The classical limit of Schr\"{o}dinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as emergent phenomenon

Abstract

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\"{o}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.