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This course introduces mathematical and numerical tools that are useful for various fields of theoretical physics. In the numerical part of the course, students will develop introductory programming skills in both Python and Mathematica. The mathematical part of the course covers three main topics: First, we will study basics of smooth manifolds, define forms on them, and then de Rham cohomology. Second, we will introduce homology theory, and explain in what sense it can be understood as dual to cohomology. Lastly, we will study homotopy theory. Throughout the course, we will emphasize many applications to physics.

The aim of this course is to revisit much of the undergraduate classical physics curriculum in preparation to the study of both General Relativity and Quantum Field Theory. We will focus on the notion of symmetry and Noether's theorem in both mechanical and field theoretical systems (stress energy tensor). We will discuss the Lagrangian, or "action principle", and Hamiltonian, or "phase space", perspectives on mechanics and their mutual relationship. We will then discuss selected topics in special relativity and electrodynamics, culminating with the derivation of the Lienard-Wiechart potentials.

The aim of the first part is to present a brief overview of selected topics in quantum theory. Schrodinger, Heisenberg and Interaction picture is discussed and applied to study time evolution. Density matrices and Feynman path integrals are introduced. The second part of the course derives the Feynman rules for scalar quantum field theory and introduces renormalization.

Structure:

We will discuss 8 papers which had huge impact in physics. One week Instructor Pedro Vieira will discuss a paper; students should read it beforehand. One week later students discuss recent papers referring to that paper (20 min each student, ~ 3 presentations; at the end of the class Pedro will grade the presentations based on “Physics”, “Presentation”, “Question handling”; and give comments).

By the end of the course, students will have explored a vast set of topics in theoretical physics — spotting potential gaps to be fixed — sharpened their presentation skills through steady practice, and sparked cross-disciplinary conversations through our shared physics language.

The papers: The papers we will cover (can find the pdfs in this spreadsheet):

Sept 1 & 8: On the Number of Prime Numbers less than a Given Quantity, Riemann, 1859

Sept 8 & 21: Limits on Massless Particles, Weinberg and Witten, 1980

Sept 22 & 28: Neural networks and physical systems with emergent collective computational abilities, Hopfield, 1982

Sept 29 & Oct 8: Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Laughlin, 1983

Oct 9 & 19: Why there is nothing rather than something, Coleman, 1988

Oct 20 & 26: Universality and Scaling in Gravitational Collapse of a Massless Scalar Field, Choptuik, 1993

Oct 27 & Nov 2: Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Mirzakhani, 2007

Nov 3 & 9: Supersymmetry and Morse Theory, Witten, 1982

Location & Building Access: Please refer to the above spreadsheet for class location as it will vary from week to week. Perimeter Institute, 31 Caroline St N, Waterloo Participants who do not have an access card for Perimeter Institute must sign in at the security desk before each session. For information on parking or accessibility please contact [email protected].

Non-local quantum computation (NLQC) is a subject within quantum information theory. NLQC considers, in a certain setting, with how local interactions can be simulated with distributed entanglement plus communication. NLQC has recently become well connected to several other areas, including communication complexity, cryptography, AdS/CFT, and computational complexity theory. This course will focus on learning the basics of NLQC, and then on understanding its applications in these other areas.

This course will cover the basics from my book, https://arxiv.org/abs/2603.12076. It is about operational probabilistic theories. The standard approach in such theories is, implicitly, from a time forward perspective. On the other hand, we will mostly take a time symmetric perspective. The course will consists of two parts: (1) a "simple part" about simple operations having simple causal structure (where all the inputs are before all the outputs); and (2) a "complex part" about complex operations that can have complicated causal structure (a complex operation comes equipped with a causal diagram). For the simple case we are able to show that the time symmetric perspective is equivalent to the time forward perspective. In each of these two parts we set up (A) operational probabilistic theories (OPTs) in terms of operations, (B) Operational Quantum Theory (OQT) in terms of operator tensors which correspond to operations, and (C) the theory of Hilbert objects which can be doubled up to give operator tensors. Operations are required to be physical. Physicality guarantees that circuits built out of operations have probabilities between 0 and 1 and that certain causality conditions are met. We prove composition theorems for both simple and complex operations -- that when we wire together operations the resulting networks are also physical (these theorems are especially interesting in the case of complex operations).The theory of complex operations can be used to model physics happening in (discrete) spacetime. We use this to address Sorkin's impossible measurements. It turns out that if the operations are physical then there is no anomalous signaling. We develop new diagrammatic notation to deal with Hilbert objects, particularly in the complex case. We discuss the conjuposition group of transformations on Hilbert objects. This includes mirrors to notate doubling up and some mirror theorems. We use this framework to prove time symmetric causal dilation theorems for a variety of causal diagrams.

Virtual Participation Link: https://pitp.zoom.us/j/93634737051?pwd=bJkB6HrVbOsrpCFInt76DNVlx7lwiS.1.

Location & Building Access: Tue, 11.00-12.30, Sky Room Wed, 11.00-12.30, Alice Room

Participants who do not have an access card for Perimeter Institute must sign in at the security desk before each session. For information on parking or accessibility please contact [email protected].

How do relativistic effects influence quantum information processing? This fundamental question has developed over the past decade into the new active field of Relativistic Quantum Information. It brings together concepts and ideas from special relativity, quantum optics, general relativity, quantum communication, and quantum computation. Its aims are to understand the relationship between relativistic physics and quantum information, to harness them for new techniques in quantum information processing and to better comprehend the foundations of relativistic quantum physics.

We will study how General Relativity (GR) is similar to and especially how it differs from other gauge theories. This will explain why, from a structural perspective, it is much harder to quantize GR than other theories without relying on any specific approach to quantization. To achieve this goal, we will introduce the so-called “Covariant Phase Space Method” and use to study in detail the symmetry structure of GR and how it is intimately related to its dynamics. Along the way we will touch on (parts of) the historical debate on whether gravity should be quantized at all, discuss how to think of time evolution when there is no absolute time, and go through Wald’s proposal of black hole entropy as a Noether charge.