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Annales De L Institut Henri Poincare-physique Theorique

Geometry of the Kepler system in coherent states approach

Maciej Horowski, Anatol Odzijewicz

Published Jan 1, 1993• 9 citations
OpenAlex

Abstract

We consider a family of holomorphic embeddings K f : C 3 → CP(M f ) of nondegenerate quadric C 3 in C 4 into complex projective Hilbert space CP(M f ) parametrized by functions f ∈ F 3 ⊂ C∞ (R + , R). We show that there is a natural correspondence between this and the Kepler system. Taking this into account we give a complete classical as well as quantum description of the system in terms of the map K f . In this way we illustrate in the paper a nontrivial application of the theory developed in our earlier publications

Related papers

Symbol calculus for the Kepler problem

S. A. Pol'shin

We construct the system of generalized coherent states for the quantum Kepler problem corresponds to the homogeneous domain $SU(2,2)/S(U(2)\times U(2))$. We show that the SU(2,2)-equivariant momentum map for this domain yields the momentum map for the classical Kepler problem via appropriate limiting passage. We also show that under this passage the average values of quantum observables in this system of coherent states pass into the functions on classical phase space and $-i/h$ times commutator pass into the Poisson bracket.

arXiv:math-ph/0505072•May 26, 2005•0 citations

Geometric Formulation of Berezin Quantization

Rasoul Rokhnizadeh, H. D. Doebner

In this paper we try to formulate the Berezin quantization on projective Hilbert space P(H) and use its geometric structure. It will be shown that the star product in Berezin quantiza- tion is equivalent to the Poisson bracket on P(H) and the Berezin method to construction a correspondence between a given classical theory and a given quantum theory is used to define a classical limit for geometric quantum mechanics. the covariant symbols. These symbols are expectation values of the observables in terms of coherent states: the holomorphic functions on classical phase space M that is assumed to be aKmanifold. Berezin (2) showed that the covariant symbols form a ∗-algebra which in limit � → 0l eads to the Poisson algebra between the corresponding classical observables: The functions on the phase space M. In this paper we will see that the Berezin ∗-algebra is in fact a Poisson algebra which is induced by the Fubini-Study 2-form on space of coherent states. This space is defined as follows: coherent states span a dense subspace H of Hilbert space H. P( H), which is denoted by M, is aKmanifold with induced symplectic structure from P(H). Therefore the covariant symbols can be considered as functions on M. It is shown (13) that there exists an embedding mapping between the classical phase space M and M, by which to any point z ∈ M is associated ap ointZ ∈M. With this construction to all of the quantum observables are associated their covariant symbols, which form a Poisson algebra on M and since the corresponding classical observables form a Poisson algebra on M , the Berezin quantization is a systematic procedure to relate these two Poisson algebras. Also the relation of Berezin quantization and geometric for- mulation of quantum mechanics will be evident as follows. The geometric quantum mechanics is a formulation of quantum mechanics in projective Hilbert spaces. With our construction one sees that the Berezin quantization is an equivalent formulation and in addition gives a prescription as classical limit for geometric quantum mechanics (9).

Presented at•Jan 1, 2001•0 citations

Coherent states of SU(l, 1) groups

D M Gitman, A L Shelepin

This work can be considered as a continuation of our previous study, in which an explicit form of coherent states (cs) for all SU(N) groups was constructed by means of representations on polynomials. Here we extend that approach to any SU(l, 1) group and construct explicitly corresponding cs. The cs are parametrized by dots of a coset space, which is, in that particular case, the open complex ball CD l . This space together with the projective space CP l , which parametrizes the cs of the SU(l + 1) group, exhaust all complex spaces of constant curvature. Thus, both sets of cs provide a possibility for an explicit analysis of the quantization problem on all the spaces of constant curvature. This is why the cs of the SU(N) and SU(l, 1) groups are of importance in connection with quantization theory. The constructed cs form an overcompleted system in the representation space and, as quantum states possessing a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator. The classical limit is investigated in terms of symbols of operators; the limit of the so called star commutator of the symbols generates the Poisson bracket in CD l , the latter plays the role of the phase space for the corresponding classical mechanics.

