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Birkhäuser Basel eBooks

On a Quantum-Classical Connection, Hidden Symmetries, and a Model of Josephson Junction

Pavel Bóna

Published Jan 1, 1990• 0 citations
Publisher

Abstract

It is shown how can a given kinematical symmetry group G of an infinite quantum system determine a set of classical-mechanical macroscopic variables with a natural Poisson structure on it. For the long-range interactions of the type introduced by Hepp and Lieb [Helvetica Phys.Acta 46, 573 (1973)] the dynamics of the large quantal system [P.Bona, J.Math.Phys. 29, 2223(1988)] leads to the classical Hamiltonian motion of the macroscopic quantities determined by a Hamilton function specified uniquely by the local quantal Hamiltonians. A model of Josephson junction is a nontrivial integrable example of this class of systems. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0348-7306-2_36

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On obtaining classical mechanics from quantum mechanics

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It is shown how can a given kinematical symmetry group G of an infinite quantum system determine a set of classical-mechanical macroscopic variables with a natural Poisson structure on it. For the long-range interactions of the type introduced by Hepp and Lieb [Helvetica Phys.Acta 46, 573 (1973)] the dynamics of the large quantal system [P.Bona, J.Math.Phys. 29, 2223(1988)] leads to the classical Hamiltonian motion of the macroscopic quantities determined by a Hamilton function specified uniquely by the local quantal Hamiltonians. A model of Josephson junction is a nontrivial integrable example of this class of systems. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
An infinite quantum system with correctly defined dynamics τQ as an automorphism group of a C*-algebra 𝒞 of observables is determined by any continuous unitary representation U(G) of a connected Lie group G, as well as by an arbitrary differentiable real function Q on the dual space 𝔤* to the Lie algebra 𝔤 of G with the canonically defined Poisson flow φQ on 𝔤*. For specific choices of Q and G, the system can be obtained as the thermodynamic limit of a net of finite lattice systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)]. A simple nontrivial model of this type is the quasispin BCS model of superconductivity in the strong coupling limit, or a corresponding model of the Josephson junction. A peculiar feature of the considered models is τQ noninvariance of the usually considered C*-algebra 𝒜 of quasilocal observables, as well as an important role of classical dynamics of a set of macroscopic (intensive) observables Qφ in the description of τQ. The work is restricted here to norm-continuous representations U(G), in which case 𝒞 is isomorphic to the tensor product 𝒜⊗𝒩, where 𝒩 is the commutative algebra of classical (intensive) observables of the considered infinite quantum system.
The general structure and existence of equilibrium and ground states of infinite spin systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)] is investigated. The form and the role of the ‘‘Bogoliubov–Haag’’ Hamiltonians is cleared. The analysis is based on a recent mathematically correct definition of the corresponding time automorphism group τQ (specified by a classical Hamiltonian Q) of an appropriately defined ‘‘physical’’ C*-algebra of observables 𝒞 containing an abelian subalgebra 𝒩 of classical macroscopic quantities as well as the conventional ‘‘bare’’ quasilocal (sub-) algebra 𝒜, 𝒞=𝒜⊗𝒩 [P. Bóna, J. Math. Phys. 29, 2223 (1988)]. A Lie group G of ‘‘hidden symmetries’’ acting nontrivially (also) on 𝒩 plays an essential role in the presented theory. The applications to the quasispin strong-coupling version of the BCS model of superconductivity, and to the corresponding model of the Josephson junction illustrate possibilities of the developed formalism as well as an easy manipulation with it. A simple mathematical description of ‘‘symmetry breaking’’ is sketched for the considered class of mathematical models of large quantal systems.
During the last three decades, Pavel Bóna developed a non-linear generalization of quantum mechanics, which is based on symplectic structures for normal states. One important application of such a generalization is a general setting that is very convenient to study the emergence of macroscopic classical dynamics from microscopic quantum processes. We propose here a new mathematical approach to Bóna’s non-linear quantum mechanics. It is based on C0-semigroup theory and has a domain of applicability that is much broader than Bóna’s original one. It highlights the central role of self-consistency. This leads to a mathematical framework in which the classical and quantum worlds are naturally entangled. In this new mathematical approach, we build a Poisson bracket for the polynomial functions on the Hermitian weak*-continuous functionals on any C*-algebra. This is reminiscent of a well-known construction for finite-dimensional Lie algebras. We then restrict this Poisson bracket to states of this C*-algebra by taking quotients with respect to Poisson ideals. This leads to densely defined symmetric derivations on the commutative C*-algebras of real-valued functions on the set of states. Up to a closure, these are proven to generate C0-groups of contractions. As a matter of fact, in generic commutative C*-algebras, even the closableness of unbounded symmetric derivations is a non-trivial issue. Some new mathematical concepts are introduced, which are possibly interesting by themselves: the convex weak* Gâteaux derivative and the state-dependent C*-dynamical systems. Our recent results on macroscopic dynamical properties of lattice-fermion and quantum-spin systems with long-range, or mean-field, interactions corroborate the relevance of the general approach we present here.
Constructing a classical mechanical system associated with a given quantum-mechanical one entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum-mechanical system naturally has the structure of an infinite-dimensional symplectic manifold ('quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straightforwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and one recovers the linear classical phase space . We report on how the procedure also allows extraction of nonlinear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin- j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (polymer quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality.
After recalling the crucial role of first integrals (or constants of the motion) in classical and quantum dynamics, we show why the concept of martingale is their natural probabilistic counterpart. This is used in the framework of Euclidean Quantum Mechanics, a probabilistic approach to quantization. The relevant Theorem of Noether is shown; it is founded on a new probabilistic interpretation of the symmetry group of some heat equations. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum mechanical system naturally has the structure of an infinite dimensional symplectic manifold (`quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straight forwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and recovers the linear classical phase space mathbbRmathrm2N\\mathbb{R}^{\\mathrm{2N}}mathbbRmathrm2N. We report on how the procedure also allows extraction of non-linear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (Bohr quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality. There is also an alternative and a priori distinct notion of effective Hamiltonian used in the context of a WKB approximation. These two notions are compared for the example of loop quantum cosmology.