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Module MA3431: Classical Field Theory
 Credit weighting (ECTS)

5 credits
 Semester/term taught

Michaelmas term 201415
 Contact Hours

11 weeks, 3 lectures including tutorials per week

 Lecturer

Prof. Tristan McLoughlin
 Learning Outcomes
 On successful completion of this module, students will be able to:
 Apply standard methods, such as orthogonal functions, to solve problems in electo and magnetostatics;
 Describe how to find the equation of motion for a scalar field using a given Lagrangian density;
 Calculate the stress tensor and evaluate its four divergence, relating it to a conservation law;
 Employ a variational principle to find the relativistic dynamics of a charged particle interacting with an electromagnetic potential;
 Use the EulerLagrange equation to show how a Lorentz scalar Lagrangian density with an interaction term leads to the Maxwell equations;
 Explain the concepts of guage invariance and tracelessness in the context of the stress tensor of a vector field;
 Demonstrate how the divergence of the symmetric stress tensor is related to the four current density of an external source;

 Module Description
 Rational and Aims  The purpose of module MA3431 is to outline the properties of a classical field theory that relate in particular to scalar and vector fields, to point out features of the tensor calculus suitable for the description of relativistic nonquantum field theories, and to indicate the importance of symmetry and invariance principles in the development of conservation laws for energy, momentum and other conserved quantities. The module is mandatory for third year undergraduate students of theoretical physics but may optionally be taken by third or fourth year undergraduate students of mathematics. Postgraduate students from other institutions have taken the module in the past. The module forms an element of the undergraduate programme in theoretical physics being built upon prerequisite first and second year courses in classical dynamics and mathmatics and leading to courses in the fourth and final year including quantum field theory. From a teaching point of view, the intention of the lecturer is to indicate how powerful analytical and formal methods can be invoked to understand and solve many problems in mathematical physics. A further intention is to provide a sense of the important role played by field theories, particularly electrodynamics, in the development of theoretical physics and its applications.

 Module Content

 Electrostatics; Green's Theorem; Solution by Green functions.
 Spherically symmetric problems; Multipole expansion; Magnetostatics.
 Maxwell equations; Gauge invariance; Transformation properties.
 Lorentz invariance; Poincare Lie algebra; Scalar, vector and tensor representations.
 Hamiltons variational principle, Lagrangian for relativistic particle.
 The Lorentz force, charged particle interaction, antisymmetric field tensor.
 Covariant field theory, tensors, scalar fields and the fourvector potential.
 Lagrangian density for a free vector field; Symmetry properties.
 Canonical stress tensor; conserved, traceless & symmetric stresstensor.
 Particle and field energymomentum & angular momentum conservation
 Indicative Textbooks
 1. Classical Electodynamcis  J. David Jackson, John Wiley (3rd edition) 1998
 2.Classical Theory of Fields  L.
D. Landau & E.
M. Lifshitz, Heinemann, 1972
 3. Classical Field Theory  Francis E. Low, J.Wiley and Sons (1st edition) 1997
 4.
Module Website 
MA3431  Classical Field Theory

 Module Prerequisite
 Advanced Classical Mechanics II (MA2342)
 Assessment Detail
 This module will be examined
in a 2hour examination in Trinity term. Assignments will contribute 15% to the final result. If supplemental exams are required it will consist of 100% exam.