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arXiv:quant-ph/0702180 · Journal of Physics A Mathematical and Theoretical

Dihedral symmetry of periodic chain: quantization and coherent states

P Luft, G Chadzitaskos, J Tolar

Published Apr 17, 2007• 0 citations
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Abstract

Our previous work on quantum kinematics and coherent states over finite configuration spaces is extended: the configuration space is, as before, the cyclic group Z_n of arbitrary order n=2,3,..., but a larger group - the non-Abelian dihedral group D_n - is taken as its symmetry group. The corresponding group related coherent states are constructed and their overcompleteness proved. Our approach based on geometric symmetry can be used as a kinematic framework for matrix methods in quantum chemistry of ring molecules.

DOI
10.1088/1751-8113/40/18/010

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Our previous work on quantum kinematics and coherent states over finite configuration spaces is extended: the configuration space is, as before, the cyclic group Z_n of arbitrary order n=2,3,..., but a larger group - the non-Abelian dihedral group D_n - is taken as its symmetry group. The corresponding group related coherent states are constructed and their overcompleteness proved. Our approach based on geometric symmetry can be used as a kinematic framework for matrix methods in quantum chemistry of ring molecules.
Our previous work on quantum kinematics and coherent states over finite configuration spaces is extended: the configuration space is, as before, the cyclic group ZnZ_{n}Zn​ of arbitrary order n=2,3,..., but as its symmetry group a larger group -- the non-Abelian dihedral group DnD_{n}Dn​ -- is taken. The corresponding group related coherent states are constructed and their overcompleteness proved. Our approach based on geometric symmetry can be used as a kinematic framework for matrix methods in quantum chemistry of ring molecules.
We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S 1 including the Aharonov–Bohm-type quantum description. Coherent states are constructed by Perelomov's method as group-related coherent states generated by Weyl operators on the quantum phase space . Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction, we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was . The coherent states constructed in this work are shown to satisfy the resolution of unity. To compare them with canonical coherent states, some of their further properties are also studied demonstrating similarities as well as substantial differences. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view. For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups. This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics. Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N. The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra. The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.
Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view.For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups.This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics.Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N .The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra.The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.
We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S^1 including the Aharonov-Bohm type quantum description. The coherent states are constructed by Perelomov's method as group related coherent states generated by Weyl operators on the quantum phase space Z x S^1. Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was Z_M x Z_M. The coherent states constructed in this work are shown to satisfy the resolution of unity. To compare them with canonical coherent states, also some of their further properties are studied demonstrating similarities as well as substantial differences.