Linder Research Group

FY3464 / FY8914 Quantum Field Theory 2024

Breaking news

I have posted a suggested solution to the 2021 exam (see below).
There was a question during the question hour regarding if it is possible to have nonlocal counterterms. The general answer seems to be that nonlocal counterterms are used in other theories: thus, although new modifications might have to be done during the renormalization procedure if the counterterms are nonlocal, there does not seem to be a general principle forbidden nonlocality.

Timetable & practical info

Lectures

Mondays 14:15-16:00 in C4-118.

Tuesdays 12:15-14:00 in C4-118.

Key topics covered in week 17 lectures

Monday

Proceed on our mission to establish a connection between the S-matrix element <f|S|i>, which is used to compute physical quantities like scattering cross sections, and higher-order correlators in a non-interacting theory (which can be computed with Wick's theorem).
Demonstrated that we can indeed write <f|S|i> in term of a free-theory expectation value of the interacting Lagrangian (in the interaction picture).

Tuesday

We argued that the non-ineracting Hamiltonian for dressed particles is in fact nothing else than the free Hamiltonian with a mass term that is the physical mass (equal to the renormalized mass in the OS scheme)
We showed that the S-matrix in general is only determined up to a phase-factor, but this factor could be uniquely determined from the requirement <0|S|0>_free = 1.
Under the assumption that all momenta of the particles involved in the scattering are different, we showed that we could reduce the S-matrix element to a series of free-theory correlators (expectation values in the ground-state). This would ultimately lead to truncated, connected diagrams describing scattering between different states.

Exercise guidance

The last 10 minutes of the Tuesday lectures will be set aside for questions regarding exercises.

Reference group

Karine Halleraker (karineeh@stud.ntnu.no) and Sebastian Siljuholtet Johansen (sebastsj@ntnu.no). A summary of the first reference group meeting. A summary of the second reference group meeting.

Exam

The exam will be a written one and takes place tba. You will be allowed to bring the mathematical formula collection by K. Rottman. No hand-written notes are allowed.

FY3464: your grade is determined 100% from the written exam. You will receive a grade A-F.

FY8914: your grade is determined 100% from the written exam. You will receive a grade A-F.

Question hour

TBA

Curriculum

Description and notes

The curriculum is defined by the lectures and belonging extra notes for each chapter (see below) which in turn will follow the structure of the following lecture notes, but with much more details. Most of the curriculum will be covered in the lectures, but some material will be self-study.

Additional useful details of certain passages are found below.

Chapter 1

Note regarding Lorentz invariant inner product and completeness relation
Note regarding branch points and branch cuts. You can find more details here here.
Here is a nice, brief intro to functional derivatives and their meaning

Chapter 2

Proof that path integrals automatically time order: page one and two.
Derivation of multidimensional Gaussian integral formulas that we use: page one and two.
Proof and discussion of the integral formula we use for multidimensional Gaussian integrals in the case of a complex, symmetric matrix with a positive-definite real part.
Relation between interacting and non-interacting vacuum state.
Discussion of Heisenberg and interaction-picture operators in interacting correlator (see pages 53-54)

Chapter 3

Note about analytic continuation.
Derivation of formula for Feyman parametrization.
The last part of the treatment of the sunset diagram.

Chapter 4

Notes on relation between first- and second-quantized Hamilton operators.
Note about charge conjugation and time-reversal symmetry for the Dirac Lagrangian.
Note on Gaussian integrals with Grassmann variables.
Generalization of fermionic path integrals to 3+1 dimensions.

Chapter 5

Notes for the entire chapter .

Some recommended books to complement the lecture notes and which can have more details about certain topics:

Quantum Fields: From the Hubble to the Planck Scale , M. Kachelriess.
Quantum Field Theory and the Standard Model, M. Schwartz.

Lecture plan

Rough plan for the semester - deviations may occur. Chapters refer to lecture notes in above link.

Week 2 - Introduction and free scalar fields part I (chapter 1)
Conventions, Green functions

Week 3 - Free scalar fields part II (chapter 1)
Scalar field theory in 3+1 dimensions, causality in correlators

Week 4 - Free scalar fields part III and Noether's theorem (chapter 1)
Causality in correlators continued, symmetries and Noether's theorem

Week 5 - Path integrals (chapter 2)
Non-relativistic path integrals, n-point correlators, Wick's theorem

Week 6 - Scalar perturbation theory part I (chapter 2+3)
Scalar field path integrals, Feynman diagrams

Week 7 - Scalar perturbation theory part II (chapter 3)
More Feynman diagrams, 2-point correlators, interacting theory in 3+1D

Week 8 - Scalar perturbation theory part III (chapter 3)
Regularization, renormalization, dimensional regularization

Week 9 - Scalar perturbation theory part IV (chapter 3)
The sunset diagram, 4-point correlators

Week 10 - Scalar perturbation theory part V + Free Dirac fields part I (chapter 3+4)
Effective coupling, Dirac equation

Week 11 - Free Dirac fields part II (chapter 4)
Lorentz algebra, solutions to Dirac equation

Week 12 - Free Dirac fields part III (chapter 4)
Quantization of Dirac field, propagator, Lagrangian

Week 13
No teaching (Easter holiday)

Week 14
No teaching (Easter holiday) Self-study this week: Symmetries of Dirac L

Week 15 - Free Dirac fields part V (chapter 4)
Grassmann variables, fermionic path integrals

Week 16 - Scattering part I (chapter 5)
Cont. fermionic path integrals, S-matrix

Week 17 - Scattering part II (chapter 5)
LSZ-reduction formula

Exercises and useful links

Exercises

The exercises are not mandatory, but highly recommended.

January 16th - Exercise #1

Problem text #1 | Solution #1

January 23th - Exercise #2

Problem text #2 | Solution #2

January 30th - Exercise #3

Problem text #3 | Solution #3

February 6th - Exercise #4

Problem text #4 | Solution #4

February 13th - Exercise #5

Problem text #5 | Solution #5

February 20th - Exercise #6

Problem text #6 | Solution #6

March 5th - Exercise #7

Problem text #7 | Solution #7

March 12th - Exercise #8

Problem text #8 | Solution #8

March 19th - Exercise #9

Problem text #9 | Solution #9

April 9th - Exercise #10

Problem text #10 | Solution #10

April 16th - Exercise #11

Problem text #11 | Solution #11

April 23th - Exercise #12

Problem text #12 | Solution #12

Lecture notes on QFT

• QFT notes for course at Cambridge

Exams from previous years

Example of a recent exam (2021) and its solution

Old exams from the period 2013-2018

Even older exams from the period 2005-2012

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