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In this paper an overview of 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 formalized in a complete algebraic manner. Additionally, since this setting allows for commutative as well as non-commutative $C^*$-algebras, and hence for classical and quantum theories, it provides an excellent framework to study SBB as an emergent phenomenon when transitioning from the quantum to the classical world by turning off a semi-classical parameter. We summarize the main results and show that this algebraic approach sheds new light on the connection between the classical and the quantum realm, where particular emphasis is placed on the role of SSB in Theory versus Nature. To this end a detailed analysis is carried out and illustrated with three different physical models: Schr\"{o}dinger operators, mean-field quantum spin systems and the Bose-Hubbard model., Comment: 30 pages

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 are in bijection with KMS states., 13 pages

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 algebra of sequences invariant under the group generated by a cyclic permutation. Our point of view is similar to the one adopted by Landsman, Moretti and van de Ven, who considered a strict deformation quantization for the case of mean-field theories. However, the methods for a local spin interaction are considerably more involved, due to the presence of a strictly smaller symmetry group., Comment: 38 pages

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 formalization in which convergence of algebraic quantum states to probability measures on phase space (typically a Poisson or symplectic manifold) is studied. In this manner we first prove the existence of the classical limit of Gibbs states illustrated with a class of Schr\"{o}dinger operators in the regime where Planck's constant $\hbar$ appearing in front of the Laplacian approaches zero. We additionally show that the ensuing limit corresponds to the unique probability measure satisfying the so-called classical or static KMS- condition. Subsequently, we conduct a similar study for the free energy of mean-field quantum spin systems in the regime of large particles, and discuss the existence of the classical limit of the relevant Gibbs states., Comment: 39 pages

The algebraic properties of a strict deformation quantization are analysed on the classical phase space $\bR^{2n}$. The corresponding quantization maps enable us to take the limit for $\hbar \to 0$ of a suitable sequence of algebraic vector states induced by $\hbar$-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on $\bR^{2n}$, defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of $C^*$ algebras) parametrized by $\hbar$. The existence of this classical limit is in particular proved for ground states of a wide class of Schr��dinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in $\bR^{2n}$ depending on the symmetry of the potential. In addition, since this $C^*$-algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off $\hbar$. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models., 44 pages, no figures, minor changes, in press in IJGMMP

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 compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of $X_k$ \`{a} la Rieffel, defined by perfectly natural quantization maps $Q_{1/N}: \tilde{A}_0\rightarrow A_{1/N}$ (where $\tilde{A}_0$ is an equally natural dense Poisson subalgebra of $A_0$). We apply this quantization formalism to the Curie--Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its $\mathbb{Z}_2$ symmetry is spontaneously broken in the thermodynamic limit $N\raw\infty$. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space $X_2\cong B^3$ (i.e\ the unit three-ball in $\R^3$). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors $\Psi_N^{(0)}$ of this model as $N\raw\infty$, in which the sequence converges to a probability measure $\mu$ on the associated classical phase space $X_2$. This measure is a symmetric convex sum of two Dirac measures related by the underlying $\Z_2$-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid., Comment: 53 pages, 13 figures/tables