(Course description last updated for academic year 2021-22).
Prerequisites

The course covers theoretical aspects of the classical dynamics of particles and fields, with emphasis on topics relevant to the transition to quantum theory.  This course is recommended only for students who have achieved a strong performance in Mathematics as well as Physics in Part IB, or an equivalent qualification. In particular, familiarity with variational principles, Euler-Lagrange equations, complex contour integration, Cauchy’s Theorem and transform methods will be assumed.  Students who have not taken the Part IB Physics B course ‘Classical Dynamics’ should familiarise themselves with the ‘Introduction to Lagrangian Mechanics’ material.

Synopsis

Lagrangian and Hamiltonian mechanics: Generalised coordinates and constraints; the Lagrangian and Lagrange's equations of motion; symmetry and conservation laws, canonical momenta, the Hamiltonian; principle of least action; velocity-dependent potential for electromagnetic forces, gauge invariance; Hamiltonian mechanics and Hamilton's equations; Liouville's theorem; Poisson brackets and the transition to quantum mechanics; relativistic dynamics of a charged particle.

Classical fields: Waves in one dimension, Lagrangian density, canonical momentum and Hamiltonian density; multidimensional space, relativistic scalar field, Klein-Gordon equation; natural units; relativistic phase space, Fourier analysis of fields; complex scalar field, multicomponent fields; the electromagnetic field, field-strength tensor, electromagnetic Lagrangian and Hamiltonian density, Maxwell's equations.

Symmetries and conservation laws: Noether's theorem, symmetries and conserved currents; global phase symmetry, conserved charge; gauge symmetry of electromagnetism; local phase and gauge symmetry; stress-energy tensor, angular momentum tensor; transition to quantum fields.

Broken symmetry: Self-interacting scalar field; spontaneously broken global phase symmetry, Goldstone's theorem; spontaneously broken local phase and gauge symmetry, Higgs mechanism.

Dirac field: [not examinable] Covariant form of Dirac equation and current; Dirac Lagrangian and Hamiltonian; global and local phase symmetry, electromagnetic interaction; stress-energy tensor, angular momentum and spin.

Phase transitions and critical phenomena: Landau theory, first order vs. continuous phase transitions, correlation functions, scaling laws and universality in simple continuous field theories.

Propagators and causality: Schrödinger propagator, Fourier representation, causality; Kramers-Kronig relations for propagators and linear response functions; propagator for the Klein-Gordon equation, antiparticle interpretation.

BOOKS

The Feynman Lectures, Feynman R P et al. (Addison-Wesley 1963) Vol. 2. Perhaps read some at the start of TP1 and re-read at the end.

Classical Mechanics, Kibble T W B and Berkshire F H (4th edn Longman 1996):  A clear basic text with many examples and electromagnetism in SI units.

Classical Mechanics, Goldstein H (2nd edn Addison-Wesley 1980): A classic text that does far more than is required for this course, but is clearly written and good for the parts that you need.

Classical Theory of Gauge Fields, Rubakov V (Princeton 2002): The earlier parts are closest to this course, with much interesting more advanced material in later chapters.

Course of Theoretical Physics, Landau L D & Lifshitz E M: Vol.1 Mechanics (3rd edn Oxford 1976-94) is all classical Lagrangian dynamics, in a structured, consistent and very brief form;

Vol.2 Classical Theory of Fields (4th edn Oxford 1975) contains electromagnetic and gravitational theory, and relativity. Many interesting worked examples.

Quantum and Statistical Field Theory, Le Bellac M, (Clarendon Press 1992): An excellent book on quantum and statistical field theory, especially applications of QFT to phase transitions and critical phenomena. The first few chapters are particularly relevant to this course.

 

Course section:

Other Information

For more information, visit the Course WebsiteWeblink

Staff
Prof Ben GripaiosCoordinator