Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Cambridge University Press eBooks

Quasi-free states

Jan Dereziński, Christian Gérard

Published Mar 6, 2013• 0 citations
Publisher

Abstract

Suppose that we have a state ψ on the polynomial algebra generated by the fields ϕ ( y ) satisfying the CCR or CAR relations. For simplicity, assume that ψ is even , that is, vanishes on odd polynomials. Clearly, this state determines a bilinear form on Y given by the “2-point function” We say that a state ψ is quasi-free if all expectation values can be expressed in terms of (17.1) by the sum over all pairings. This chapter is devoted to a study of (even) quasi-free states, both bosonic and fermionic. This is an important class of states, often used in physical applications. Fock vacuum states belong to this class. It also includes Gibbs states of quadratic Hamiltonians. Representations obtained by the GNS construction from quasi-free states will be called quasi-free representations . They are usually reducible. Many interesting concepts from the theory of von Neumann algebras can be nicely illustrated in terms of quasi-free representations. Quasi-free states can be easily realized on Fock spaces, using the so-called Araki–Woods, resp. Araki–Wyss representations in the bosonic, resp. fermionic case. Under some additional assumptions, in particular in the case of a finite number of degrees of freedom, these representations can be obtained as follows. First we consider a Fock space equipped with a quadratic Hamiltonian. Then we perform the GNS construction with respect to the corresponding Gibbs state. Finally, we apply an appropriate Bogoliubov rotation.

DOI
10.1017/cbo9780511894541.018

Related papers

Among Quadratic Hamiltonians, Bogoliubov Transformations and Non-Regular States on CCRs *-Algebra. I. Pure and Invariant States. (In the Mood for the Manuceau Verbeure Theorems about Quasi-free States and Automorphisma of the CCR Algebra)

Sergej A. Choroszavin

The features of the paper are these: 1) we discuss {\bf especially} quadratic (alias bilinear) Bose-Hamiltonians, the related Bogoliubov transformations and {\bf especially} quasi-free-like (alias coherent or Fock-like) states 2) we discuss {\bf any} quadratic Bose-Hamiltonians and Bogoliubov transformations, whether diagonalizable or not, whether proper or improper, and {\bf arbitrary} quasi-free-like states, whether regular or non-regular they are 3) we associate notions and terms of the CCRs theory {CCRs = Canonical Commutation Relations} with notions and terms of the indefinite inner product spaces theory. Then, we apply the corresponding `bilingual dictionary' so as to construct invariant states of some of the quadratic Hamiltonians.

arXiv:math-ph/0301027•Jan 20, 2003•0 citations

CAR on Fock spaces

Jan Dereziński, Christian Gérard

CAR on Fock spacesThis chapter is devoted to the study of Fock representations of the canonical anticommutation relations, a basic tool of quantum many-body theory and quantum field theory.It is parallel to Chap. 9, where Fock CCR representations were studied.The basic framework is almost the same as in Chap.9. Throughout this chapter Z is a Hilbert space, called the one-particle space.We will consider the Fock CAR representation acting on the fermionic Fock space Γ a (Z).As in Sect.1.3, we introduce the space Y = Re(Z ⊕ Z) := {(z, z) : z ∈ Z}, which will serve as the dual phase space of our system.Recall that in the bosonic case we equipped Y with the structure of the Kähler space consisting of the anti-involution j, the Euclidean scalar product • and the symplectic form ω: j(z, z) := (iz, iz), (13.1) (z, z) • (w, w) := 2Re(z|w), (13.2) (z, z)•ω(w, w) := 2Im(z|w) = -(z, z) • j(w, w).(13.3)In our presentation of the fermionic case, we will need the Kähler anti-involution j.The symplectic form ω will not be used.Instead of the scalar product (13.2) we will use another scalar product,Again we will avoid identifing Z with Y. CY is identified with Z ⊕ Z by the mapY # , the space dual to Y, is canonically identified with Re(Z ⊕ Z) by using the scalar product (9.2), and CY # is identified with Z ⊕ Z. Fock CAR representationConsider the fermionic Fock space Γ a (Z).Recall that, for z ∈ Z, a * (z), resp.a(z) denote the corresponding creation, resp.annihilation operators defined in Sect.3.4.

