# Ten-week online school

Perimeter Institute will host a 10-week online school from **May 21 to July 25, 2024**. Participants will learn about cutting-edge topics in modern theoretical physics through four courses, group projects on specialized topics, and other special events such as presentations from researchers and panel discussions. The school will involve 4.5 hours of synchronous content and 1-2 hours of independent work or study each week.

Courses for the 2024 online school are tentatively:

- Path integrals
- Symmetries
- Topics in quantum science
- Numerical methods

In order to accommodate students from different time zones, the synchronous online content will be offered on Tuesdays and Thursdays at two different times each day: 9:00 am to 11:15 am, and 3:15 pm to 5:30 pm (Eastern Time). Be sure to check these times in your own time zone using the links above, because the collaborative nature of the online school relies on your active participation in the synchronous activities.

Students will indicate their time preferences in the online application but the preference can be changed once the school starts if necessary.

### Group mini-projects

During two of the ten weeks of the school, participants will have the opportunity to work in small groups on a mini-project related to the content of courses. The format of the mini-project could range from learning about a more advanced topic as a group to a more hands-on research project to which all students contribute. Mini-project assignments will take into account students' preferences.

Participation in this aspect is reserved for the online school students who are not part of the research internship program.

Interactions with the project supervisor will generally happen during the regular school time slots, although some projects may have more flexible times to accommodate the supervisor’s schedule.

### Events

The online school will also include a series of organized events, including but not limited to:

- An ice-breaking event to start off the school. This will be an informal way for you to meet the other students and some of the instructors, play some games, and get to know each other.
- A keynote presentation (speaker to be announced)
- An information session about the PSI master’s program
- A closing event including group presentations on their mini-project

## Sample course outlines

Note: The sample course outlines below are from previous versions of the program and do not necessarily reflect the topics that will be covered in the 2023 PSI Start program.

**Course summary**

This course is designed to introduce students to fundamental programming skills for use within theoretical physics. The course will also discuss selected research areas that rely on numerical methods. Students will become familiar with the programming language Python.

**Instructor information**

**Learning outcomes**

By the end of this course, students should be able to:

- Write code in Python using if statements, for loops, while loops, and functions
- Import and use Python libraries such as NumPy
- Describe Monte Carlo methods
- Write code to make plots and perform curve fitting
- Describe machine learning, its general categories, and the architecture of a neural network

**References**

- Python Software Foundation, "The Python Tutorial" (Python 2 version, Python 3 version)
- Python Software Foundation, "The Python Standard Library" (Python 2 version, Python 3 version)
- The SciPy community, "NumPy User Guide"
- The Matplotlib development team, "matplotlib.pyplot: Pyplot function overview"
- Peter Young, “Everything you wanted to know about Data Analysis and Fitting but were afraid to ask”, arXiv:1210.3781

**Outline of topics**

- Introduction to Programming in Python
- Monte Carlo methods
- Data visualization and curve fitting
- Machine learning

#### Course description

The goal of this course is to introduce the path integral formulation of quantum mechanics. We will derive the path integral representation of the propagator. Perturbation theory in the path integral formulation will be developed. Path integrals will be used to compute tunneling rates and understand particle statistics.

Each session will include roughly equal amounts of lecture time and activities. The activities are designed to enhance your learning experience and allow you to assess your own level of understanding.

**Instructor information**

**Learning outcomes**

By the end of this course students should be able to:

- Prove that the path integral expression for the propagator is equivalent to the bra-ket expression
- Use the semi-classical approximation to perturbatively evaluate path integrals
- Use the instanton method to calculate decay rates
- Explain why anyons exist only in two dimensions

**Resources**

Recordings of the lectures will be available to PSI Start participants, so you may re-watch them; these are not a substitute for attending the live course sessions on Zoom.

Lecture notes will also be posted before each course session, but they may not agree perfectly with the actual course content .

#### Tentative course schedule

Lecture 1 | Introduction to path integrals |

Lecture 2 | Path integral expression for the propagator |

Lecture 3 | Classical limit and perturbation theory |

Lecture 4 | Imaginary time and statistical physics |

Lecture 5 | Instantons and tunneling |

Lecture 6 |
Path integrals and particle statistics |

Lecture 7 | Path integrals and relativity |

#### Course summary

The goal of this course is to introduce foundational ideas in quantum information theory, including entanglement, the Schmidt decomposition, mixed quantum states, and density matrices.

If time permits, we will also cover some applications, possibly including quantum teleportation and quantum algorithms. The intended course schedule is below, though we may go more slowly and drop some topics from the final day if needed.

#### Instructor information

Aaron Szasz

#### Learning goals

By the end of this course students should:

- Understand tensor product Hilbert spaces
- Understand the meaning of entanglement
- Be familiar with the Schmidt decomposition, and know how to compute it using software such as Python or Mathematica
- Understand the idea of mixed quantum states, and their formulation as density matrices
- Be able to compute reduced density matrices
- (Time permitting) understand some simple applications of entanglement, such as teleportation and small quantum circuits

#### Tentative course schedule

Lecture 1 | Review two-level systems, including change of basis; tensor product spaces |

Lecture 2 | Schmidt decomposition; several perspectives on entanglement |

Lecture 3 | Mixed quantum states; density matrices and reduced density matrices |

Lecture 4 | Quantum teleportation; SWAP test algorithm; Grover’s algorithm |

#### Course description

The aim of this course is to explore some of the many ways in which symmetries play a role in physics. We’ll start with an overview of the concept of symmetries and their description in the language of group theory. We will then discuss continuous symmetries and infinitesimal symmetries, their fundamental role in Noether’s theorem, and their formalisation in terms of Lie groups and Lie algebras. In the last part of the course) we will focus on symmetries in quantum theory.

Each session will include roughly equal amounts of lecture time and activities. The activities are designed to enhance your learning experience and allow you to assess your own level of understanding.

#### Instructor information

#### Learning outcomes

By the end of this course students should be able to:

- Evaluate the symmetries of an action functional and construct the associated Noether’s conserved quantities
- Invert the Noether's theorem process to find symmetries starting from conserved quantities
- Construct the Lie algebras for the classical Lie groups and specify their structure constants
- Justify the appearance of spin in quantum mechanics from the point of view of representation theory

#### Resources

These are some of the resources that we will use during the lectures/activities:

- Socrative and Slido. These are online apps that we will use for activities. They don’t need to be installed (they can run from a browser) and don’t require an account.
- GeoGebra. I will use this to show you interactive simulations or visualisations during the lectures, and I’ll then share the applets I create with you. You can either use the online version or install it, which I recommend since it’s a very useful piece of software. It’s open source and cross platform.

#### Tentative course schedule

Lecture 1 | Overview/definition of symmetry, elements of group theory, examples of applications of symmetries in physical problems |

Lecture 2 | Continuous and discrete symmetries, infinitesimal symmetries, Noether's theorem |

Lecture 3 | Noether’s theorem (continued), Lie groups and Lie algebras |

Lecture 4 | Symmetries in quantum mechanics |

## Main Navigation

*The PSI Start program is supported by Michael Serbinis and Laura Adams. *