## Trinity College Dublin, The University of Dublin

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# Module MA3431: Classical Field Theory

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2014-15
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 magneto-statics;
• 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 Euler-Lagrange 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 non-quantum 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 four-vector potential.
• Lagrangian density for a free vector field; Symmetry properties.
• Canonical stress tensor; conserved, traceless & symmetric stress-tensor.
• Particle and field energy-momentum & 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