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arXiv:2109.05653 · Expositiones Mathematicae

The classical limit and spontaneous symmetry breaking in algebraic quantum theory

Christiaan J. F. van de Ven

Published Feb 24, 2022• 9 citations
arXivHTMLPDF
DOI
10.1016/j.exmath.2022.02.002

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The algebraic properties of a strict deformation quantization are analyzed on the classical phase space R2. The corresponding quantization maps enable us to take the limit for ħ→0 of a suitable sequence of algebraic vector states induced by ħ-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on R2, 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 ∗ algebras) parametrized by ħ. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger 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 R2 depending on the symmetry of the potential. In addition, since this ∗-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 ħ. 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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The algebraic properties of a strict deformation quantization are analyzed on the classical phase space [Formula: see text]. The corresponding quantization maps enable us to take the limit for [Formula: see text] of a suitable sequence of algebraic vector states induced by [Formula: see text]-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on [Formula: see text], 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 [Formula: see text] algebras) parametrized by [Formula: see text]. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger 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 [Formula: see text] depending on the symmetry of the potential. In addition, since this [Formula: see text]-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 [Formula: see text]. 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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The algebraic properties of a strict deformation quantization are analyzed on the classical phase space R2. The corresponding quantization maps enable us to take the limit for ħ→0 of a suitable sequence of algebraic vector states induced by ħ-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on R2, 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 ∗ algebras) parametrized by ħ. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger 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 R2 depending on the symmetry of the potential. In addition, since this ∗-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 ħ. 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.
The algebraic properties of a strict deformation quantization are analyzed on the classical phase space [Formula: see text]. The corresponding quantization maps enable us to take the limit for [Formula: see text] of a suitable sequence of algebraic vector states induced by [Formula: see text]-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on [Formula: see text], 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 [Formula: see text] algebras) parametrized by [Formula: see text]. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger 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 [Formula: see text] depending on the symmetry of the potential. In addition, since this [Formula: see text]-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 [Formula: see text]. 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.
This essay expounds the algebraic framework describing general physical the-ories, within which the phenomenon of spontaneous symmetry breaking (SSB) makes its appearance in infinite quantum systems. This is in contrast with the fact that a large class of theories- both classical and quantum, finite and infinite- are termed, in the conventional account of classical and quantum mechanics, as exhibiting SSB. This discrepancy will be understood in the light of an interpretation that finds the symmetry breaking to be in some respects stronger in the algebraic account than is generally the case in the conventional picture. The case of SSB in the standard account of quantum field theory (QFT) will then be discussed, and it will be argued that, although one would expect a connection with the algebraic account to be possible, this turns out to be problematic. Finally the role of the idealisation of infinite systems, crucial to algebraic SSB, will be dis-cussed.