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Jordan Algebraic Formulation of Quantum Mechanics and The Non-commutative Landau Problem

Tekin Dereli, Ekin Sıla Yörük

Abstract

We present a Jordan algebraic formulation of the non-commutative Landau problem coupled to a harmonic potential. To achieve this, an alternative formulation of the Hilbert space version of quantum mechanics is presented. Using this construction, the Hilbert space corresponding to the non-commutative Landau problem is obtained. Non-commutative parameters are then described in terms of an associator in the Jordan algebraic setting. Pure states and density matrices arising from this problem are characterized. This in turn leads us to the Jordan-Schrödinger time-evolution equation for the state vectors for this specific problem.

Jordan Algebraic Formulation of Quantum Mechanics and The Non-commutative Landau Problem

Abstract

We present a Jordan algebraic formulation of the non-commutative Landau problem coupled to a harmonic potential. To achieve this, an alternative formulation of the Hilbert space version of quantum mechanics is presented. Using this construction, the Hilbert space corresponding to the non-commutative Landau problem is obtained. Non-commutative parameters are then described in terms of an associator in the Jordan algebraic setting. Pure states and density matrices arising from this problem are characterized. This in turn leads us to the Jordan-Schrödinger time-evolution equation for the state vectors for this specific problem.
Paper Structure (9 sections, 6 theorems, 117 equations)

This paper contains 9 sections, 6 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Hilbert-Schmidt Operator Formulation
  3. Jordan Algebraic Formulation
  4. Hilbert Space Construction
  5. Example: Non-commutative Landau Problem
  6. Density Matrices in the JBW-algebra Setting
  7. Concluding Remarks
  8. Acknowledgement
  9. Data Availability

Key Result

Proposition 1

Every self-adjoint element $H\in\mathcal{A}$ can be written as a finite sum of commutators in the form where ${H_L}_j$, and ${H_R}_j$ are self-adjoint operators in $\mathcal{A}$.

Theorems & Definitions (13)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Theorem
  • Definition 1
  • Theorem 2
  • proof
  • Definition 2
  • Theorem 3
  • ...and 3 more