Generalized coherent state approach to star products and applications to the fuzzy sphere

We construct a new class of scalar noncommutative multi-solitons on an arbitrary Kähler manifold by using Berezin's geometric approach to quantization and its generalization to deformation quantization. We analyze the stability condition which arises from the leading 1/ correction to the soliton energy and for homogeneous Kähler manifolds obtain that the stable solitons are given in terms of generalized coherent states. We apply this general formalism to a number of examples, which include the sphere, hyperbolic plane, torus and general symmetric bounded domains. As a general feature we notice that on homogeneous manifolds of positive curvature, solitons tend to attract each other, while if the curvature is negative they will repel each other. Applications of these results are discussed.

Section: Jhep03(2002)011mentioning

confidence: 99%

Noncommutative solitons on Kahler manifolds

Spradlin¹,

Volovich²

2002

“…A possible approach for recovering the spatial coordinates used in the previous sections is to apply the modified coherent states of [9]. Here we can try truncating the usual coherent states Now it is easy to see how the noncommutative plane is attained.…”

Section: )mentioning

confidence: 99%

“…So in this limit we recover the standard coherent states for R 2 . Furthermore, using techniques of [9] we can construct the Voros star product, which is equivalent to the Moyal star product. [9], [10] The other limit N → ∞ and → 0 with N = const is more complicated because it demands a more accurate definition of the coherent states in that case.…”

Section: )mentioning

confidence: 99%

Absence of the holographic principle in noncommutative Chern-Simons theory

Pinzul

Stern

2001

We examine noncommutative Chern-Simons theory on a bounded spatial domain. We argue that upon 'turning on' the noncommutativity, the edge observables, which characterized the commutative theory, move into the bulk. We show this to lowest order in the noncommutativity parameter appearing in the Moyal star product. If one includes all orders, the Hamiltonian formulation of the gauge theory ceases to exist, indicating that the Moyal star product must be modified in the presence of a boundary. Alternative descriptions are matrix models. We examine one such model, obtained by a simple truncation of Chern-Simons theory on the noncommutative plane, and express its observables in terms of Wilson lines.

“…The generalization of the notion of coherent states developed by [21] for non commutative theories has been considered by many authors [22,23,24,25]. Having a deformation of the usual position-momentum commutation relations, one constructs operators obeying the relation…”

Section: High Dimensional Extensions a Generalized Coherent Statesmentioning

confidence: 99%

Functional approach to squeezed states in non commutative theories

Lubo

2004

We review here the quantum mechanics of some noncommutative theories in which no state saturates simultaneously all the non trivial Heisenberg uncertainty relations. We show how the difference of structure between the Poisson brackets and the commutators in these theories generically leads to a harmonic oscillator whose positions and momenta mean values are not strictly equal to the ones predicted by classical mechanics.This raises the question of the nature of quasi classical states in these models. We propose an extension based on a variational principle. The action considered is the sum of the absolute values of the expressions associated to the non trivial Heisenberg uncertainty relations. We first verify that our proposal works in the usual theory i.e we recover the known Gaussian functions. Besides them, we find other states which can be expressed as products of Gaussians with specific hyper geometrics.We illustrate our construction in two models defined on a four dimensional phase space: a model endowed with a minimal length uncertainty and the non commutative plane. Our proposal leads to second order partial differential equations. We find analytical solutions in specific cases. We briefly discuss how our proposal may be applied to the fuzzy sphere and analyze its shortcomings.