# Equations of motion from field action

1. Nov 9, 2009

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 $$X^{\mu}(\tau)$$ and D Majorana fermions $$\psi^{\mu}(\tau).$$ The action is $$S_{0}=\int d\tau \left(\frac{1}{2}\dot{X}^\mu \dot{X}_{\mu} - i\psi^{\mu}\dot{\psi}_{\mu} \right).$$

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

2. The attempt at a solution
I have no idea how to start part (ii) - what exactly are $$\delta_1, \delta_2$$? 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 $$\X^\mu \to X^\mu + \delta X^\mu$$ and $$\psi^\mu \to \psi^\mu + \delta\psi^\mu$$.

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

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

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!