Introduction to Maxwell’s equations and their symmetries

¨ The ultimate importance of the Maxwell theory is far greater than its immediate achievement in explaining and unifying the phenomena of electricity and magnetism. Its ultimate importance is to be the prototype for all the great triumphs of twentieth-century physics. It is the prototype for Einstein’s theories of relativity, for quantum mechanics, for the Yang-Mills theory of generalised gauge invariance, and for the unified theory of fields and particles that is known as the Standard Model of particle physics…We may hope that a deep understanding of Maxwell’s theory will result in dispersal of the fog of misunderstanding that still surrounds the interpretation of quantum mechanics. And we may hope that a deep understanding of Maxwell’s theory will help to lead the way toward further triumphs of physics in the twenty-first century.¨—-Freeman Dyson

Electrodynamics is a very exciting subject to learn. Unfortunately, Maxwell’s equations are often taught in a rather dull manner, which usually involves solving lots of boring problems without emphasizing the pivotal role symmetry principles play in the formal structure of electromagnetism. This may lead to the rather unpleasant feeling of not truly understanding! So this course is entirely devoted to trying to understand  Maxwell’s equations through their symmetries.


Prerequisites: The course is aimed at advanced undergraduates and beginning graduates. Basic quantum mechanics and familiarity with tensor calculus are desirable. 

Lecture Notes: Lecture Notes currently in progress!


SYLLABUS

Electrostatics and Magnetostatics 

  • I  Electrostatics, magnetostatics, and the Helmholtz theorem.

Electrodynamics

  • II Maxwell’s equations and the Helmholtz theorem for retarded fi elds.
  • III Gauge symmetry in Maxwell’s equations.
  • IV Conservation laws: charge, energy, momentum, and angular momentum.
  • V CPT Symmetry in Maxwell’s equations
  • VI “Inventing” Maxwell’s equations from charge conservation

Electrodynamics and relativity 

  • VII Special relativity and Lorentz tensors.
  • VIII Covariant formulation of Electrodynamics.

Action Principle in Electrodynamics

  • IX The principle of least action.
  • X Lagrangian formulation of electrodynamics.
  • XI Noether’s theorem and conservation laws.

Electrodynamics and Quantum mechanics

  • XII Gauge symmetry in quantum mechanics.
  • XIII “Inventing” Maxwell’s equations from quantum mechanics.

Duality Symmetry in Electrodynamics

  • XIV Duality symmetry in Maxwell’s equations and the Dirac quantization condition.

Useful textbooks

  • J. D. Jackson, 1999, Classical Electrodynamics, 3rd edn (New York: Wiley)
  • A. Zangwill, 2012, Modern Electrodynamics (Cambridge University Press: Cambridge)
  • J. Schwinger et al., 1998,  Classical Electrodynamics (Westview press)
  • B. Felsager, 1981, Geometry, particles, and fields, 3ed, Odense University Press.
  • F. E. Low, 1997, Classical Field Theory: Electromagnetism and Gravitation (Wiley)

Recommended Articles for the course

  • R. Heras, 2017, Alternative routes to the retarded potentials, Eur. J. Phys 38, 055203 [PDF].
  • R. Heras, 2016, The Helmholtz theorem and retarded fields, Eur. J. Phys. 37, 065204 [PDF].
  • R. Heras, 2016, Lorentz transformations and the wave equation, Eur. J. Phys. 37, 025603  [PDF].
  • F. J. Dyson, Why is Maxwell’s Theory so Hard to Understand? [PDF].
  • F. J. Dyson, 1991, Feynman’s proof of the Maxwell’s equations, Am. J. Phys. 58, 209 [PDF].
  • T. T. Wu and C. N. Yang, 1975, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D 12, 3845 [PDF].
  • P. A. M. Dirac, 1931, Quantised Singularities in the Electromagnetic Field, Proc. R. Soc. Lond. A 133, 60 [PDF].
  • J. D. Jackson, From Lorenz to Coulomb and other explicit gauge transformations, Am.  J. Phys. 70, 917 (2002) [PDF].
  • D. H. Kobe, 1984,  Helmholtz theorem for antisymmetric second-rank tensor fields and electromagnetism with magnetic monopoles,  Am. J. Phys. 52, 354 [PDF].
  • C. N. Yang, The conceptual origins of Maxwell’s equations and gauge theory, Phys. Today 67 (11), 45 (2014) [PDF].
  • D. J. Gross, 1996, The role of symmetry in fundamental physics, PNAS, 93, 14256 [PDF].
  • J. Schwinger, 1969,  A magnetic model of matter, Science, 165, 3895 [PDF].
  • E. Witten, 1997, Duality, Spacetime, and Quantum mechanics, Phys.Today 50, 28 [PDF].