## Trinity College Dublin, The University of Dublin

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# Module MAU34401: Classical Field Theory

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2019-20
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Learning Outcomes
On successful completion of this module, students will be able to:
• Apply standard methods, to solve problems in electro- 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 traceless ness 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

Rationale and Aims - The purpose of module MAU34401 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 classical 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 and may optionally be taken by third or fourth year undergraduate students of mathematics.  The module forms a core element of the undergraduate programme in theoretical physics being built upon prerequisite first and second year courses in classical dynamics and mathematics and leading to courses in the fourth and final year including quantum field theory. A key goal is to elucidate  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;  Magnetostatics.
• Maxwell equations; Gauge invariance; Transformation properties.
• Lorentz invariance; Scalar, vector and tensor representations.
• Hamilton 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 https://www.maths.tcd.ie/~ryan/3431.html

Module Prerequisite
MAU23402
Assessment Detail
This module will be examined in a 2-hour examination in Michaelmas term. Assignments will contribute 20% . Re-assessments if required will consist of 100% exam.
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