# The classical limit of SchrÃ¶dinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as emergent phenomenon

March 22, 2021

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.

Keywords:

Strict quantization deformation, Spontaneous Symmetry Breacking