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Quantum Bifurcations

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Quantum bifurcations (QB) are qualitative phenomena occurring in quantum systems under the variation of some internal or external parameters. Inorder to make this definition a little more precise we add the additional requirement: The qualitative modification of the “behavior” ofa quantum system can be described as QB if it consists of the manifestation for the quantum system of the classical bifurcation presented inclassical dynamic systems which is the classical analog of the initial quantum system. Quantum bifurcations are typical elementary steps leading from thesimplest in some way effective Hamiltonian to more complicated ones underthe variation of external or internal parameters. As internal parameters one may consider the values of exact or approximate integrals of motion. Theconstruction of an effective Hamiltonian is typically based on theaveraging and/or reduction procedure which results in the appearance of “good” quantum...

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Abbreviations

Classical limit:

The classical limit is the classical mechanical problem which can be constructed from a given quantum problem by some limiting procedure. During such a construction the classical limiting manifold should be defined which plays the role of classical phase space. As soon as quantum mechanics is more general than classical mechanics, going to the classical limit from a quantum problem is much more reasonable than discussing possible quantizations of classical theories [73].

Energy‐momentum map:

In classical mechanics for any problem which allows the existence of several integrals of motion (typically energy and other integrals often named as momenta) the Energy‐Momentum (EM) map gives the correspondence between the phase space of the initial problem and the space of values of all independent integrals of motion. The energy‐momentum map introduces the natural foliation of the classical phase space into common levels of values of energy and momenta [13,35]. The image of the EM map is the region of the space of possible values of integrals of motion which includes regular and critical values. The quantum analog of the image of the energy‐momentum map is the joint spectrum of mutually commuting quantum observables.

Joint spectrum:

For each quantum problem a maximal set of mutually commuting observables can be introduced [16]. A set of quantum wave functions which are mutual eigenfunctions of all these operators exists. Each such eigenfunction is characterized by eigenvalues of all mutually commuting operators. The representation of mutual eigenvalues of n commuting operators in the n‑dimensional space gives the geometrical visualization of the joint spectrum.

Monodromy :

In general, the monodromy characterizes the evolution of some object after it makes a close path around something. In classical Hamiltonian dynamics the Hamiltonian monodromy describes for completely integrable systems the evolution of the first homology group of the regular fiber of the energy‐momentum map after a close path in the regular part of the base space [13].

For a corresponding quantum problem the quantum monodromy describes the modification of the local structure of the joint spectrum after its propagation along a close path going through a regular region of the lattice.

Quantum bifurcation:

Qualitative modification of the joint spectrum of the mutually commuting observables under the variation of some external (or internal) parameters and associated in the classical limit with the classical bifurcation is named quantum bifurcation [59]. In other words the quantum bifurcation is the manifestation of the classical bifurcation presented in the classical dynamic system in the quantum version of the same system.

Quantum‐classical correspondence:

Starting from any quantum problem the natural question consists of defining the corresponding classical limit, i. e. the classical dynamic variables forming the classical phase space and the associated symplectic structure. Whereas in simplest quantum problems defined in terms of standard position and momentum operators with commutation relation \( { [q_i,p_j]= i \hbar\delta_{ij} } \), \( { [q_i,q_j] } {= [p_i,p_j] = 0 } \) (\( { i,j = 1\ldots n } \)) the classical limit phase space is the 2n‑dimensional Euclidean space with standard symplectic structure on it, the topology of the classical limit manifold in many other important for physical applications cases can be rather non‐trivial [73,87].

Quantum phase transition :

Qualitative modifications of the ground state of a quantum system occurring under the variation of some external parameters at zero temperature are named quantum phase transitions [65]. For finite particle systems the quantum phase transition can be considered as a quantum bifurcation [60].

Spontaneous symmetry breaking:

Qualitative modification of the system of quantum states caused by perturbation which has the same symmetry as the initial problem. Local symmetry of solutions decreases but the number of solutions increases. In the energy spectra of finite particle systems the spontaneous symmetry breaking produces an increase of the “quasidegeneracy”, i. e. formation of clusters of quasi‐degenerate levels whose multiplicity can be much higher than the dimension of the irreducible representations of the global symmetry group [51].

Symmetry breaking :

Qualitative changes in the properties (dynamical behavior, and in particular in the joint spectrum) of quantum systems which are due to modifications of the global symmetry of the problem caused by external (less symmetrical than original problem) perturbation can be described as symmetry breaking effects. Typical effects consist of splitting of degenerate energy levels classified initially according to irreducible representation of the initial symmetry group into less degenerate groups classified according to irreducible representation of the subgroup (the symmetry group of the perturbation) [47].

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  1. Université du Littoral, Dunkerque, France

    Boris Zhilinskií

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Zhilinskií, B. (2009). Quantum Bifurcations. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_425

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