The classical limit of Schr\"{o}dinger operators in the framework of
Berezin quantization and spontaneous symmetry breaking as emergent phenomenon
- URL: http://arxiv.org/abs/2103.11914v2
- Date: Mon, 4 Oct 2021 11:37:16 GMT
- Title: The classical limit of Schr\"{o}dinger operators in the framework of
Berezin quantization and spontaneous symmetry breaking as emergent phenomenon
- Authors: Valter Moretti and Christiaan J.F.van de Ven
- Abstract summary: A strict deformation quantization is analysed on the classical phase space $bR2n$.
The existence of this classical limit is in particular proved for ground states of a wide class of Schr"odinger operators.
The support of the classical state is included in certain orbits in $bR2n$ depending on the symmetry of the potential.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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