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Wave function in one-dimensional Dirac operator

  1. Jan 17, 2014 #1
    How to understand the case in which one-dimensional Dirac operator includes a wave function?
    Such as
    \left( \begin{array}{ccc}
    1+p & 0 \\
    0 & 1-p \end{array} \right)
    \frac{\partial}{\partial x}
    \left( \begin{array}{ccc}
    0 & \psi^{*} \\
    \psi & 0 \end{array} \right)
    [/itex], where [itex]
    It is called one-dimensional Dirac operator in [2] and used when solving nonlinear Schrodinger equation [1]:
    \frac{\partial\psi}{\partial t}
    \frac{\partial^2\psi}{\partial x^2}
    First equation also referred as "Zakharov-Shabat operator".
    In simple cases, when operator consisting of 4x4 matrices [itex]\gamma^\mu[/itex] (constructed from Pauli and identity 2x2 matrices), constants and partial derivatives is applied to wave function, it yields Dirac equation:
    [itex](i\hbar c\gamma^\mu\partial_\mu - mc^2)\psi = 0[/itex]
    Is there a generalization that allows to use more complex notation?

    [1] V. F . Zakharov; A. B, Shabat. "Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Wave in Nonlinear Media" Zh. Eksp. Teor. Fiz. 61, 118-134 (July, 1971)
    [2] V.E. Zakharov; S.V. Manakov. "On the complete integrability of a nonlinear Schrödinger equation" Journal of Theoretical and Mathematical Physics 19 (3). Originally in: Teoreticheskaya i Matematicheskaya Fizika 19 (3): 332–343. June 1974
  2. jcsd
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