# Quantum Field Theory I & II

### 24 Lectures Dr P. Mansfield and Dr N. Iqbal

This course will describe some of the basic ingredients of quantum field theories in the language of path integrals, and as such is complementary to the canonical quantisation approach in the Introductory Field Theory course. Path integrals are first introduced in the familiar context of quantum mechanics, before being applied to field theory where techniques are developed to allow the calculation of scattering amplitudes in gauge theories. Most of the content of this course can be found in Peskin and Schroeder, particularly in chapters 9, 10 and 16, although other textbooks are also useful, especially Feynman and Hibbs for an introduction to path integral quantum mechanics.

### Outline of the course

**Path Integrals:** derivation of path-integal formulation of quantum mechanics. Examples including the simple harmonic oscillator. Formulation of perturbation theory, and Feynman diagram representation. Introduction to Grassmann numbers. Extension to ﬁeld theory.

**Green functions:** LSZ reduction formulae for scalar and Dirac fields. Green functions. Perturbation theory and derivation of Feynman rules. Generating functionals for disconnected and connected Green functions.

**Renormalisation**: One-loop renormalisation of φ^{4} theory. General discussions of perturbative renormalisability. Brief discussion of regulators.

**Gauge theory**: Abelian and non-Abelian gauge invariance, Yang-Mills action. Gauge ﬁxing by Faddeev-Popov method, ghosts, Feynman rules. BRST invariance, simple consequences. Brief discussion of regulators.

### Books for the course

R. P. Feynman and A. R. Hibbs, **Quantum Mechanics and Path Integrals** (Dover Books on Physics, 2010)

M. Peskin and D. Schroeder, Quantum Field Theory (Addison-Wesley, 1995)

R.J.Rivers, Path Integral methods in QFT (CUP, 1987)