arXiv:hep-th/9308157•Journal of Physics A Mathematical and General•Dec 7, 1993•18 citations

The Geometry Underlying the Quantum Harmonic Oscillator

Alexander D. Popov

We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with $T^*{\mathbb R}^{2}={\mathbb C}^2$ as classical phase space. We show that the eigenfunctions $ψ_n$ of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space ${\mathbb C}^2/{\mathbb Z}_n\subset{\mathbb C}^2$. They describe ${\mathbb Z}_n$-invariant motion of particle along a circle $S^1$ in lens space $S^3/{\mathbb Z}_n\subset{\mathbb C}^2/{\mathbb Z}_n$, where ${\mathbb Z}_n$ is the cyclic group of rotation by an angle $2π/n$ on the circle $S^1$, $n=1,2,...\,$. Thus the general solution of the Schrödinger equation carries information about an infinite number of admissible classical states $ψ_n$ that can be mapped to other states after lifting into the quantum bundle. We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.

math-phCross-listedarXiv:2604.21373•Apr 24, 2026•0 citations

Coherent states and the Kepler-Coulomb problem

Christopher C. Gerry

The Kustaanheino-Stiefel transformation is used to transform the Coulomb problem to a four-dimensional isotropic harmonic oscillator with a constraint. Ordinary coherent states are introduced over this oscillator and are shown to evolve in a fictitious time. When projected back into the physical space, the states follow the classical Kepler orbits in physical time. The resulting quasiclassical picture is obtained without the Bohr correspondence principle of taking only contributions from high principal quantum numbers.

Physical Review A•Jan 1, 1986•76 citations

Coherent states of SU(N) groups

Dmitri M Gitman, A. L. Shelepin

An explicit and uniform construction for coherent states (CS) for all the SU(N) groups is given and, on this basis, their properties are investigated. The CS are parametrized by the dots of a coset space, which is, in this particular case, the projective space CP N-1 and which plays the role of the phase space in the corresponding classical mechanics. The logarithm of the modulus of the CS overlap, being interpreted as assymmetric in the space, gives the Fubini-Study metric in CP N-1 . The classical limit is investigated in terms of operator symbols. h=P -1 (where P is the signature of the representation) plays the role of Planck's constant. The classical limit of the so called star commutator of the symbols generates the Poisson bracket in the corresponding phase space. The CS form an overcompleted system in the representation space and, as quantum states possess a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator.

