Strings from position-dependent noncommutativity

Abstract

We introduce a new set of noncommutative space-time commutation relations in two space dimensions. The space-space commutation relations are deformations of the standard flat noncommutative space-time relations taken here to have position dependent structure constants. Some of the new variables are non-Hermitian in the most natural choice. We construct their Hermitian counterparts by means of a Dyson map, which also serves to introduce a new metric operator. We propose PT like symmetries, i.e. antilinear involutory maps, respected by these deformations. We compute minimal lengths and momenta arising in this space from generalized versions of Heisenberg's uncertainty relations and find that any object in this two dimensional space is string like, i.e. having a fundamental length in one direction beyond which a resolution is impossible. Subsequently we formulate and partly solve some simple models in these new variables, the free particle, its PT -symmetric deformations and the harmonic oscillator.

Minimal areas fromq-deformed oscillator algebras

“…Taking now a less trivial limit, we may obtain string like relations from (3.36)-(3.41) similar to those proposed in [18]. Parameterizing q = e 2τ κ 2 5 with τ ∈ R + and taking the limit κ 5 → 0 we obtain yet simpler relations.…”

“…the constant θ becomes position and possibly also momentum dependent. A set of consistent commutation relations for such a scenario was introduced in [18] […”

“…In [18] it was demonstrated explicitly that they also result in simultaneous x, y-measurements as a consequence of the dynamical noncommutativity of space-time. Whereas the algebra investigated in [18] only gave rise to a minimal length in one direction, i.e. "string like" objects, we demonstrate here that the algebras provided in section 3 will lead to minimal lengths in two direction, i.e.…”

Dirac equation, hydrogen atom spectrum and the Lamb shift in dynamical non-commutative spaces

“…The new Hermitian variables x, y, and can be expressed in terms of noncommutative -space as follows [6] :…”

Entangled squeezed states in noncommutative spaces with minimal length uncertainty relations

“…In section 3, we assemble various generalities on squeezed coherent states from section 2 and construct the nonlinear coherent states and the squeezed states for a specific harmonic oscillator in a noncommutative space [14][15][16][17][18][19]. In section 4, we measure the quantum entanglement of noncommutative systems by computing the linear entropy of the corresponding models and provide their comparative analysis with the ordinary quantum systems.…”