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Outline:

The module is aimed at students who want to understand and possibly in future pursue research in theoretical high energy physics. The approach is mathematical and complements other particle physics courses which give a more intuitive approach, more physics results and comparison of data and theory. Although presenting the topic from a particle physics based approach, much of the material will be of use to those interested in, or intending to do research in theoretical condensed matter physics as well. A reminder of various relevant aspects of quantum mechanics and classical field theory will be given. Then the necessary background will be built up to enable the calculation of cross-sections, or Green’s functions for a given process. The use of a significant number of mathematical methods to describe and make predictions for physical processes will be developed, which will be useful in a wide variety of potential future areas of research.

Aims:

The module aims at forming an appreciation of the main concepts underpinning the construction of a fundamental quantum field theory: relativistic covariance, symmetry and the construction of perturbative expansions to be related to experimental scattering cross sections. Thus it intends to put the students in a position to explore further, more sophisticated field theories, relevant to both high-energy and condensed matter physics.Ìý

Intended Learning Outcomes:

On completion of the module, students should be able to:

• For a particular particle content, identify the form of the Lagrangian for the corresponding Quantum Field Theory
• Obtain, with justification, the Feynman rules for the Theory
• Construct operators for basic physical quantities for simple Field Theories, and obtain and interpret expectation values
• Calculate matrix elements and cross-sections at tree level, and in the most basic cases at one loop

Teaching and Learning Methodology:

This module is delivered via weekly lectures supplemented by a series of workshops and additional discussion. In addition to timetabled lecture hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.

Indicative Topics:

1. Introduction and Relativistic Wave Equations: Outline, conventions and introduction to quantum mechanics for relativistic particles.
2. Reminder of Lagrangian and Hamiltonian Mechanics: Introduction of Lagrangian and Hamiltonian field theory; Importance of Poisson brackets
3. Quantisation of a free scalar field theory: Introduction of creation and annihilation operators and canonical quantisation
4. Introduction of point interactions: Calculation of scattering matrix elements and LSZ reduction formula; Wicks theorem and normal ordering; Generating functionals and origin of Feynman diagrams
5. Quantisation for fermionic fields: Recap of relativistic quantum mechanics and symmetries in fermionic systems
6. Creation and annihilation operators for fermions and anti-commutation relations
7. Quantisation of photon field and necessity for gauge fixing
8. Origin of Feynman rules in QED: Simple examples.
9. Introduction to renormalisation: Ultraviolet divergences and their regularization; Counterterms and origin of running masses and couplings; Implications for physics