Interpretation of Quantum Hamiltonian Monodromy in Terms of Lattice Defects
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Interpretation of Quantum Hamiltonian Monodromy in Terms of Lattice Defects B. ZHILINSKII Université du Littoral, UMR du CNRS 8101, 59140 Dunkerque, France. e-mail: [email protected] Abstract. The analogy between monodromy in dynamical (Hamiltonian) systems and defect in crystal lattices is used in order to formulate some general conjectures about possible types of qualitative features of quantum systems which can be interpreted as a manifestation of classical monodromy in quantum finite particle (molecular) problems. Mathematics Subject Classifications (2000): 37J05, 70H33, 81Q70. Key words: monodromy, defects, molecules.
1. Introduction The purpose of this paper is to demonstrate amazing similarity between apparently different subjects: defects of regular periodic lattices, monodromy of classical Hamiltonian integrable dynamical systems, and qualitative features of joint quantum spectra of several commuting observables for quantum finite-particle systems. The physical motivation comes from the qualitative analysis of quantum finite particle systems (molecules). Many important features of the system of quantum eigenstates can be understood if one finds a physically reasonable system of approximate “good quantum numbers” or equivalently a set of mutually commuting operators. In the classical limit such approximate models correspond to integrable models which nevertheless can possess nontrivial qualitative features like Hamiltonian monodromy. Monodromy, which is the topological obstruction to the existence of global action-angle variables, naturally, manifests itself in the joint spectrum of mutually commuting quantum operators. Thus it is important to find the correspondence between typical singularities which are present for classical integrable models and specific patterns in the joint spectra of corresponding commuting observables for associated quantum problems. In the case of a regular classical integrable fibration, the joint spectrum of the corresponding quantum observables can be represented as a system of points labeled by quantum numbers (n1 , . . . , nk ) taking integer values. They form regular lattice of quantum states which can be interpreted as a part of a standard k-dimensional lattice, Z k . In a
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more general case of quantum problems which in the classical limit lead to integrable fibrations with singularities the joint spectrum can be mapped on Z k lattice only locally. Globally, it can be described as a lattice with defects. Following the general idea of solid state physics to describe different defects as a result of “cutting and gluing” of an “ideal” lattice we apply the same construction to the “quantum state lattice” which represents the joint spectrum of a set of mutually commuting operators. First of all we describe several “elementary monodromy” defects which appear naturally in the qualitative description of “quantum state lattices”. Starting with integer monodromy defects we generalize them to fractional monodromy defects and describe their rel
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