MPhys Physics with Study in Europe

This unit introduces quantum field theory as the framework that unifies quantum mechanics and special relativity. Students begin with the classical foundations of field theory, including Lagrangian dynamics, continuous symmetries, and Noether’s theorem, before developing the formalism of canonical quantisation for scalar and complex fields, with attention to causality and antiparticles. The course then focuses on practical tools for calculating physical processes, including the S-matrix, propagators, Wick’s theorem, and Feynman rules, with applications to particle decays and scattering cross sections. Core aspects of quantum electrodynamics are covered, including the Dirac equation, spinor fields, gauge symmetry, and gauge fixing. The unit concludes with an introduction to renormalisation, dimensional regularisation, and the renormalisation group, illustrated through key physical predictions such as the anomalous magnetic moment and the Lamb shift. 

The unit aims to introduce the unifying framework of quantisation of fundamental forces and particles in agreement with special relativity, and apply this to a variety of physical phenomena, including particle decay widths, reaction cross-sections and anomalous magnetic moment of a fermion. 

On the successful completion of the course, students will be able to:  

• Explain the concept of global and local symmetries in Quantum Field Theory and their implications. 
• Explain the concept of canonical quantization for scalar, vector and fermion fields. 

• Derive the Feynman rules from the Lagrangian formalism, use these to calculate S-matrix elements, and understand their physical significance.

   

• Calculate the lifetime of unstable particles and cross sections of reactions that occur in the lowest order of perturbation theory
• Explain the concept of renormalization and apply this to quantum field theories.

1. Preliminaries (3 Lectures)

Classical Lagrangian Dynamics; Lagrangian Field Theory; Global and Local Symmetries;  Noether's Theorem.

2. Canonical Quantization (3 lectures + 2 Examples)

From Classical to Quantum Mechanics; Quantum Fields and Causality; Canonical Quantization of Scalar Field Theory; Complex Fields and Anti-Particles.

3. The S-Matrix in Quantum Field Theory (6 lectures + 3 Examples)

Time Evolution of Quantum States and the S-Matrix; Feynman Propagator and Wick's Theorem; Transition Amplitudes and Feynman Rules; Particle Decays and Cross Sections;  Unitarity and the Optical Theorem.

4. Quantum Electrodynamics (6 lectures + 3 Examples)

Dirac Equation and its Nonrelativistic Limit, Dirac and Clifford Algebras, Dirac and Weyl Spinors; Quantization of the Fermion Field; Gauge Symmetry; Quantization of the Electromagnetic Field; Photon Propagator and Gauge Fixing; Becchi--Rouet—Stora—Tyutin (BRST) Transformations*, Feynman Rules for Quantum Electrodynamics.

5.  Renormalization (6 lectures + 1 Examples)

Renormalizability; Dimensional Regularization, Renormalization of a Scalar Theory;

Displacement Operator Formalism of Renormalization to All Orders*; Renormalization Group Equation; Anomalous Magnetic Moment and the Lamb shift.

*indicates a topic for further reading. 

Synchronous learning: lectures, example classes

Asynchronous learning: online material, collated on Canvas 

1. J.D. Bjorken and S.D. Drell, Quantum Mechanics, McGraw-Hill Inc, 1964.

2. F. Mandl and G. Shaw, Quantum Field Theory, Wiley, 1992.

3. M. E. Peskin and D. V. Schroeder, Quantum Field Theory, Perseus Books Group, 1995.

4. S. Pokorski, Gauge Field Theories, Cambridge University Press, 2000, Second Edition.

5. J. Zinn–Justin, Quantum Field Theory and Critical Phenomena, Oxford Science Publications,  

   2002, Fourth Edition.