373 research outputs found
Noncommutative deformation of four dimensional Einstein gravity
We construct a model for noncommutative gravity in four dimensions, which
reduces to the Einstein-Hilbert action in the commutative limit. Our proposal
is based on a gauge formulation of gravity with constraints. While the action
is metric independent, the constraints insure that it is not topological. We
find that the choice of the gauge group and of the constraints are crucial to
recover a correct deformation of standard gravity. Using the Seiberg-Witten map
the whole theory is described in terms of the vierbeins and of the Lorentz
transformations of its commutative counterpart. We solve explicitly the
constraints and exhibit the first order noncommutative corrections to the
Einstein-Hilbert action.Comment: LaTex, 11 pages, comments added, to appear in Classical and Quantum
Gravit
Quantum Mechanics as an Approximation to Classical Mechanics in Hilbert Space
Classical mechanics is formulated in complex Hilbert space with the
introduction of a commutative product of operators, an antisymmetric bracket,
and a quasidensity operator. These are analogues of the star product, the Moyal
bracket, and the Wigner function in the phase space formulation of quantum
mechanics. Classical mechanics can now be viewed as a deformation of quantum
mechanics. The forms of semiquantum approximations to classical mechanics are
indicated.Comment: 10 pages, Latex2e file, references added, minor clarifications mad
Cosmological perturbations and short distance physics from Noncommutative Geometry
We investigate the possible effects on the evolution of perturbations in the
inflationary epoch due to short distance physics. We introduce a suitable non
local action for the inflaton field, suggested by Noncommutative Geometry, and
obtained by adopting a generalized star product on a Friedmann-Robertson-Walker
background. In particular, we study how the presence of a length scale where
spacetime becomes noncommutative affects the gaussianity and isotropy
properties of fluctuations, and the corresponding effects on the Cosmic
Microwave Background spectrum.Comment: Published version, 16 page
Group Theory and Quasiprobability Integrals of Wigner Functions
The integral of the Wigner function of a quantum mechanical system over a
region or its boundary in the classical phase plane, is called a
quasiprobability integral. Unlike a true probability integral, its value may
lie outside the interval [0,1]. It is characterized by a corresponding
selfadjoint operator, to be called a region or contour operator as appropriate,
which is determined by the characteristic function of that region or contour.
The spectral problem is studied for commuting families of region and contour
operators associated with concentric disks and circles of given radius a. Their
respective eigenvalues are determined as functions of a, in terms of the
Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in
Hilbert space carrying the positive discrete series representations of the
algebra su(1,1)or so(2,1). The explicit relation between the spectra of
operators associated with disks and circles with proportional radii, is given
in terms of the dicrete variable Meixner polynomials.Comment: 11 pages, latex fil
A finite model of two-dimensional ideal hydrodynamics
A finite-dimensional su() Lie algebra equation is discussed that in the
infinite limit (giving the area preserving diffeomorphism group) tends to
the two-dimensional, inviscid vorticity equation on the torus. The equation is
numerically integrated, for various values of , and the time evolution of an
(interpolated) stream function is compared with that obtained from a simple
mode truncation of the continuum equation. The time averaged vorticity moments
and correlation functions are compared with canonical ensemble averages.Comment: (25 p., 7 figures, not included. MUTP/92/1
Perturbation theory of the space-time non-commutative real scalar field theories
The perturbative framework of the space-time non-commutative real scalar
field theory is formulated, based on the unitary S-matrix. Unitarity of the
S-matrix is explicitly checked order by order using the Heisenberg picture of
Lagrangian formalism of the second quantized operators, with the emphasis of
the so-called minimal realization of the time-ordering step function and of the
importance of the -time ordering. The Feynman rule is established and is
presented using scalar field theory. It is shown that the divergence
structure of space-time non-commutative theory is the same as the one of
space-space non-commutative theory, while there is no UV-IR mixing problem in
this space-time non-commutative theory.Comment: Latex 26 pages, notations modified, add reference
The Fuzzy Sphere: From The Uncertainty Relation To The Stereographic Projection
On the fuzzy sphere, no state saturates simultaneously all the Heisenberg
uncertainties. We propose a weaker uncertainty for which this holds. The family
of states so obtained is physically motivated because it encodes information
about positions in this fuzzy context. In particular, these states realize in a
natural way a deformation of the stereographic projection. Surprisingly, in the
large limit, they reproduce some properties of the ordinary coherent states
on the non commutative plane.Comment: 18 pages, Latex. Minor changes in notations. Version to appear in
JHE
Mixed Weyl Symbol Calculus and Spectral Line Shape Theory
A new and computationally viable full quantum version of line shape theory is
obtained in terms of a mixed Weyl symbol calculus. The basic ingredient in the
collision--broadened line shape theory is the time dependent dipole
autocorrelation function of the radiator-perturber system. The observed
spectral intensity is the Fourier transform of this correlation function. A
modified form of the Wigner--Weyl isomorphism between quantum operators and
phase space functions (Weyl symbols) is introduced in order to describe the
quantum structure of this system. This modification uses a partial Wigner
transform in which the radiator-perturber relative motion degrees of freedom
are transformed into a phase space dependence, while operators associated with
the internal molecular degrees of freedom are kept in their original Hilbert
space form. The result of this partial Wigner transform is called a mixed Weyl
symbol. The star product, Moyal bracket and asymptotic expansions native to the
mixed Weyl symbol calculus are determined. The correlation function is
represented as the phase space integral of the product of two mixed symbols:
one corresponding to the initial configuration of the system, the other being
its time evolving dynamical value. There are, in this approach, two
semiclassical expansions -- one associated with the perturber scattering
process, the other with the mixed symbol star product. These approximations are
used in combination to obtain representations of the autocorrelation that are
sufficiently simple to allow numerical calculation. The leading O(\hbar^0)
approximation recovers the standard classical path approximation for line
shapes. The higher order O(\hbar^1) corrections arise from the noncommutative
nature of the star product.Comment: 26 pages, LaTeX 2.09, 1 eps figure, submitted to 'J. Phys. B.
Quantum Mechanics on the cylinder
A new approach to deformation quantization on the cylinder considered as
phase space is presented. The method is based on the standard Moyal formalism
for R^2 adapted to (S^1 x R) by the Weil--Brezin--Zak transformation. The
results are compared with other solutions of this problem presented by
Kasperkovitz and Peev (Ann. Phys. vol. 230, 21 (1994)0 and by Plebanski and
collaborators (Acta Phys. Pol. vol. B 31}, 561 (2000)). The equivalence of
these three methods is proved.Comment: 21 pages, LaTe
On plane wave and vortex-like solutions of noncommutative Maxwell-Chern-Simons theory
We investigate the spectrum of the gauge theory with Chern-Simons term on the
noncommutative plane, a modification of the description of the Quantum Hall
fluid recently proposed by Susskind. We find a series of the noncommutative
massive ``plane wave'' solutions with polarization dependent on the magnitude
of the wave-vector. The mass of each branch is fixed by the quantization
condition imposed on the coefficient of the noncommutative Chern-Simons term.
For the radially symmetric ansatz a vortex-like solution is found and
investigated. We derive a nonlinear difference equation describing these
solutions and we find their asymptotic form. These excitations should be
relevant in describing the Quantum Hall transitions between plateaus and the
end transition to the Hall Insulator.Comment: 17 pages, LaTeX (JHEP), 1 figure, added references, version accepted
to JHE
- …