NONEXISTENCE OF QUANTUM NAMBU MECHANICS

1994 ◽  
Vol 09 (29) ◽  
pp. 2727-2732 ◽  
Author(s):  
DEBENDRANATH SAHOO ◽  
M. C. VALSAKUMAR
Keyword(s):  
Phase Space ◽  
Classical Mechanics ◽  
Nambu Mechanics ◽  
Quantum Analog

We investigate the problem of quantization of Nambu mechanics — a problem posed by Nambu [Phys. Rev.D7, 2405 (1973)] — along the line of Wigner–Weyl–Moyal (WWM) phase-space quantization of classical mechanics and show that the quantum analog of Nambu mechanics does not exist.

scholarly journals Non-Smooth Geodesic Flows and Classical Mechanics

1969 ◽  
Vol 12 (2) ◽  
pp. 209-212 ◽  
Author(s):  
J. E. Marsden
Keyword(s):  
Phase Space ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Configuration Space ◽  
Smooth Function ◽  
Cotangent Bundle ◽  
Classical Mechanics ◽  
Riemannian Metric ◽  
Geodesic Flows ◽  
Image Position

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In fact, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q-space or the configuration space), we may define a smooth function Tg on the cotangent bundle T*M (q-p-space, or the phase space). This function is the kinetic energy of q, and locally is given by

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.

scholarly journals Semi-classical mechanics in phase space: the quantum target of minimal strings

2005 ◽  
Vol 2005 (11) ◽  
pp. 049-049 ◽  
Author(s):  
César Gómez ◽  
Sergio Montañez ◽  
Pedro Resco
Keyword(s):  
Phase Space ◽  
Classical Mechanics

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.

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 [

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 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.

QUANTUM-DEFORMED GEOMETRY ON PHASE-SPACE

1993 ◽  
Vol 08 (15) ◽  
pp. 1433-1442 ◽  
Author(s):  
E. GOZZI ◽  
M. REUTER
Keyword(s):  
Vector Field ◽  
Phase Space ◽  
Mechanical System ◽  
Time Evolution ◽  
Cotangent Bundle ◽  
Symplectic Manifolds ◽  
Hamiltonian Vector ◽  
Lie Derivative ◽  
Hamiltonian Vector Field ◽  
Quantum Analog

In this paper we extend the standard Moyal formalism to the tangent and cotangent bundle of the phase-space of any Hamiltonian mechanical system. In this manner we build the quantum analog of the classical Hamiltonian vector-field of time evolution and its associated Lie-derivative. We also use this extended Moyal formalism to develop a quantum analog of the Cartan calculus on symplectic manifolds.

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