The classical limit and spontaneous symmetry breaking in algebraic quantum theory

Abstract

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 $C^{\ast }$-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 $C^{\ast }$-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: Schrödinger operators, mean-field quantum spin systems and the Bose-Hubbard model.

arXiv comment30 pages

References in corpus (14)

• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$
• Understanding quantum measurement from the solution of dynamical models
• Less is Different: Emergence and Reduction Reconciled
• Symmetry Breaking and Finite Size Effects in Quantum Many-Body Systems
• Spontaneous Symmetry Breaking in Quantum Mechanics
• Coherent States of the SU(N) groups
• Quantum Dynamics in the Thermodynamic Limit
• Long-range order, "tower" of states, and symmetry breaking in lattice quantum systems
• The Phase Diagram of the Quantum Curie-Weiss Model
• Strict deformation quantization of the state space of $M_k(\mathbb{C})$ with applications to the Curie-Weiss model
• Quantum spin systems versus Schroedinger operators: A case study in spontaneous symmetry breaking
• Ground states for mean field models with a transverse component
• Bulk-boundary asymptotic equivalence of two strict deformation quantizations
• Injective tensor products in strict deformation quantization

Cited by in corpus (3)

• DLR-KMS correspondence on lattice spin systems
• Injective tensor products in strict deformation quantization
• Strict deformation quantization and local spin interactions