Introduction to representations of the canonical commutation and anticommutation relations

@article{Dereziski2005IntroductionTR,
  title={Introduction to representations of the canonical commutation and anticommutation relations},
  author={Jan Dereziński},
  journal={Lecture Notes in Physics},
  year={2005},
  volume={695},
  pages={63-143}
}
  • J. Dereziński
  • Published 8 November 2005
  • Mathematics
  • Lecture Notes in Physics
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. 

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References

SHOWING 1-10 OF 83 REFERENCES

On factor representations and theC*-algebra of canonical commutation relations

A newC*-algebra,A, for canonical commutation relations, both in the case of finite and infinite number of degrees of freedom, is defined. It has the property that to each, not necessarily continuous,

Free states of the canonical anticommutation relations

Each gauge invariant generalized free state ωA of the anticommutation relation algebra over a complex Hilbert spaceK is characterized by an operatorA onK. It is shown that the cyclic representations

On Quasifree States of the Canonical Commutation Relations (II)

A nece?sar> and sufficient condition for the qu?H-equivalence of two quasifree primary representations of the canonical commutation relations is derived. A quasifree state of the self-dual CCR

Quasi-equivalence of quasi-free states on the Weyl algebra

A necessary and sufficient condition for quasi-equivalence of quasi-free factor states over the Weyl algebra is proved. The essential part of this paper is closely related to the work of Powers and

Quantum electrodynamics in external fields from the spin representation

Systematic use of the infinite‐dimensional spin representation simplifies and rigorizes several questions in quantum field theory. This representation permutes ‘‘Gaussian’’ elements in the fermion

Spectral Theory of Pauli–Fierz Operators

Abstract We study spectral properties of Pauli–Fierz operators which are commonly used to describe the interaction of a small quantum system with a bosonic free field. We give precise estimates of

On the equilibrium states in quantum statistical mechanics

AbstractRepresentations of theC*-algebra $$\mathfrak{A}$$ of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential μ are studied. Both for finite

Type of von Neumann Algebra Associated with Free Field

A class of von Neumann algebras associated with the normal representation (i.e. the Fock representation) of the canonical commutation relations has been studied in an earlier paper. > The von Neumann

REPRESENTATIONS OF THE CANONICAL COMMUTATION RELATIONS DESCRIBING A NONRELATIVISTIC INFINITE FREE BOSE GAS

The existence of inequivalent representations of the canonical commutation relations which describe a nonrelativistic infinite free Bose gas of uniform density is investigated, with a view to

A Lattice of Von Neumann Algebras Associated with the Quantum Theory of a Free Bose Field

Von Neumann algebras associated with the normal representation of canonical commutation relations are studied. Corresponding to each subspace of a real Hilbert space (test function space), a von
...