This is the webpage for the Michaelmas 2019 lecture course on "Advanced Quantum Theory". This counts as part of both the C6 (Theory) MPhys option, as well as the Advanced Quantum Theory course within the Oxford Masters Course in Mathematical and Theoretical Physics (MMathPhys). This course will focus on path integral techniques and many-particle quantum mechanics, and provide an introduction to Landau's theory of phases and phase transitions. This course thus serves as a broad overview of the basic tools and concepts of theoretical physics.

The course comprises a total of 26 hours of lectures, in Weeks 1-8. Of these, roughly hours 1-20 count for AQT; hours 21-26 will cover stochastic processes, which is an examinable part of the C6 course. Lectures last for an hour on Mondays but two on Thursdays lectures for two hours. There will be a 5 minute break roughly mid-way though Thursday lectures; you are encouraged to use this time to ask questions, take a short walk outside the lecture theatre to get some fresh air, etc.

12/6/2020: The revision class Zoom link is now disabled as the exam is completed.

31/5/2020: The revision classes were held in Week 5. A video recording of the Zoom session (from the second revision class) covering the four revision problems is now available on the C6 recordings page. (now disabled)

18/4/2020 Covid-19 Update:

We understand that this is a difficult and uncertain time, and hope that you are all well and safe.

Owing to the ongoing crisis, there are several changes to the teaching and assessment of this lecture course for Trinity Term 2020. Please read the instructions below that explains how these will affect you, depending on how you are enrolled in the course:

• MPhys Students taking the C6 Theory Option: you will be bound by the physics examination conventions. Accordingly, there will be no formal exam-based assessment. However, you will have to complete the usual Trinity Term revision problem sheets and then - in week 7 or 8 - one past paper, as described on the main C6 course page. Any questions should be addressed to Prof. Andrei Starinets in his role as the C6 Option Coordinator. General academic questions for Part C (not specific to C6) should be addressed to Carrie Leonard-McIntyre in the Physics Teaching Office.

• MMathPhys Students taking Advanced Quantum Theory: you will sit a take-home examination in Trinity term for which you will receive detailed instructions from the MMathPhys Examiners. You will have revision classes to prepare you for this examination. The exam will be broadly similar to previous instances, with suitable modifications to ensure that there is no unfair advantage to anyone from the fact that it is now a take-home assessment. Any questions should be addressed to Prof. Sid Parameswaran in his role as the AQT Lecturer. General academic questions for MMathPhys (not specific to AQT) should be addressed to Jasmine Smith, the MMathPhys Coordinator.

• Students enrolled in a different manner: please consult your course coordinator(s) for instructions, as they will depend on the form of assessment deemed appropriate by your relevant examination board. We cannot presume to know what those requirements will be; whether you can be formally assessed via a take-home exam depends on various factors and may require additional arrangements, so please contact the relevant coordinator(s) as soon as possible.

Revision work will be marked by Filippo Revello. Remote revisions will be led by Filippo Revello (C6) and Sid Parameswaran (AQT). They will likely be organized via Zoom/Microsoft Teams. You do not need a paid account for Zoom (as the conference will be set up by the instructor as a host). Microsoft Teams is available as a free download to students. We suggest that you install and test both pieces of software before Week 1 of term to ensure that you are prepared for remote learning.

Given the many administrative complexities due to these sudden course changes, we request that you follow the above guidance on consulting the relevant course instructors depending on your enrolment status as above. If you contact the wrong person there is a very good chance (given the volume of inquiries we have been dealing with) that it gets lost in the shuffle.

***This section updated to reflect a schedule change due to a lecture conflict that emerged in Week 6***

The transition from AQT to C6-only material will occur sometime during the two hour block on Thursday of week 7. This is because the AQT topics require 20+ε hours while the stochastic processes material requires 6-ε hours, where 0≲ε≲1, and can vary depending on how the course progresses, digressions due to questions, etc. Therefore AQT students are advised to plan on attending the entire block of lectures in week 7 if necessary.

AQT lectures will continue until the end of Week 7.

The five six hours of lecture in week 8 will be C6-only unless there are some exceptional circumstances. MMathPhys students interested in the material are of course welcome to continue to attend the final six hours of lectures, but may also choose to take the Hilary Term MMathPhys course on Non-Equilibrium Statistical Physics for credit. Lectures will be based on notes written by Prof. Fabian Essler, but I may edit portions of them during the year. The most up-to-date set of lecture notes is available here.

Problems are included "inline" in the notes, but for convenience will also be collected into problem sheets posted on this site. You will have problem classes in Weeks 4, 6, and 8. C6 students will continue in HT 2019 with the particle physics component of the course, and will have an additional sheet including material on stochastic processes in HT.

Any concerns with the AQT course may be addressed to me, either in class or via email to sid.parameswaran_at_physics.ox.ac.uk. For C6 students I can answer questions about the lecture material, but organizational matters are primarily decided by the course coordinator, Prof. Starinets (andrei.starinets_at_physics.ox.ac.uk)

Topics covered will include:

Hours 1-(20+ε): (AQT and C6)

Mathematical background on functional methods; path integrals in quantum mechanics and statistical physics; their connection to Feynman diagrams; the Ising model and transfer matrices in 1D; second quantization and application to many-particle systems of fermions, bosons, and spins; phase transitions, critical phenomena, and Landau theory.

Hours (20+ε)-26: (C6 only):

Random variables; stochastic processes; Markov processes and Markov chains; Brownian motion, Langevin and Fokker-Planck equations; diffusion.

Very Rough Plan of Lectures for 2019. This may change somewhat depending on the flow of the lectures. The actual progress of the course will be indicated in boldface as the course progresses.

Announcements:

The 2-hour lecture on Thursday, November 7 is canceled, and will be replaced by a pair of lectures from 12:00-13:00 on Friday Nov 1 and Monday Nov 4.

21/11: The second half of the 2-hour lecture on Thursday, November 21 had a conflict with the Laser Optionand had to be canceled. We will therefore continue with AQT lectures all the way to the end of Week 7.

We will then have an extra hour of lecture on Monday of Week 8, making 6 lectures in all in Week 8 to complete the stochastic processes component of the course.

All lectures take place in the Dennis Sciama Lecture Theatre (in the Denys Wilkinson Building).

*except for the lectures on Tuesday and Wednesday of week 8 which will take place in the Lindemann Lecture Theatre.

†Non-examinable topics.

Notes and Problems: (may be updated during the lectures)

Lecture notes - path integrals and many-particle physics

Lecture notes - Stochastic processes (C6 only)

All the problems are included in the lecture notes, but for convenience, they are posted here in collated form. Each problem class will cover one problem sheet's worth of material.

Problem Sheet 1 [Problems 1-10, covering path integrals and transfer matrices; problem class in MT week 4]

Problem Sheet 2 [Problems 11-18, covering second quantization and applications; problem class in MT week 6 -- NB problem 17 should be submitted with PS3 instead.]

Problem Sheet 3 [Problems 19-21, covering Landau theory; problem class in MT week 8.]

Problem Sheet 4 [Problems 22-26, covering stochastic processes; problem class in HT, C6 only]

Revision Problems [N.B. Question 4 on Brownian motion is outside of the AQT syllabus and need not be attempted]