MaPhySto
Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

## Second-quantized Hamiltonians

### 9-10 August, 1999 in Auditorium D3 Department of Mathematical Sciences, University of Aarhus

Since the thirties physicists have developed a very efficient formalism to describe systems of many identical particles. Depending on the context, this formalism goes under the name of second quantization or quantum field theory. Some of its key words are canonical commutation/anticommutation relations, Bosonic/Fermionic Fock spaces, creation/annihilation operators.

In the first part of the course an introduction to a variety of mathematical structures underlying second quantization was given. Then a number of problems formulated within this framework - some of them solved only recently, some of them still open - was described.

Below follows the list of topics that was be covered in the course

	I. Canonical Commutation Relations.

1. Symplectic spaces, Heisenberg algebra and group, Weyl algebra.
2. Bosonic Fock spaces, Wick quantization.
3. Schrodinger representation, Kohn-Nirenberg quantization.
4. Gaussian processes, Q-representation.
5. Weyl quantization, representation of symplectic and metaplectic group.
6. Representations with the Fock property and number operator.
7. Quasi-free states.
8. Coherent state representations.

II. Canonical anticommutation relations.

1. Grassmann algebra, orthogonal spaces, Clifford algebra.
2. Fermionic Fock spaces and super Fock spaces.
3. Representation of orthogonal and Spin groups.

III. Basic examples of second-quantized Hamiltonians with a localized
interaction.

1. Exactly solvable Hamiltonians.
2. Spin-boson   and Pauli-Fierz Hamiltonian.
3. Nonrelativistic QED.
4. Lee models.
5. Spacially cut-off $P(\phi)_2.$

IV. Scattering theory for Hamiltonians with a localized interaction.

1. Existence of asymptotic fields.
2. Fock property of asymptotic fields in the massive case.
3. Asymptotic completeness.

V. Relativistic quantum field theory.

1. Representations of the Lorentz group.
2. Representations of the Poincare group.
3. Free fields.
4. Algebraic approach -- Haag-Kastler axioms.
5. Haag-Ruelle scattering theory.



Registered participants:

• Jan Derezinski (lecturer)
• Søren Fournais
• Arne Jensen
• Michael Melgaard
• Jacob Schach Møller
• Erik Skibsted

Please make further inquiries to MaPhySto (maphysto@mi.aau.dk) or to the organizer Erik Skibsted.