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arXiv:2103.11914 · International Journal of Geometric Methods in Modern Physics

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

Valter Moretti, Christiaan J. F. van de Ven

Published Nov 30, 2021• 0 citations
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Abstract

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.

DOI
10.1142/s0219887822500037

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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.
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 theory of strict deformation quantization of the two-sphere S2⊂R3 is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by HN, where N indicates the number of sites. Indeed, since the fibers A1/N=MN+1(C) and A0 = C(S2) form a continuous bundle of C*-algebras over the base space I={0}∪1/N*⊂[0,1], one can define a strict deformation quantization of A0 where quantization is specified by certain quantization maps Q1/N:Ã0→A1/N, with Ã0 being a dense Poisson subalgebra of A0. Given now a sequence of such HN, we show that under some assumptions, a sequence of eigenvectors ψN of HN has a classical limit in the sense that ω0(f) ≔ limN→∞⟨ψN, Q1/N(f)ψN⟩ exists as a state on A0 given by ω0(f)=1n∑i=1nf(Ωi), where n is some natural number. We give an application regarding spontaneous symmetry breaking, and moreover, we show that the spectrum of such a mean-field quantum spin system converges to the range of some polynomial in three real variables restricted to the sphere S2.
In this paper we try to formulate the Berezin quantization on projective Hilbert space P(H) and use its geometric structure. It will be shown that the star product in Berezin quantiza- tion is equivalent to the Poisson bracket on P(H) and the Berezin method to construction a correspondence between a given classical theory and a given quantum theory is used to define a classical limit for geometric quantum mechanics. the covariant symbols. These symbols are expectation values of the observables in terms of coherent states: the holomorphic functions on classical phase space M that is assumed to be aKmanifold. Berezin (2) showed that the covariant symbols form a ∗-algebra which in limit � → 0l eads to the Poisson algebra between the corresponding classical observables: The functions on the phase space M. In this paper we will see that the Berezin ∗-algebra is in fact a Poisson algebra which is induced by the Fubini-Study 2-form on space of coherent states. This space is defined as follows: coherent states span a dense subspace H of Hilbert space H. P( H), which is denoted by M, is aKmanifold with induced symplectic structure from P(H). Therefore the covariant symbols can be considered as functions on M. It is shown (13) that there exists an embedding mapping between the classical phase space M and M, by which to any point z ∈ M is associated ap ointZ ∈M. With this construction to all of the quantum observables are associated their covariant symbols, which form a Poisson algebra on M and since the corresponding classical observables form a Poisson algebra on M , the Berezin quantization is a systematic procedure to relate these two Poisson algebras. Also the relation of Berezin quantization and geometric for- mulation of quantum mechanics will be evident as follows. The geometric quantum mechanics is a formulation of quantum mechanics in projective Hilbert spaces. With our construction one sees that the Berezin quantization is an equivalent formulation and in addition gives a prescription as classical limit for geometric quantum mechanics (9).
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.
Abstract Berezin and Weyl quantization are renowned procedures for mapping commutative Poisson algebras of observables to their non-commutative, quantum counterparts. The latter is famous for its use on Weyl algebras, while the former is more appropriate for continuous functions decaying at infinity. In this work, we define a variant of the Berezin quantization map, which acts on the classical Weyl algebra W(E,0)\mathcal {W}(E,0)W(E,0) W ( E , 0 ) and constitutes a positive strict deformation quantization . This construction provides a natural framework to compare classical and quantum thermal equilibrium states of a Bose gas through the computation of their semi-classical limit. To this end, we first introduce a purely algebraic notion of KMS states for the classical Weyl algebra and establish that, in finite volume, there exists a unique such state, which can be interpreted as the Fourier transform of a Gibbs measure on a Hilbert space. We then construct a new class of classical KMS states that realize representations of the canonical commutation relations with infinite local density. These states arise as the semi-classical high-density limit of the quantum equilibrium states originally studied by Araki and Woods [5]. A key feature of our approach is that it preserves the macroscopic ground-state occupation of the Bose gas in the classical regime. Finally, we demonstrate that the infinite-volume classical states can be obtained as thermodynamic limits of finite-volume Gibbs states.
In this paper we try to formulate the Berezin quantization on projective Hilbert space P(H) and use its geometric structure to construct a correspondence between a given classical theory and a given quantum theory. It wil be shown that the star product in berezin quantization is equivalent to the Posson bracket on coherent states manifold M, embodded in P(H), and the Berezin method is used to define a classical limit for geometric quantum mechnics. With this construction to all of the quantum observables are associated their covariant symbols, which form a poisson algebra on P(H) and since the corresponding classical phase space has a natural Poisson structure, the Berezin quantization is then a systematic procedure to relate these tow piosson algebras.