Instructor: Robert Brout
Type: Mini-course, non credit
Dates: October 18 - November 30, 2007, each Thursday from 10:00am - 12:00pm
Location: Perimeter Institute, Bob Room

A series of 8 lectures, 2 hours each on Spontaneous Broken Symmetry

Lecture 1 Historical Introduction

• The atomic hypothesis and the concept of spin gave birth to the notion of “molecular field”. There arose self consistent fields in the absence of external fields which would give structures that did not possess the symmetry of the underlying dynamics.
  Example: The Weiss molecular theory of ferromagnetism. Other examples will be given but ferromagnetism will be pursued in detail because it exemplifies the concept particularly well.
• Exchange coupling and ferromagnetism.
• Ising and Heisenberg models exemplifying discrete and continuous symmetries respectively.
• Ising Model.

Lecture 2 Ising Model Continued

• The Ising Model viewed as a field theory.
• Introduction to infra-red singular behaviour. The Wilson – Fuhr renormalization group will be motivated.
• Spin – spin correlations and field theoretic Green’s functions are essentially equivalent and will be analyzed as random walks. Field – field interactions come about from intersections of walks.

Lecture 3  Heisenberg Model

• Low lying excitations of the Heisenberg are massless spin waves. Those of the Ising model are massy.
• Analogies with the Debeye versus Einstein theory of specific heats of solids are made.
• These considerations are extended to classical theory i.e. classical statistical mechanics, euclidean quantum field theory
• The notion of longitudinal and transverse susceptibility is introduced. When expressed in field theory the vanishing of the latter is the vanishing of the mass of a collective mode (or in phenomenology of scalar fields a linear combination of field amplitudes). It is then called Goldstone’s theorem. The principle is always the same: it costs zero energy to “rotate” an order parameter when the dynamics are symmetric with respect to that rotation.

Lecture 4 Bose Einstein Condensation

• The condensation of the non interacting Bose gas is reviewed.
• Elementary excitations a la Bogolyubov are constructed. They are massless as a consequence of the Goldstone theorem where the symmetry in question is that of the rotation of the quantum mechanical phase of the wave function.
• Mention is made of the refinement of Bogolyubov’s analysis and independently Feynman with his more physical interpretation.
• The calculation of Penrose and Onsagen of the fraction of liquid helium that is condensed is presented.
• From this emerges the notion of the superfluid single particle wave function and the notion of superfluid flow.

Lectures 5 & 6 Superconductivity

• The nature of the phenomenon is presented, the Meissner effect and London’s conclusion that superconductivity is a thermodynamic phase.
• The BCS theory (a landmark!) where once more the broken symmetry concerns the quantum mechanical phase.
• Ginzburg – London theory, penetration lengths, correlation lengths and connection with BCS indicated.
• Superconductivity of the 2nd kind and flux tubes. Gauge theory will play a major role.
  
  This part of the course is conceived as my homage to P G de Gennes who died in June 2007 at a relatively young age (74). de Gennes elucidated brilliantly many of the features of superconductivity).

Lecture 7 Spontaneous Broken Chival Symmetry

• The Dirac equation is discussed in terms of chivality.
• The Nambu form Lasinio construction of the fermions mass is given with emphasis on spontaneously broken chival symmetry.
• Soft pions and the Goldberger – Treiman relation.

Lecture 8 Yang-Mills and Spontaneous Broken Symmetry

• YM theory reviewed.
• How a gauge field picks up a mass in the presence of SBS.
• Yet the theory (and the vacuum) is invariant under gauge transformations, the secret of renormalizability.
• A brief sketch of the electroweak theory.
• The significance of the LHC.