Bohr-Sommerfeld-Heisenberg Quantization of the Mathematical Pendulum
In this paper we give the Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum.
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arXiv subjects
Jedrzej Sniatycki
Publications and source records attributed to Jedrzej Sniatycki.
Shifting operators in geometric quantization
In a series of papers on Bohr-Sommerfeld-Heisenberg quantization of completely integrable systems we interpreted shifting operators as quantization of functions ${\mathrm{e}}^{ \pm iθ_j}$ , where $(I_j , θ_j )$ are action angle coordinates. The aim of this paper is to show how these operators occur in geometric quantization.
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Differentiable spaces that are subcartesian
We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is continuously reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differenial forms, which statisfies Smith's version of de Rham's theorem. Because the orbit space is a locally closed subcartesian space, it has vector fields and their flows.
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De Donder Construction for Higher Jets
In this paper, we generalize De Donder approach to construct boundary forms that depend on the adapted coordinate system used. In continuum mechanics, use of boundary forms leads to splitting of the total force acting on the body into body force and surface traction. Moreover, this splitting is independent of the choice of the boundary form used. In calculus of variations, use of boundary forms leads to equations in exterior differential forms that are equivalent to the Euler-Lagrange equations. Infinitesimal symmetries of the theory lead to conservation laws valid for any choice of the boundary form used. In an example, we show that the boundary conditions lead to independence of constants of motion of the choice of the boundary form.
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On jets, almost symmetric tensors, and traction hyper-stresses
The paper considers the formulation of higher-order continuum mechanics on differentiable manifolds devoid of any metric or parallelism structure. For generalized velocities modeled as sections of some vector bundle, a variational kth order hyper-stress is an object that acts on jets of generalized velocities to produce power densities. The traction hyper-stress is introduced as an object that induces hyper-traction fields on the boundaries of subbodies. Additional aspects of multilinear algebra relevant to the analysis of these objects are reviewed.
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Differential spaces in integrable Hamiltonian systems
This paper uses differential spaces to obtain some new results in integrable Hamiltonian systems
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Bohr-Sommerfeld-Heisenberg Quantization of the 2-dimensional Harmonic Oscillator
We study geometric quantization of the harmonic oscillator in terms of a singular real polarization given by fibres of the energy momentum map.
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Classical and quantum spherical pendulum
This paper extends the Bohr-Sommerfeld quantization of the spherical pendulum to a full quantum theory. This the first application of geometric quantization to a classical system with monodromy.
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The continuous transition of Hamiltonian vector fields through manifolds of constant curvature
We ask whether Hamiltonian vector fields defined on spaces of constant Gaussian curvature $κ$ (spheres, for $κ>0$, and hyperbolic spheres, for $κ<0$), pass continuously through the value $κ=0$ if the potential functions $U_κ, κ\in\mathbb R$, that define them satisfy the property $\lim_{κ\to 0}U_κ=U_0$, where $U_0$ corresponds to the Euclidean case. We prove that the answer to this question is positive, both in the 2- and 3-dimensional cases, which are of physical interest, and then apply our conclusions to the gravitational $N$-body problem.
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Not-quite-Hamiltonian reduction
The not-quite-Hamiltonian theory of singular reduction and reconstruction is described. This includes the notions of both regular and collective Hamiltonian reduction and reconstruction.
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On Bohr-Sommerfeld-Heisenberg Quantization
This paper presents the theory of Bohr-Sommerfeld-Heisenberg quantization of a completely integrable Hamiltonian system in the context of geometric quantization. The theory is illustrated with several examples.
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Polite actions of non-compact Lie groups
Based mainly on examples of interest in mechanics, we define the notion of a polite group action. One may view this as not only trying to give a more general notion than properness of a group action, but also to more fully understand the role of invariant functions in describing just about everything of interest in reduction. We show that a polite action of a symmetry group of a dynamical system admits reduction and reconstruction.
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An extension of the Dirac theory of constraints
Constructions introduced by Dirac for singular Lagrangians are extended and reinterpreted to cover cases when kernel distributions are either nonintegrable or of nonconstant rank, and constraint sets need not be closed.
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Bohr-Sommerfeld-Heisenberg Theory in Geometric Quantization
In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1-dimensional harmonic oscillator and the coadjoint orbits of the rotation group.
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Singular reduction of Dirac structures
The regular reduction of a Dirac manifold acted upon freely and properly by a Lie group is generalized to a nonfree action. For this, several facts about $G$-invariant vector fields and one-forms are shown.
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Geometric Quantization of Algebraic Reduction
An analogue of geometric quantization of Poisson algebras obtained by algebraic reduction of symmetries is developed. Interpretation of the obtained results and their application to the problem of commutativity of quantization and reduction are given
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Singular reduction for nonlinear control systems
We discuss smooth nonlinear control systems with symmetry. For a free and proper action of the symmetry group, the reduction of symmetry gives rise to a reduced smooth nonlinear control system. If the action of the symmetry group is only proper, the reduced nonlinear control system need not be smooth. Using the smooth calculus on nonsmooth spaces, provided by the theory of differential spaces of Sikorski, we prove a generalization of Sussmann's theorem on orbits of families of smooth vector fields.
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