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Equations of motion from field action

  1. Nov 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Consider a massless supersymmetric particle (or superparticle) propagating in D-dimensional Minkowski space-time. It is described by D bosonic fields [tex]X^{\mu}(\tau)[/tex] and D Majorana fermions [tex]\psi^{\mu}(\tau).[/tex] The action is [tex]S_{0}=\int d\tau \left(\frac{1}{2}\dot{X}^\mu \dot{X}_{\mu} - i\psi^{\mu}\dot{\psi}_{\mu} \right).[/tex]

    (i) Derive the field equations for [tex]X^{\mu}, \psi^{\mu}.[/tex]
    (ii) Suppose that [tex]\delta_{1}[/tex] and [tex]\delta_{2}[/tex] are two infinitesimal supersymmetry transformations with parameters [tex]\epsilon_1[/tex] and [tex]\epsilon_2[/tex], respectively. Show that the commutator [tex][\delta_1, \delta_2][/tex] gives a [tex]\tau[/tex] translation by an amount [tex]\delta\tau[/tex]. Determine [tex]\delta\tau[/tex] and explain why [tex]\delta\tau[/tex] is real.

    2. The attempt at a solution
    I have no idea how to start part (ii) - what exactly are [tex]\delta_1, \delta_2[/tex]? Once I know what they are, should I explicitly calculate the commutator?

    As for part (i), I don't know if my approach is correct. I split the action into a bosonic term and a fermionic one, then varied each individually using [tex]\X^\mu \to X^\mu + \delta X^\mu[/tex] and [tex]\psi^\mu \to \psi^\mu + \delta\psi^\mu[/tex].

    For the bosonic term, I found:
    [tex]\delta S_0 = \frac{1}{2}\int d\tau \left( 2\dot{X}^\mu \delta\dot{X}_\mu + \delta \dot{X}^\mu\delta\dot{X}_\mu \right) \to \ddot{X}^\mu = 0,[/tex]

    And for the fermionic term,
    [tex]\delta S_0 = -i\int d\tau \left( \psi^\mu \delta\dot{\psi}_\mu + \delta\psi^\mu \dot{\psi}_\mu + \delta\psi^\mu \delta\dot{\psi_\mu}\right).[/tex]

    However, when I integrate by parts, the entire variation in the action vanishes, and I have no equations of motion. What am I doing wrong?

    Thanks!
     
  2. jcsd
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