Prerequisites

As well as Part IB NST mathematics, the course will assume a basic knowledge of Lagrangian, statistical, and quantum mechanics. Exposure to the Part II theory courses (TP1 and TP2) and Quantum Condensed Matter course is useful but not essential. 

Learning Outcomes and Assessment

By the end of this course, you will be familiar with the basic foundations of quatum field theory including the method of second quantisation, the Feynman path integral, and field integral techniques. On the Feynman path integral, you will be able to address the quantum mechanics of single particle systems from the physics of bound state systems to the estimation of tunnelling rates in unbound systems. On the field integral, you will be able to formulate the coherent state path integral of many-particle bosonic and fermionic systems. In particular, you will be able to address the quantum mechanics of superfluid and superconducting systems. Finally, you will have an appreciation of how the concepts of quantum field theory provide a common language to address phase transitions and collective phenomena in both high and low energy quan tum many-particle systems.

Synopsis

Collective Phenomena - From Particles to Fields:  Linear harmonic chain and free scalar field theory; functional analysis; quantisation of the classical field; phonons; relation to quantum electrodynamics; concept of broken symmetry, collective modes, elementary excitations and universality.

Second Quantisation:  Fock states; creation and annihilation operators for Bose and Fermi systems; represention of one and two-body operators; canonical transformations; Applications to the interacting electron gas; Wannier states, strong correlation and the Mott transition; †quantum ferromagnetism and antiferromagnetism, spin representations and SU(2) spin algebra, spin-waves; †spin liquids; weakly interacting Bose gas.

Path Integral Methods:  Propagators and the construction of the Feynman Path integral; Gaussian functional integration, stationary phase and saddle-point analyses; relation to semi-classics and statistical mechanics; quantum harmonic oscillator and the single well; double well, instantons, and tunnelling phenomena; †metastability.

Many-Body Field Integral:  Bose and Fermi coherent states; Grassmann algebra; coherent state path integral; quantum partition function; Bogoluibov theory of the weakly interacting Bose gas and superfluidity; Cooper pair instability, and the BCS theory of superconductivity; Ginzburg-Landau phenomenology and the connection to classical statistical field theory; †Gauge theory and the Anderson-Higgs mechanism; †Resonance superfluidity in ultracold atomic gases and the BEC to BCS crossover.

Italics denote specific mathematical topics.

Items marked † will be largely used as source material for problem sets and supervision.

BOOKS

Condensed Matter Field Theory, Altland A and Simons B D (CUP 2006).

Statistical Mechanics, Feynman R P and Hibbs A R (McGraw-Hill 1965).

Quantum Field Theory in Condensed Matter Physics, Nagaosa N (Springer 1999).

Quantum Many-Particle Systems, Negele J W and Orland H (Addison-Wesley 1988).

Techniques and Applications of Path Integration, Schulman L S (Wiley 1981).

The Physics of Quantum Fields, Stone M. (Springer 2000).

Path Integrals in Quantum Mechanics, Zinn-Justin J (Oxford Graduate Texts 2004).

Lectures on Statistical Physics, Levitov L S (http://www.mit.edu/~levitov/8.334/)

Contact

ccw50 at cam.ac.uk

office 5.39 Mott building