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Quasicoherent states of noncommutative D2-branes, Aharonov-Bohm effect and quantum Mobius strip

David Viennot

Abstract

We find an analytical formula for the quasicoherent states of 3D fuzzy spaces defined by algebras generated by bosonic creation and annihilation operators. This one is expressed in a representation onto the coherent states of the CCR algebra. Such a fuzzy space can be assimilated to a noncommutative D2-brane of the M-theory (but also as a model of a qubit in contact with a bosonic environment). We apply this formula onto a D2-brane wrapped around an axis to obtain the geometry of a noncommutative cylinder. We show that the adiabatic transport of its quasicoherent states exhibits a topological effect similar to the Aharonov-Bohm effect. We study also a D2-brane wrapped and twisted to have the geometry of a noncommutative Mobius strip. Finally we briefly present the other two examples of a noncommutative torus and of a noncommutative Klein bottle.

Quasicoherent states of noncommutative D2-branes, Aharonov-Bohm effect and quantum Mobius strip

Abstract

We find an analytical formula for the quasicoherent states of 3D fuzzy spaces defined by algebras generated by bosonic creation and annihilation operators. This one is expressed in a representation onto the coherent states of the CCR algebra. Such a fuzzy space can be assimilated to a noncommutative D2-brane of the M-theory (but also as a model of a qubit in contact with a bosonic environment). We apply this formula onto a D2-brane wrapped around an axis to obtain the geometry of a noncommutative cylinder. We show that the adiabatic transport of its quasicoherent states exhibits a topological effect similar to the Aharonov-Bohm effect. We study also a D2-brane wrapped and twisted to have the geometry of a noncommutative Mobius strip. Finally we briefly present the other two examples of a noncommutative torus and of a noncommutative Klein bottle.

Paper Structure

This paper contains 34 sections, 1 theorem, 152 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Noncommutative D2-branes and their quasicoherent states
  3. Definition of a noncommutative D2-brane
  4. Quasicoherent states
  5. Adiabatic dynamics
  6. CCR noncommutative D2-branes
  7. CCR D2-branes and $|\alpha\rangle$-representation
  8. Quasicoherent states of a CCR D2-brane
  9. Example: the noncommutative plane
  10. Magnetic potential onto a CCR D2-brane
  11. Noncommutative cylinder and Aharonov-Bohm effect
  12. The noncommutative cylinder
  13. Magnetic field of the noncommutative cylinder
  14. Quantum Möbius strip
  15. The noncommutative Möbius strip
  16. ...and 19 more sections

Key Result

Theorem 1

Let $\slashed D_x = \left(\right)$ be a Dirac operator of a CCR D2-brane $\mathfrak M$ ($A,X^3 \in \mathrm{Env}(a,a^+)$). Let $\varphi_A,\varphi_{X^3} \in \mathbb C^{\mathbb C}$ be the diagonal $|\alpha\rangle$-representations of $A$ and $X^3$. $\ker \slashed D_x \not= \{0\}$ if and only if In other words, the eigensurface is $M_\Lambda = \{(\Re\mathrm{e}(\varphi_A(\beta)),\Im\mathrm{m}(\varphi_A

Figures (4)

  • Figure 1: The topological index of the noncommutative cylinder (solenoid current) with respect to the wrap parameter $\ell$ for different values of $\frac{R}{L}$ (ratio between the cylinder radius and the scale factor).
  • Figure 2: Matrix elements $\langle \uparrow |\rho_\Lambda|\uparrow \rangle$ (population of the state $\uparrow$) and $\Re\mathrm{e}(\langle \uparrow |\rho_\Lambda |\downarrow \rangle)$ (coherence) of the quasicoherent density matrix of the quantum Klein bottle with $R=2$, $r=1$ and $\ell=10^{2}$. The integrations have been numerically computed.
  • Figure 3: Deformation field $\delta_A = 2\int_{\mathbb C} \frac{\beta|\beta|^2e^{-|\beta|^2}}{|\Delta\varphi_A|^2+|\Delta \varphi_{X^3}|^2} \frac{d^2\beta}{\pi N_x^2}$ of the quantum Klein bottle with $R=2$, $r=1$ and $\ell=10^2$. The integrations have been numerically computed. $\Re\mathrm{e}(\delta_A)=0$.
  • Figure 4: The magnetic potential $\mathrm A_{imm} = \mathrm A_{imm,\theta^1}d\theta^1+\mathrm A_{imm,\theta^2}d\theta^2$ issuing from the space immersion of the quantum Klein bottle with $R=2$, $r=1$ and $\ell=10^{2}$. The integrations have been numerically computed.

Theorems & Definitions (3)

  • Definition 1: Noncommutative D2-brane
  • Definition 2: Quasicoherent state
  • Theorem 1: Quasicoherent states of a CCR D2-brane