The classical limit of Schrödinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as emergent phenomenon

Valter Moretti, Christiaan J. F. van de Ven

Published 2021-03-22Version 1

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 converges to a probability measure on $\mathbb{R}^{2n}$, 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 proven 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 $\mathbb{R}^{2n}$ depending on the symmetry of the potential. This yields a notion of spontaneous symmetry breaking (SSB) as an emergent phonomenon when passing from the quantum realm to the classical world by switching off $\hbar$. A detailed mathematical description is outlined, and it is shown how the present algebraic approach sheds new light on the connection between quantum and classical theory.