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Coherent states on the circle

Goce Chadzitaskos, P Luft, J. Tolar

Published Aug 14, 2016• 5 citations
OpenAlex

Abstract

We present a possible construction of coherent states on the unit circle as configuration space. In our approach the phase space is the product 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 over a finite periodic chain where the phase space was Z_M x Z_M. Properties of the coherent states constructed in this way are studied and the coherent states are shown to satisfy the resolution of unity.

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We present a possible construction of coherent states on the unit circle as configuration space. In our approach the phase space is the product × 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 over a finite periodic chain where the phase space was M × M . Properties of the coherent states constructed in this way are studied and the coherent states are shown to satisfy the resolution of unity.

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We present a possible construction of coherent states on the unit circle as configuration space. In our approach the phase space is the product 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 over a finite periodic chain where the phase space was Z_M x Z_M. Properties of the coherent states constructed in this way are studied and the coherent states are shown to satisfy the resolution of unity.
We present a possible construction of coherent states on the unit circle as configuration space. In our approach the phase space is the product × 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 over a finite periodic chain where the phase space was M × M . Properties of the coherent states constructed in this way are studied and the coherent states are shown to satisfy the resolution of unity.
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.
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'.
Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.
In this paper, the Higgs-like approach is used to analyze the quantum dynamics of a harmonic oscillator constrained on a circle. We obtain the Hamiltonian of this system as a function of the Cartesian coordinate of the tangent line through the gnomonic projection and then quantize it in the standard way. We then recast the Hamiltonian in a shape-invariant form and derive the spectrum energy of the confined harmonic oscillator on the circle. With help of the f-deformed oscillator algebra, we construct the coherent states on the circle and investigate their quantum statistical properties. We find that such states show nonclassical features like squeezing and sub-Poissonian statistics even in small curvatures of the circle.
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.