scholarly journals COMPLEX MODULI OF PHYSICAL QUANTA

2005 ◽  
Vol 20 (12) ◽  
pp. 869-874
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Symplectic Structure ◽  
Classical Mechanics ◽  
Differentiable Structure ◽  
Complex Moduli ◽  
Classical Phase Space

Classical mechanics can be formulated using a symplectic structure on classical phase space, while quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This paper is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space.

scholarly journals A QUANTUM IS A COMPLEX STRUCTURE ON CLASSICAL PHASE SPACE

2005 ◽  
Vol 02 (04) ◽  
pp. 633-655
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Degrees Of Freedom ◽  
Complex Structure ◽  
Classical Mechanics ◽  
Elementary Excitation ◽  
Differentiable Structure ◽  
Duality Transformations ◽  
Kähler Potential ◽  
Classical Phase Space

Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase space. Under an observer we understand, as in general relativity, a local coordinate chart. While classical mechanics can be formulated using a symplectic structure on classical phase space, quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This article is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space. For that purpose we consider Kähler phase spaces, endowed with a dynamics whose Hamiltonian equals the local Kähler potential.

scholarly journals QUANTUM-MECHANICAL DUALITIES ON THE TORUS

2004 ◽  
Vol 19 (23) ◽  
pp. 1733-1744 ◽  
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Differentiable Structure ◽  
Recent Developments ◽  
The Creation ◽  
Classical Phase Space ◽  
Creation Operators

On classical phase spaces admitting just one complex-differentiable structure, there is no indeterminacy in the choice of the creation operators that create quanta out of a given vacuum. In these cases the notion of a quantum is universal, i.e. independent of the observer on classical phase space. Such is the case in all standard applications of quantum mechanics. However, recent developments suggest that the notion of a quantum may not be universal. Transformations between observers that do not agree on the notion of an elementary quantum are called dualities. Classical phase spaces admitting more than one complex-differentiable structure thus provide a natural framework to study dualities in quantum mechanics. As an example we quantise a classical mechanics whose phase space is a torus and prove explicitly that it exhibits dualities.

Invariance Groups in Classical and Quantum Mechanics

1973 ◽  
Vol 28 (3-4) ◽  
pp. 538-540 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Energy Levels ◽  
Classical Mechanics ◽  
Symmetry Groups ◽  
Casimir Invariants ◽  
Classical Phase Space ◽  
Invariance Groups ◽  
Classical And Quantum Mechanics

AbstractThis is a report on some new relations and analogies between classical mechanics and quantum mechanics which arise out of the work of Kostant and Souriau. Topics treated are i) the role of symmetry groups; ii) the notion of elementary system and the role of Casimir invariants; iii) energy levels; iv) quantisation in terms of geometric data on the classical phase space. Some applications are described.

Phase-Space Approach to Berry Phases

2006 ◽  
Vol 13 (01) ◽  
pp. 67-74 ◽  
Author(s):  
Dariusz Chruściński
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Wigner Function ◽  
Berry Phase ◽  
Space Formulation ◽  
Berry Phases ◽  
Classical Phase Space ◽  
New Formula ◽  
Space Approach

We propose a new formula for the adiabatic Berry phase which is based on phase-space formulation of quantum mechanics. This approach sheds a new light onto the correspondence between classical and quantum adiabatic phases — both phases are related with the averaging procedure: Hannay angle with averaging over the classical torus and Berry phase with averaging over the entire classical phase space with respect to the corresponding Wigner function.

Phase Space, Density Matrices, Energy Densities, and Exchange Holes

2001 ◽  
Vol 215 (10) ◽  
Author(s):  
M. Springborg ◽  
J. P. Perdew ◽  
K. Schmidt
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Momentum Space ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Phase Space Density ◽  
Density Matrices ◽  
Space Density

In the general case, quantum-mechanical quantities are represented by operators in position- or momentum-space representations, but in phase space they are represented by functions. The correspondence between classical mechanics and quantum mechanics is non-unique as a consequence of [

Geometric Hamiltonian quantum mechanics and applications

2016 ◽  
Vol 13 (Supp. 1) ◽  
pp. 1630017 ◽  
Author(s):  
Davide Pastorello
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Projective Space ◽  
Star Product ◽  
Hamiltonian Formulation ◽  
Classical Mechanics ◽  
Point Of View ◽  
Quantum Theories ◽  
Geometric Point ◽  
Set Up

Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed.

scholarly journals On the Geometry of Relativistic Anyon

1997 ◽  
Vol 12 (24) ◽  
pp. 1783-1789 ◽  
Author(s):  
A. Nersessian
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Irreducible Representations ◽  
Hamiltonian Reduction ◽  
Poincare Group ◽  
Covariant Model ◽  
Lobachevsky Plane ◽  
Spin Generator

A twistor model is proposed for the free relativistic anyon. The Hamiltonian reduction of this model by the action of the spin generator leads to the minimal covariant model; whereas that by the action of spin and mass generators leads to the anyon model with free phase space which is a cotangent bundle of the Lobachevsky plane with twisted symplectic structure. Quantum mechanics of that model is described by irreducible representations of the (2+1)-dimensional Poincaré group.

scholarly journals QUANTUM MECHANICS IN INFINITE SYMPLECTIC VOLUME

2004 ◽  
Vol 19 (05) ◽  
pp. 349-355
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Phase Space ◽  
Projective Space ◽  
Bergman Metric ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Classical Phase Space ◽  
Dimensional Projective Space

We quantise complex, infinite-dimensional projective space CP(ℋ). We apply the result to quantise a complex, finite-dimensional, classical phase space [Formula: see text] whose symplectic volume is infinite, by holomorphically embedding it into CP(ℋ). The embedding is univocally determined by requiring it to be an isometry between the Bergman metric on [Formula: see text] and the Fubini–Study metric on CP(ℋ). Then the Hilbert-space bundle over [Formula: see text] is the pullback, by the embedding, of the Hilbert-space bundle over CP(ℋ).

scholarly journals A GAUGE THEORY OF QUANTUM MECHANICS

2007 ◽  
Vol 22 (03) ◽  
pp. 191-200 ◽  
Author(s):  
JOSÉ M. ISIDRO ◽  
MAURICE A. DE GOSSON
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theory ◽  
Phase Space ◽  
Gauge Group ◽  
Path Integrals ◽  
Invariance Group ◽  
Feynman Path ◽  
Cartan Form ◽  
Feynman Path Integrals ◽  
Classical Phase Space

An Abelian gerbe is constructed over classical phase space. The two-cocycles defining the gerbe are given by Feynman path integrals whose integrands contain the exponential of the Poincaré–Cartan form. The U(1) gauge group on the gerbe has a natural interpretation as the invariance group of the Schrödinger equation on phase space.

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