arXiv:hep-th/9208017•Journal of Physics A Mathematical and General•Jan 21, 1993•39 citations
We consider a family of holomorphic embeddings K f : C 3 → CP(M f ) of nondegenerate quadric C 3 in C 4 into complex projective Hilbert space CP(M f ) parametrized by functions f ∈ F 3 ⊂ C∞ (R + , R). We show that there is a natural correspondence between this and the Kepler system. Taking this into account we give a complete classical as well as quantum description of the system in terms of the map K f . In this way we illustrate in the paper a nontrivial application of the theory developed in our earlier publications
We construct the system of generalized coherent states for the quantum Kepler problem corresponds to the homogeneous domain SU(2,2)/S(U(2)×U(2))SU(2,2)/S(U(2)\times U(2))SU(2,2)/S(U(2)×U(2)). We show that the SU(2,2)-equivariant momentum map for this domain yields the momentum map for the classical Kepler problem via appropriate limiting passage. We also show that under this passage the average values of quantum observables in this system of coherent states pass into the functions on classical phase space and −i/h-i/h−i/h times commutator pass into the Poisson bracket.
In this paper we try to formulate the Berezin quantization on projective Hilbert space P(H) and use its geometric structure. It will be shown that the star product in Berezin quantiza- tion is equivalent to the Poisson bracket on P(H) and the Berezin method to construction a correspondence between a given classical theory and a given quantum theory is used to define a classical limit for geometric quantum mechanics. the covariant symbols. These symbols are expectation values of the observables in terms of coherent states: the holomorphic functions on classical phase space M that is assumed to be aKmanifold. Berezin (2) showed that the covariant symbols form a ∗-algebra which in limit � → 0l eads to the Poisson algebra between the corresponding classical observables: The functions on the phase space M. In this paper we will see that the Berezin ∗-algebra is in fact a Poisson algebra which is induced by the Fubini-Study 2-form on space of coherent states. This space is defined as follows: coherent states span a dense subspace H of Hilbert space H. P( H), which is denoted by M, is aKmanifold with induced symplectic structure from P(H). Therefore the covariant symbols can be considered as functions on M. It is shown (13) that there exists an embedding mapping between the classical phase space M and M, by which to any point z ∈ M is associated ap ointZ ∈M. With this construction to all of the quantum observables are associated their covariant symbols, which form a Poisson algebra on M and since the corresponding classical observables form a Poisson algebra on M , the Berezin quantization is a systematic procedure to relate these two Poisson algebras. Also the relation of Berezin quantization and geometric for- mulation of quantum mechanics will be evident as follows. The geometric quantum mechanics is a formulation of quantum mechanics in projective Hilbert spaces. With our construction one sees that the Berezin quantization is an equivalent formulation and in addition gives a prescription as classical limit for geometric quantum mechanics (9).
This work can be considered as a continuation of our previous study, in which an explicit form of coherent states (cs) for all SU(N) groups was constructed by means of representations on polynomials. Here we extend that approach to any SU(l, 1) group and construct explicitly corresponding cs. The cs are parametrized by dots of a coset space, which is, in that particular case, the open complex ball CD l . This space together with the projective space CP l , which parametrizes the cs of the SU(l + 1) group, exhaust all complex spaces of constant curvature. Thus, both sets of cs provide a possibility for an explicit analysis of the quantization problem on all the spaces of constant curvature. This is why the cs of the SU(N) and SU(l, 1) groups are of importance in connection with quantization theory. The constructed cs form an overcompleted system in the representation space and, as quantum states possessing a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator. The classical limit is investigated in terms of symbols of operators; the limit of the so called star commutator of the symbols generates the Poisson bracket in CD l , the latter plays the role of the phase space for the corresponding classical mechanics.
We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with T∗R2=C2T^*{\mathbb R}^{2}={\mathbb C}^2T∗R2=C2 as classical phase space. We show that the eigenfunctions ψnψ_nψn​ of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space C2/Zn⊂C2{\mathbb C}^2/{\mathbb Z}_n\subset{\mathbb C}^2C2/Zn​⊂C2. They describe Zn{\mathbb Z}_nZn​-invariant motion of particle along a circle S1S^1S1 in lens space S3/Zn⊂C2/ZnS^3/{\mathbb Z}_n\subset{\mathbb C}^2/{\mathbb Z}_nS3/Zn​⊂C2/Zn​, where Zn{\mathbb Z}_nZn​ is the cyclic group of rotation by an angle 2π/n2π/n2π/n on the circle S1S^1S1, n=1,2,... n=1,2,...\,n=1,2,.... Thus the general solution of the Schrödinger equation carries information about an infinite number of admissible classical states ψnψ_nψn​ that can be mapped to other states after lifting into the quantum bundle. We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.
The Kustaanheino-Stiefel transformation is used to transform the Coulomb problem to a four-dimensional isotropic harmonic oscillator with a constraint. Ordinary coherent states are introduced over this oscillator and are shown to evolve in a fictitious time. When projected back into the physical space, the states follow the classical Kepler orbits in physical time. The resulting quasiclassical picture is obtained without the Bohr correspondence principle of taking only contributions from high principal quantum numbers.
An explicit and uniform construction for coherent states (CS) for all the SU(N) groups is given and, on this basis, their properties are investigated. The CS are parametrized by the dots of a coset space, which is, in this particular case, the projective space CP N-1 and which plays the role of the phase space in the corresponding classical mechanics. The logarithm of the modulus of the CS overlap, being interpreted as assymmetric in the space, gives the Fubini-Study metric in CP N-1 . The classical limit is investigated in terms of operator symbols. h=P -1 (where P is the signature of the representation) plays the role of Planck's constant. The classical limit of the so called star commutator of the symbols generates the Poisson bracket in the corresponding phase space. The CS form an overcompleted system in the representation space and, as quantum states possess a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator.