Equations of motion from field action

[adartsesirhc]

Homework Statement

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!

 

[Jhero]

Thank you for your post. I am happy to help you with your questions.

For part (ii), \delta_1 and \delta_2 are two infinitesimal supersymmetry transformations with parameters \epsilon_1 and \epsilon_2, respectively. In other words, they are transformations that involve small changes in the fields X^\mu and \psi^\mu. To find the commutator [\delta_1, \delta_2], you can explicitly calculate it using the definition of the commutator, which is [\delta_1, \delta_2] = \delta_1\delta_2 - \delta_2\delta_1. This will give you a new transformation, which in this case is a \tau translation. The value of \delta\tau and why it is real can be found by solving for the commutator and analyzing its properties.

For part (i), your approach is correct. However, when you integrate by parts in the fermionic term, make sure to keep track of the boundary terms. These boundary terms will contribute to the equations of motion and will not necessarily vanish. Also, remember that the variations \delta X^\mu and \delta\psi^\mu are independent, so you should vary each term separately and then combine them to get the full equation of motion.

I hope this helps. Let me know if you have any further questions.