Course Outline
This course was given in Jan-March 2013.

It discusses quantum field theory in dimensions less than four, and principally in two dimensions. Topics covered are:

Mathematical preliminaries
Massive and Massless theories in two dimensions
Operator Formalism
Algebraic Structures
The Renormalisation Group, the Wilson-Fisher fixed of the φ4 theory
Conformal field theory and the breaking of conformal invariance
Integrability and its consequences in two dimensions

Lecture notes

Lecture 1
For an introduction to the scalar field, see section 2.3 of [PS]

Lecture 2
This follows quite closely the ideas in [IZ], which are summarised in section 13.2.3. Since we are only consider one loop corrections, we do not have a wave-function renormalisation.

[PS] uses a different, but equivalent, way of relating bare and renormalised couplings, and a slightly different renormalisation condition - see section 10.2.

Lecture 2 - extras File
Euclidean integrals and angular integration

Lecture 3
The section on scale transformations and the calculation of the β functions follows [C]

Lecture 4
Typo on page 4.7 - not yet corrected - the dimensionless coupling is μi=ayiλi, not μi=a−yiλi.

Lecture 5
Typo on page 5.1 - not yet corrected -  the factor in front of Δ4 in the product :ϕ4(x)::ϕ4(y): should be 16, not 12.

Lecture 6 - solving the fixed point equations File
Correct derivation of the perturbative expansion of the position of the Wilson-Fisher fixed point.

Lecture 6 - The derivation of the scale dimensions at a non-trivial fixed-point
Note, there is a typo in the hand-written notes. The left-eigenvectors are correct here.

Lecture 7

Lecture 8

Lecture 9

Lecture 10

Notes on vanishing curves
Some extra notes on vanishing curves together with a few plots

Lecture 11

References

There are numerous sets of lecture notes on the renormalisation group, conformal field theory and integrable field theory in two dimensions as well as on three dimensional field theories. A few are listed here

Any good book on QFT (for regularisation and renormalisation)
The recommended book is "An Introduction to Quantum Field Theory" by Peskin and Schroeder
For scaling and renormalisation: Scaling and Renormalisation in Statistical Physics, J L Cardy, CUP (1996)
For two-dimensional CFT: Conformal Field Theory, Ph Di Francesco, P Mathieu and D Senechal (1997) Springer
For two-dimensional CFT: Conformal Field Theory and Statistical Mechanics, John Cardy (2008)
For two-dimensional models: Statistical Field Theory - an introduction to Exactly Solved models in Statistical Physics. G Mussardo, Oxford Graduate Texts (2010)
For solitons: Solitons and instantons: an introduction to solitons and instantons in quantum field theory, R. Rajaraman, North Holland (1987)