Non-commutative geometry and physics: a review of selected recent results

Castellani, Leonardo

doi:10.1088/0264-9381/17/17/301

Cited by 51 publications

(65 citation statements)

References 58 publications

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“…Noncommutative Quantum Field Theories 1 have recently attracted renewed attention, not only because of their relevance to String Theory [1,3], but also in the Condensed Matter Physics context, since they have been proposed as effective descriptions of the Laughlin states in the Quantum Hall Effect [4,5,6]. Noncommutativity has also been introduced to describe the skyrmionic excitations of the Quantum Hall ferromagnet at ν = 1 [7,8].…”

Section: Introductionmentioning

confidence: 99%

Planar field theories with space-dependent noncommutativity

Fosco¹,

Torroba²

2005

J. Phys. A: Math. Gen.

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We study planar noncommutative theories such that the spatial coordinatesx 1 ,x 2 verify a commutation relation of the form: [x 1 ,x 2 ] = i θ(x 1 ,x 2 ). Starting from the operatorial representation for dynamical variables in the algebra generated byx 1 andx 2 , we introduce a noncommutative product of functions corresponding to a specific operator-ordering prescription. We define derivatives and traces, and use them to construct scalar-field actions. The resulting expressions allow one to consider situations where an expansion in powers of θ and its derivatives is not necessarily valid. In particular, we study in detail the case when θ vanishes along a linear region. We show that, in that case, a scalar field action generates a boundary term, localized around the line where θ vanishes.

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Section: Introductionmentioning

confidence: 99%

Planar field theories with space-dependent noncommutativity

Fosco¹,

Torroba²

2005

J. Phys. A: Math. Gen.

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“…Consider now monomials of the form q m p n with m, n positive integers. To define the corresponding operator product it is possible to use the Weyl ordering prescription [6]…”

Section: Introductionmentioning

confidence: 99%

“…Thus, Weyl ordering is an invertible map from the space of functions on the phase-space to the space of quantum operators. We can use now Weyl ordering to define a new product between functions on the phase-space [4], [6]:…”

Section: Introductionmentioning

confidence: 99%

The noncommutative harmonic oscillator in more than one dimension

Hatzinikitas

Smyrnakis

2002

Journal of Mathematical Physics

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The noncommutative harmonic oscillator in arbitrary dimension is examined. It is shown that the ⋆-genvalue problem can be decomposed into separate harmonic oscillator equations for each dimension. The noncommutative plane is investigated in greater detail. The constraints for rotationally symmetric solutions and the corresponding twodimensional harmonic oscillator are solved. The angular momentum operator is derived and its ⋆-genvalue problem is shown to be equivalent to the usual eigenvalue problem. The ⋆-genvalues for the angular momentum are found to depend on the energy difference of the oscillations in each dimension. Furthermore two examples of assymetric noncommutative harmonic oscillator are analysed. The first is the noncommutative two-dimensional Landau problem and the second is the three-dimensional harmonic oscillator with symmetrically noncommuting coordinates and momenta. †

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“…The relations (3.3) and (3.4) imply that the WQO belongs to the class of models of non-commutative quantum oscillators [34]- [38] and, more generally, to theories with noncommutative geometry [39,40]. The literature on this subject is vast.…”

Section: Known Properties Of 3d Wqosmentioning

confidence: 99%

The non-commutative and discrete spatial structure of a 3D Wigner quantum oscillator

King

Palev

Stoilova

et al. 2003

J. Phys. A: Math. Gen.

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The properties of a non-canonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl(1|3), are further investigated. Within each state space W (p), p = 1, 2, . . ., the energy E q , q = 0, 1, 2, 3, takes no more than 4 different values. If the oscillator is in a stationary state ψ q ∈ W (p) then measurements of the non-commuting Cartesian coordinates of the particle are such that their allowed values are consistent with it being found at a finite number of sites, called "nests". These lie on a sphere centered on the origin of fixed, finite radius ̺ q . The nests themselves are at the vertices of a rectangular parallelepiped. In the typical cases (p > 2) the number of nests is 8 for q = 0 and 3, and varies from 8 to 24, depending on the state, for q = 1 and 2. The number of nests is less in the atypical cases (p = 1, 2), but it is never less than two. In certain states in W (2) (resp. in W (1)) the oscillator is "polarized" so that all the nests lie on a plane (resp. on a line). The particle cannot be localized in any one of the available nests alone since the coordinates do not commute. The probabilities of measuring particular values of the coordinates are discussed. The mean trajectories and the standard deviations of the coordinates and momenta are computed, and conclusions are drawn about uncertainty relations. †

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Non-commutative geometry and physics: a review of selected recent results

Cited by 51 publications

References 58 publications

Planar field theories with space-dependent noncommutativity

Planar field theories with space-dependent noncommutativity

The noncommutative harmonic oscillator in more than one dimension

The non-commutative and discrete spatial structure of a 3D Wigner quantum oscillator

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