, Ashtekar-Barbero variables, loop quantum gravity, quantum geometries, path integral formulation and spinfoam models, the Ponzano-Regge model.

**Explorations in Machine Learning for Many-Body Physics**

Supervised learning (feedforward neural networks, convolutional neural networks) and unsupervised learning (data visualization and clustering, generative modelling) with applications to classical and quantum many-body physics.