# Jan Philip Solovej (U Copenhagen): Many-body quantum physics

Lecture Course (260172 VO), March - May 2014:

Lectures: Tuesday 16:15 - 18:15 hrs,

Exercise Class: Thursday 13:00 -14:00 hrs Changes to the Schedule

at ESI, Schrödinger Lecture Hall. Course Website (includes Lecture Notes)

Description of Course Content: Flyer

- Tensor products and the formulation of many-body quantum mechanics
- Identical particles and statistics
- Fock spaces and second quantization
- Bogolubov transformations
- Quasi-free states
- Quadratic Hamiltonians
- The Bogolubov variational principle
- The Bogolubov approximation for bosonic systems
- Fermionic systems and the BCS and Hartree-Fock-Bogolubov models

The lectures introduce the formalism and concepts of many-body quantum mechanics. The basic notions of Hilbert spaces and operators as well as the principles of quantum mechanics will be reviewed. In particular, the concept of many-body systems and the notion of indistinguishable particles will be discussed and bosons and fermions will be introduced. This requires introducing tensor products of Hilbert spaces and simple representations of the permutation group. Some time will be spent on semibounded operators and the min-max principle for describing their eigenvalues below the continuum. The general theory will be applied to Schrödinger operators. The lectures will, however, not give a full account of spectral theory. The formalism of second quantization will be introduced and basic properties of many-body quantum states will be analyzed. In particular we will discuss their one- and two-particle reduced density matrices and their eigenvalues. Different classes of states will be discussed and, in particular, the notion of quasi-free states and Bogolubov transformations will be introduced and analyzed. The role of quasi-free states as equilibrium states of quadratic Hamiltonians will be emphasized and the method of diagonalizing these using Bogolubov transformations will be described. This leads to introducing the Bardeen-Cooper-Schrieffer theory or more generally the Bogolubov-Hartree-Fock theory for fermions and the Bogolubov theory for bosons. Hopefully time will permit to consider the theory applied to simple physical systems to illustrate such phenomena as super-conductivity and super-fluidity.