The Perimeter Scholars International (PSI) program starts in September and runs for 10 months to June of the following calendar year. Successful graduates of PSI will receive both a master's degree in physics from the University of Waterloo and a Perimeter Scholars International Certificate from the Perimeter Institute for Theoretical Physics. All courses take place at Perimeter Institute.

### Curriculum

The coursework is divided into two overlapping phases and a short research project in the form of an essay.

- Core Topics: foundational subjects — classical physics. quantum theory, relativity, quantum field theory, statistical physics, mathematical physics, numerical methods. (Four five-week sessions, each with two courses running in parallel).
- Elective Courses: subdisciplinary subjects — such as particle physics, cosmology, quantum information, quantum foundations, and quantum matter physics — and courses on specialized fields which are currently "hot", such as Machine Learning for Many-Body Physics and AdS/CFT. Students are required to take at least six elective courses.

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 English courses (if needed), training in science communication, and presentation workshops. As an example, the 2022/2023 schedule is below but it is subject to change each year.

### Assessment

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

## PSI courses

September to January. The core courses cover foundational graduate-level subjects, and each course is five weeks long. Students take 7 out of the 8 core courses. The following are the descriptions of recent iterations of the core courses. Note that the courses and their topics are subject to change and may not fully reflect the content of the upcoming year.

**Classical Physics**

This course aims to review the basics of theoretical mechanics, special relativity, and classical field theory, with the emphasis on geometrical notions and relativistic formalism.

**Quantum Theory**

This course on quantum mechanics is divided in two parts:

The aim of the first part is to review the basis of quantum mechanics. The course aims to provide an overview of the perturbation theory to handle perturbations in quantum systems. Time evolution of quantum systems using the Schrodinger, Heisenberg and interaction pictures will be covered. Basics of quantum statistical mechanics for distinguishable particles, bosons, and fermions will be covered. A brief overview of the density matrix approach and quantum systems interacting with the environment will be given.

The second part of the course is an introduction to scalar quantum field theory. The Feynman diagram technique for perturbation theory is developed and applied to the scattering of relativistic particles. Renormalization is briefly discussed.

**Quantum Field Theory I**

The course starts by looking for a quantum theory that is compatible with special relativity, without assuming fields are fundamental. Nevertheless fields turn out to be a very good, maybe inevitable mathematical tool for formulating and studying such a relativistic quantum theory. The second part of the course introduces the Dirac theory and canonically quantizes it. It also quantizes the Maxwell field theory. The Feynman diagram technique for perturbation theory is developed and applied to the scattering of relativistic fermions and photons. Renormalization of quantum electrodynamics is done to one-loop order.

**Relativity**

This is an introductory course on general relativity (GR). We shall cover the basics of differential geometry and its applications to Einstein’s theory of gravity. The plan is to discuss black holes, gravitational waves, and observational evidence for GR, as well as to cover some of the more advanced topics.

**Quantum Field Theory II**

The course has three parts. In the first part of the course, the path integral formulation of non-relativistic quantum mechanics and the functional integral formulation of quantum field theory are developed. The second part of the course covers renormalization and the renormalization group. Finally, non-abelian gauge theories are quantized using functional integral techniques.

**Statistical physics**

The course begins by discussing several topics in equilibrium statistical physics including phase transitions and the renormalization group. The second part of the course covers non-equilibrium statistical physics including kinetics of aggregation, spin dynamics, population dynamics, and complex networks.

January to April. Elective courses introduce students to modern topics and cover cutting-edge research topics from various specialized subfields. Students take at least six courses. The total number of elective courses changes each year, and is generally between twelve and fifteen. Each course is five weeks long. The following are the descriptions of elective courses from recent years. Note that the courses and their topics are subject to change and may not fully reflect the content of the upcoming year.

Recent courses and topics include:

**Condensed Matter**

Spontaneous symmetry breaking of continuous symmetries. Goldstone modes. Nonlinear sigma model. 2+epsilon and large-N expansion. 2D melting. Hexatic phase. Quantum spin chains. Bosonization. Luttinger liquid. Fermi liquid. Cooper instability. Shankar-Polchinski RG. BCS superconductivity.