Cambridge University Press eBooks•Nov 4, 2022•0 citations

Fermionic Dynamical Systems

Robert Alicki, M. Fannes

This chapter focuses on the mathematical description of Fermionic systems. It first describes the usual formalism of second quantization in terms of operators on anti-symmetric Fock space. It then studies quadratic Hamiltonians and their corresponding equilibrium states. It presents a more abstract algebraic approach, suitable for the description of infinite systems: the CAR-formalism. It discusses the main properties of quasi-free states and automorphisms; they are the Fermionic counterparts of classical Gaussian measures and are widely applicable in approximate or effective treatments.

Oxford University Press eBooks•Jul 12, 2001•0 citations

Spin-1 vector boson structure of free spin-1/2 quantum field

Piotr Garbaczewski

Construction of a Fock representation of the CAR algebra in a given Fock representation of the CCR algebra is extended to include physically admissible quantum fields: an example of a free spin-1/2 quantum field constructed from a spin-1 vector boson quantum field is considered. (auth)

•Jan 1, 1975•1 citation

Bosonic Coherent States

Didier Robert, Monique Combescure

In a first part we give a brief presentation of general Fock space setting to describe quantum field theory. Bosons are quantum particles with integer spin and have symmetric wave functions; Fermions are quantum particles with half-integer spin and are represented by anti-symmetric wave functions. The functional setting is given by a symmetric or anti-symmetric tensor product of Hilbert spaces. We describe these spaces and transformations between these spaces. We shall follow the references [23, 42]. Coherent states are defined by translating the vacuum states with the Weyl operators. This is easily done here for bosons. We shall see in the next chapter how to deal with fermions. In a second part we give an interesting application of bosonic coherent states to the study of the classical limit as \(\hbar \searrow 0\) of non-relativistic boson systems with two body interaction in the neighborhood of a solution of the classical system (here the Hartree equation). The classical limit corresponds here to the mean-field limit as the number of particles goes to infinity. As we have done for finite systems, we use here Hepp’s method, which is a linearization procedure of the quantum Hamiltonian around the classical field. The fluctuations around this solution are controlled by a purely quadratic Hamiltonian. In a series of several important papers [114–116] Ginibre and Velo have proven an asymptotic expansion and remainder estimates for these quantum fluctuations. Finally, following the paper [225] one can show that, in the limit \(\hbar \rightarrow 0\), the marginal distribution of the time-evolved coherent states tends in trace-norm to the projector onto the solution of the classical fields equation (Hartree equation) with a uniform remainder estimates in time.

