296 Phillips Hall, 962-7213, khvesh@physics.unc.edu

**Lectures**: TTh 1:30-2:45 PM, 277 Phillips

**Office hours**: before and after class

**Introduction**:

In many areas of Physics that include High Energy, Nuclear, and Condensed
Matter, the understanding of the essential

physical phenomena requires a consideration of the collective effects of a
large number of degrees of freedom.

Quantum Many-Body Theory is the tool as well as the language that has been
developed to

describe the physics of problems in such apparently dissimilar fields. This
course is designed to provide the basic conceptual

and computational tools of Quantum Many-Body Theory to students (both theorists
and experimentalists)

with a wide range of interests in Physics. This material will also be used in
subsequent and more specialized

courses (e.g., Phys822/823). As a prerequisite, all the students should take Phys721/722
(Quantum Mechanics).

Grading will be based on the homework and students' participation in the discussions of the course material.

**Syllabus:**

Excerpts from the AGD textbook are cited as, e.g., "AGD_1.10-1.12"

which stands for Chapter 1, parts 10 to 12.

1. Non-relativistic boson/fermion many-body systems:

1.1 Creation/annihilation operators, Fock space, ground and excited states of interacting
bosons/fermions **(AGD_1.3-1.5)
** 1.2 Interaction representation, Green function, S-matrix. (

1.4 Self-energy and polarization operator, Example of Coulomb interacting fermions

1.5 Thermal Green function, Matsubara representation, Thermodynamic potential.

2. Introduction to relativistic field theory:

2.1 Classical equations of motion, Lagrangians and actions, canonical quantization.

2.2 Discrete/continuous and global/local symmetries, conservation laws, Noether's theorem. (

2.2. Free scalar and vector fields, Klein-Gordon and Maxwell equations. (

2.3 Dirac equation, (bi)spinors and Grassmann algebra. (

3. Path integral techniques:

3.1 Path integral in single-particle quantum mechanics. Harmonic oscillator. (**F_Ch.5.1-5.2)**

3.2 Path integral for a relativistic bosonic
field. . (**F_Ch.5.3-5.6)**

** **3.3 Gaussian path integrals. Perturbation theory and cluster expansion.

3.4 Grassmanian numbers. Fermion path integral

4. Further applications of many-body techniques:

4.1 Interacting non-relativistic fermions: Hartree-Fock and Random Phase Approximations.

4.2 Interacting non-relativistic bosons: Bogoliubov transformation, superfluidity and superconductovity.

4.3 Landau theory of phase transition: mean field and
fluctuations. Poor man's scaling.

4.4.Fermion and spin chains, Jordan-Wigner
transformation and bosonization, solitons
and kinks.

4.5 Quantum statistical mechanics, Matsubara sums, thermal Green
functions **(AGD_3.11-3.16) .**

4.6 Response and correlations functions, fluctuation-dissipation
theorem, quantum critical behavior.

**Bibliography:**

**Concise introduction to the subject**:

(AGD) A. Abrikosov, L. Gorkov
and I. Dzyaloshinski. ``Methods of
Quantum Field Theory in Statistical Physics",

**Online resources:**

(F) General field theory course by E.Fradkin (UIUC):

*http://webusers.physics.uiuc.edu/~efradkin/phys582/physics582.html*

(B) A similar course - with emphasis on particle physics - by I.Balitsky (ODU):

*http://www.physics.odu.edu/~balitsky/phy842*

Other useful textbooks:

-Short:

M. Stone, "The Physics of Quantum Fields" , Springer.

P.Ramond, "Field Theory : a Modern
Primer",

-Comprehensive:

G. Mahan, "Many Particle Physics", Plenum Press.

A. Fetter and J. D. Walecka. ``Quantum
Theory of Many Particle Systems", McGraw-Hill.

-More up to date:

E. Fradkin. ``Field Theories of
Condensed Matter Systems", Addison-Wesley.

S.Sachdev,
"Quantum Phase transitions", Cambridge Univ. Press.

A. M. Tsvelik, "Quantum field theory in
condensed matter physics", Cambridge Univ. Press.

**
Homework assignments**: see the