**Quantum Foundations**

The problem with the "textbook interpretation" of quantum mechanics. Operationalism vs. realism. The quantum measurement problem and a classification of research programs in terms of their proposed solution to it. Formulating quantum theory as an operational theory: density operators, positive operator-valued measures, completely positive trace-preserving maps, purification, Naimark extension, Stinespring dilation. Operational axiomatizations of quantum theory. Hidden variable models and the distinction between psi-ontic and psi-epistemic models. Hidden variable models based on an epistemic restriction. Bell’s theorem and locality. The Kochen-Specker theorem and noncontextuality. The deBroglie-Bohm interpretation. Dynamical Collapse Theories. The Everett Interpretation.

**Gravitational Physics**

The main objective of this course is to discuss some advanced topics in gravitational physics and its applications to high energy physics. Necessary mathematical tools will be introduced on the way.

**Quantum Information**

Formulating quantum theory as an operational theory. Reversible computation. Quantum gates. Complexity. Algorithms. Error correction. Cryptography and information theory.

**Cosmology**

FRW universe. Dark energy. Cosmic Microwave Background. Big Bang Nucleosynthesis. Dark Matter. LCDM cosmology. Inflation. Primordial perturbations. QFT in curved space background.

**Machine Learning for Many Body Physics**

This course is designed to introduce machine learning techniques for studying classical and quantum many-body problems encountered in quantum matter, quantum information, and related fields of physics. Lectures will emphasize relationships between statistical physics and machine learning. Tutorials and homework assignments will focus on developing programming skills for machine learning using Python. Students will be introduced to topics such as training techniques for supervised learning, feedforward neural networks, convolutional neural networks, unsupervised learning for data visualization and clustering, and research frontiers at the intersection of physics and machine learning. generative modelling.

**Quantum Fields and Strings**

Conformal field theory. Anomalies. Introduction to string theory.

**Relativistic Quantum Information (RQI)**

What is RQI? Research topics in RQI. Measuring quantum fields; The measurement problem and relativity. What is a particle detector? The light-matter interaction form first principles. Unruh-DeWitt detectors; practical quantum information on QFTs. Relativity, covariance and quantum optics; A critical view on the single-mode and rotating wave approximations. Measuring vacuum fluctuations. Non-inertial particle detectors and qubits in curved spacetimes. The Unruh effect: Original derivation with quantum information perspective. Thermality. Gibbs vs KMS. When Gibbs is not enough: The KMS condition in QFT. Thermalization of accelerated particle detectors: The Unruh effect, modern derivation. Entanglement structure of a QFT. Introduction to entanglement harvesting from the vacuum. Quantum Energy Teleportation and violation of energy conditions.

**Chern-Simons**

Review of differential forms and a very brief discussion of cohomology. Review of gauge fields, gauge invariance, field strength, and Wilson lines. The Chern-Simons Lagrangian. Gauge invariance and quantization of the level. Wilson lines and their topological invariance. Knot invariants from Wilson lines. Jones polynomial, and the relation between Wilson lines and the Jones polynomial. The phase space and the Hilbert space on a Riemann surface. Braid group actions on the Hilbert space in the presence of Wilson lines, and the Knizhnik-Zamolodchikov equation.

**Quantum Gravity**

Canonical formulation of constrained systems. The Dirac program. First order formalism of gravity. Loop Quantum Gravity. Spinfoam models. Discussions with invited speakers on: Space and Time in Quantum Gravity, Loop Quantum Cosmology, Black Holes, Other approaches of Quantum Gravity, Numerical methods in Spinfoam models, Research at PI.

**Standard Model**

Standard model particle content and Lagrangian with application of Yang-Mills theory. Spontaneous symmetry breaking Standard model effective theory. Right handed neutrino. Application to particle physics: Feynman rules for the standard model.

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*The PSI program is supported by the Savvas Chamberlain Family Foundation, the Hellenic Heritage Foundation, the Marsland Family, and members of the Emmy Noether Circle.*