Theoretical and mathematical physics•Jan 1, 2021•0 citations

Classical and quantum quadratic Hamiltonians

Philip Broadbridge

This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real

Bulletin of the Australian Mathematical Society•Jun 1, 1983•1 citation
Suppose that we have a state ψ on the polynomial algebra generated by the fields ϕ ( y ) satisfying the CCR or CAR relations. For simplicity, assume that ψ is even , that is, vanishes on odd polynomials. Clearly, this state determines a bilinear form on Y given by the “2-point function” We say that a state ψ is quasi-free if all expectation values can be expressed in terms of (17.1) by the sum over all pairings. This chapter is devoted to a study of (even) quasi-free states, both bosonic and fermionic. This is an important class of states, often used in physical applications. Fock vacuum states belong to this class. It also includes Gibbs states of quadratic Hamiltonians. Representations obtained by the GNS construction from quasi-free states will be called quasi-free representations . They are usually reducible. Many interesting concepts from the theory of von Neumann algebras can be nicely illustrated in terms of quasi-free representations. Quasi-free states can be easily realized on Fock spaces, using the so-called Araki–Woods, resp. Araki–Wyss representations in the bosonic, resp. fermionic case. Under some additional assumptions, in particular in the case of a finite number of degrees of freedom, these representations can be obtained as follows. First we consider a Fock space equipped with a quadratic Hamiltonian. Then we perform the GNS construction with respect to the corresponding Gibbs state. Finally, we apply an appropriate Bogoliubov rotation.
The features of the paper are these: 1) we discuss {\bf especially} quadratic (alias bilinear) Bose-Hamiltonians, the related Bogoliubov transformations and {\bf especially} quasi-free-like (alias coherent or Fock-like) states 2) we discuss {\bf any} quadratic Bose-Hamiltonians and Bogoliubov transformations, whether diagonalizable or not, whether proper or improper, and {\bf arbitrary} quasi-free-like states, whether regular or non-regular they are 3) we associate notions and terms of the CCRs theory {CCRs = Canonical Commutation Relations} with notions and terms of the indefinite inner product spaces theory. Then, we apply the corresponding `bilingual dictionary' so as to construct invariant states of some of the quadratic Hamiltonians.
CAR on Fock spacesThis chapter is devoted to the study of Fock representations of the canonical anticommutation relations, a basic tool of quantum many-body theory and quantum field theory.It is parallel to Chap. 9, where Fock CCR representations were studied.The basic framework is almost the same as in Chap.9. Throughout this chapter Z is a Hilbert space, called the one-particle space.We will consider the Fock CAR representation acting on the fermionic Fock space Γ a (Z).As in Sect.1.3, we introduce the space Y = Re(Z ⊕ Z) := {(z, z) : z ∈ Z}, which will serve as the dual phase space of our system.Recall that in the bosonic case we equipped Y with the structure of the Kähler space consisting of the anti-involution j, the Euclidean scalar product • and the symplectic form ω: j(z, z) := (iz, iz), (13.1) (z, z) • (w, w) := 2Re(z|w), (13.2) (z, z)•ω(w, w) := 2Im(z|w) = -(z, z) • j(w, w).(13.3)In our presentation of the fermionic case, we will need the Kähler anti-involution j.The symplectic form ω will not be used.Instead of the scalar product (13.2) we will use another scalar product,Again we will avoid identifing Z with Y. CY is identified with Z ⊕ Z by the mapY # , the space dual to Y, is canonically identified with Re(Z ⊕ Z) by using the scalar product (9.2), and CY # is identified with Z ⊕ Z. Fock CAR representationConsider the fermionic Fock space Γ a (Z).Recall that, for z ∈ Z, a * (z), resp.a(z) denote the corresponding creation, resp.annihilation operators defined in Sect.3.4.
This chapter focuses on the mathematical description of Fermionic systems. It first describes the usual formalism of second quantization in terms of operators on anti-symmetric Fock space. It then studies quadratic Hamiltonians and their corresponding equilibrium states. It presents a more abstract algebraic approach, suitable for the description of infinite systems: the CAR-formalism. It discusses the main properties of quasi-free states and automorphisms; they are the Fermionic counterparts of classical Gaussian measures and are widely applicable in approximate or effective treatments.
In a first part we give a brief presentation of general Fock space setting to describe quantum field theory. Bosons are quantum particles with integer spin and have symmetric wave functions; Fermions are quantum particles with half-integer spin and are represented by anti-symmetric wave functions. The functional setting is given by a symmetric or anti-symmetric tensor product of Hilbert spaces. We describe these spaces and transformations between these spaces. We shall follow the references [23, 42]. Coherent states are defined by translating the vacuum states with the Weyl operators. This is easily done here for bosons. We shall see in the next chapter how to deal with fermions. In a second part we give an interesting application of bosonic coherent states to the study of the classical limit as ℏ↘0\hbar \searrow 0ℏ↘0 of non-relativistic boson systems with two body interaction in the neighborhood of a solution of the classical system (here the Hartree equation). The classical limit corresponds here to the mean-field limit as the number of particles goes to infinity. As we have done for finite systems, we use here Hepp’s method, which is a linearization procedure of the quantum Hamiltonian around the classical field. The fluctuations around this solution are controlled by a purely quadratic Hamiltonian. In a series of several important papers [114–116] Ginibre and Velo have proven an asymptotic expansion and remainder estimates for these quantum fluctuations. Finally, following the paper [225] one can show that, in the limit ℏ→0\hbar \rightarrow 0ℏ→0, the marginal distribution of the time-evolved coherent states tends in trace-norm to the projector onto the solution of the classical fields equation (Hartree equation) with a uniform remainder estimates in time.
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real