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
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.

Strict quantization deformation, Spontaneous Symmetry Breacking