Introduction to representations of the canonical commutation and anticommutation relations

TL;DR

This paper provides an overview of the mathematical representations of canonical commutation and anticommutation relations, focusing on quasifree states and von Neumann algebras in quantum field theory.

Contribution

It offers a detailed introduction to the structures and representations of CCR and CAR, including quasifree states and specific representations like Araki-Woods and Araki-Wyss.

Findings

Detailed discussion of quasifree states and their properties

Analysis of the Araki-Woods and Araki-Wyss representations

Examination of von Neumann algebra lattices in Fock spaces

Abstract

Lecture notes of a minicourse given at the Summer School on Large Coulomb Systems - QED in Nordfjordeid, 2003, devoted to representations of the CCR and CAR. Quasifree states, the Araki-Woods and Araki-Wyss representations, and the lattice of von Neumenn algebras in a bosonic/fermionic Fock space are discussed in detail.