Christiaan van de Ven

University of Tübingen | EKU Tübingen

Spontaneous symmetry breaking (SSB) is mathematically tied to some limit, but must physically occur, approximately, before the limit. Approximate SSB has been independently understood for Schroedinger operators with double well potential in the classical limit (Jona-Lasinio et al, 1981; Simon, 1985) and for quantum spin systems in the thermodynamic...

Continuous fields (or bundles) of $C^*$-algebras form an important ingredient for describing emergent phenomena, such as phase transitions and spontaneous symmetry breaking. In this work, we consider the continuous $C^*$-bundle generated by increasing symmetric tensor powers of the complex $\ell\times\ell$ matrices $M_\ell(\mathbb{C})$, which can b...

We study the classical and quantum KMS conditions within the context of spin lattice systems. Specifically, we define a strict deformation quantization (SDQ) for a S^2-valued spin lattice system over Z^d generalizing the renown Berezin SDQ for a single sphere. This allows to promote a classical dynamics on the algebra of classical observables to a...

We study the classical and quantum KMS conditions within the context of spin lattice systems. Specifically, we define a strict deformation quantization (SDQ) for a $\mathbb{S}^2$-valued spin lattice system over $\mathbb{Z}^d$ generalizing the renown Berezin SDQ for a single sphere. This allows to promote a classical dynamics on the algebra of class...

We define a strict deformation quantization which is compatible with any Hamiltonian with local spin interaction (e.g. the Heisenberg Hamiltonian) for a spin chain. This is a generalization of previous results known for mean-field theories. The main idea is to study the asymptotic properties of a suitably defined algebra of sequences invariant unde...

We construct C*-dynamical systems for the dynamics of classical infinite particle systems describing harmonic oscillators interacting with arbitrarily many neighbors on lattices, as well on more general structures. Our approach allows particles with varying masses, varying frequencies, irregularly placed lattice sites and varying interactions subje...

A continuous bundle of [Formula: see text]-algebras provides a rigorous framework to study the thermodynamic limit of quantum theories. If the bundle admits the additional structure of a strict deformation quantization (in the sense of Rieffel) one is allowed to study the classical limit of the quantum system, i.e. a mathematical formalism that exa...

The existence of various physical phenomena stems from the concept called asymptotic emergence , that is, they seem to be exclusively reserved for certain limiting theories. Important examples are spontaneous symmetry breaking (SSB) and phase transitions: these would only occur in the classical or thermodynamic limit of underlying finite quantum sy...

The Dobrushin–Lanford–Ruelle condition (Dobrushin in Theory Prob Appl 17:582–600, 1970. https://doi.org/10.1137/1115049 ; Lanford and Ruelle in Commun Math Phys 13:194–215, 1969. https://doi.org/10.1007/BF01645487 ) and the classical Kubo–Martin–Schwinger (KMS) condition (Gallavotti and Verboven in Nuov Cim B 28:274–286, 1975. https://doi.org/10.10...

The Dobrushin-Lanford-Ruelle (DLR) condition and the classical Kubo-Martin-Schwinger (KMS) condition are considered in the context of classical lattice systems. In particular, we prove that these conditions are equivalent for the case of a lattice spin system with values in a compact symplectic manifold by showing that infinite volume Gibbs states...

A continuous bundle of $C^*$-algebras provides a rigorous framework to study the thermodynamic limit of quantum theories. If the bundle admits the additional structure of a strict deformation quantization (in the sense of Rieffel) one is allowed to study the classical limit of the quantum system, i.e. a mathematical formalism that examines the conv...

We define a strict deformation quantization which is compatible with any Hamiltonian with local spin interaction (e.g. the Heisenberg Hamiltonian) for a spin chain with periodic boundary conditions. This is a generalization of previous results known for mean-field theories. The main idea is to study the asymptotic properties of a suitably defined a...

The existence of various physical phenomena stems from the concept called asymptotic emergence, that is, they seem to be exclusively reserved for certain limiting theories. Important examples are spontaneous symmetry breaking (SSB) and phase transitions: these would only occur in the classical or thermodynamic limit of underlying finite quantum sys...

The aim of this paper is two-fold. Firstly we provide necessary and sufficient criteria for the existence of a strict deformation quantization of algebraic tensor products of Poisson algebras, and secondly we discuss the existence of products of KMS states. As an application, we discuss the correspondence between quantum and classical Hamiltonians...

In this paper an overview on some recent developments on the classical limit and spontaneous symmetry breaking (SSB) in algebraic quantum theory is given. In such works, based on the theory of C∗-algebras, the concept of the classical limit has been formalised in a complete algebraic manner. Additionally, since this setting allows for commutative a...

The algebraic properties of a strict deformation quantization are analyzed on the classical phase space [Formula: see text]. The corresponding quantization maps enable us to take the limit for [Formula: see text] of a suitable sequence of algebraic vector states induced by [Formula: see text]-dependent eigenvectors of several quantum models, in whi...

In this paper an overview on some recent developments on the classical limit and spontaneous symmetry breaking (SSB) in algebraic quantum theory is given. To this end, a neat framework based on the theory of C* algebras is introduced. Consequently, a novel point of view on the concept of the classical limit and the natural emergent phenomenon of sp...

The algebraic properties of a strict deformation quantization are analyzed on the classical phase space $\mathbb{R}^{2n}$. The corresponding quantization maps enable us to take the limit for $\hbar \to 0$ of a suitable sequence of algebraicvector states induced by $\hbar$-dependent eigenvectors of several quantum models, in which the sequence conve...

The theory of strict deformation quantization of the two-sphere S2⊂R3 is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by HN, where N indicates the number of sites. Indeed, since the fibers A1/N=MN+1(C) and A0 = C(S²) form a continuous bundle of C*-algebras over the base...

The existence of a strict deformation quantization of Xk=S(Mk(C)), the state space of the k×k matrices Mk(C) which is canonically a compact Poisson manifold (with stratified boundary), has recently been proved by both authors and Landsman (Rev Math Phys 32:2050031, 2020. 10.1142/S0129055X20500312). In fact, since increasing tensor powers of the k×k...

Since the fibers $A_{1/N}=M_{N+1}(\mathbb{C})\cong B(\text{Sym}^N(\mathbb{C}^2))$ and $A_0=C(S^2)$ form a continuous bundle of $C^*$-algebras over the base space $I=\{0\}\cup 1/\mathbb{N}\subset[0,1]$, one can define a strict deformation quantization of $A_0$ where quantization is specified by certain quantization maps $Q_{1/N}: \tilde{A}_0 \righta...

The existence of a strict deformation quantization of $X_k=S(M_k({\mathbb{C}}))$, the state space of the $k\times k$ matrices $M_k({\mathbb{C}})$ which is canonically a compact Poisson manifold (with stratified boundary) has recently been proven by both authors and K. Landsman \cite{LMV}. In fact, since increasing tensor powers of the $k\times k$ m...

Increasing tensor powers of the [Formula: see text] matrices [Formula: see text] are known to give rise to a continuous bundle of [Formula: see text]-algebras over [Formula: see text] with fibers [Formula: see text] and [Formula: see text], where [Formula: see text], the state space of [Formula: see text], which is canonically a compact Poisson man...

Increasing tensor powers of the $k\times k$ matrices $M_k({\mathbb{C}})$ are known to give rise to a continuous bundle of $C^*$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset[0,1]$ with fibers $A_{1/N}=M_k({\mathbb{C}})^{\otimes N}$ and $A_0=C(X_k)$, where $X_k=S(M_k({\mathbb{C}}))$, the state space of $M_k({\mathbb{C}})$, which is canonically a co...