The Phase Space Associated to the Heisenberg Group

We present an embedding of the Tsallis entropy into the three-dimensional Heisenberg group, in order to understand the meaning of generalized independence as encoded in the Tsallis entropy composition property. We infer that the Tsallis entropy composition induces fractal properties on the underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we justify why the underlying configuration/phase space of systems described by the Tsallis entropy has polynomial growth for both discrete and Riemannian cases. We provide a geometric framework that elucidates Abe's formula for the Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian spaces.

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Convolution equations and the local Pompeiu property on symmetric spaces and on phase space associated to the Heisenberg group

Journal d Analyse Mathématique ◽

10.1007/s11854-008-0031-7 ◽

2008 ◽

Vol 105 (1) ◽

pp. 43-123 ◽

Cited By ~ 6

Author(s):

V. V. Volchkov ◽

Vit. V. Volchkov

Keyword(s):

Phase Space ◽

Heisenberg Group ◽

Symmetric Spaces ◽

Pompeiu Property ◽

Convolution Equations

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Weyl-Heisenberg group and magnetic translations in finite phase space

Physical Review B ◽

10.1103/physrevb.39.694 ◽

1989 ◽

Vol 39 (1) ◽

pp. 694-700 ◽

Cited By ~ 20

Author(s):

J. Zak

Keyword(s):

Phase Space ◽

Heisenberg Group

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On the essential spectrum of phase-space anisotropic pseudodifferential operators

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004112000321 ◽

2012 ◽

Vol 154 (1) ◽

pp. 29-39 ◽

Cited By ~ 1

Author(s):

MARIUS MĂNTOIU

Keyword(s):

Phase Space ◽

Heisenberg Group ◽

Essential Spectrum ◽

Unitary Representation ◽

Pseudodifferential Operators ◽

Previous Analysis ◽

Adjoint Operator ◽

Order Zero ◽

Resolvent Family

AbstractA phase-space anisotropic operator in=L2(ℝn) is a self-adjoint operator whose resolvent family belongs to a naturalC*-completion of the space of Hörmander symbols of order zero. Equivalently, each member of the resolvent family is norm-continuous under conjugation with the Schrödinger unitary representation of the Heisenberg group. The essential spectrum of such a phase-space anisotropic operator is the closure of the union of usual spectra of all its “phase-space asymptotic localizations”, obtained as limits over diverging ultrafilters of ℝn×ℝn-translations of the operator. The result extends previous analysis of the purely configurational anisotropic operators, for which only the behavior at infinity in ℝnwas allowed to be non-trivial.

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A one-dimensional self-gravitating stellar gas

Symposium - International Astronomical Union ◽

10.1017/s0074180900105261 ◽

1966 ◽

Vol 25 ◽

pp. 46-48 ◽

Cited By ~ 2

Author(s):

M. Lecar

Keyword(s):

Gravitational Field ◽

Phase Space ◽

Motion Picture ◽

Space Distribution ◽

Two Dimensional ◽

One Dimensional ◽

Phase Space Distribution ◽

Dimensional Phase Space ◽

The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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