What is going on with the indices

[vertices]

If:

$$\delta x^\mu = a^\mu +w^{\mu}_{\nu}x^\nu$$

How would we go about working out

$$\delta_1 \delta_2 x^\mu$$

This is apparently equal to:

$$(w^{\mu}_{1}_\nu a^{\nu}_{2} + w^{\mu}_{1}_\lambda w^{\lambda}_{2}_\nu) x^\nu$$

but I can't understand what is going on with the indices - can someone help me out?

 

[homology]

What's the context, the answer depends on what these things are. I'm assuming that $$\delta x^{\mu}$$ is some sort of variation of a coordinate and you're expanding it about some point $$a^{\mu}$$ and only keeping terms up to order 1 in x. What do the subscripts on your deltas mean? I can get some results similar to what you propose as an answer but it depends on how I interpret.

For example if I apply delta on the first line: $$\delta(a^{\mu}+\omega^{\mu}_{\nu}x^{\nu})=\delta(a^{\mu})+\omega^{\mu}_{\nu}\delta(x^{\nu})$$

where I've assumed the matrix $$\omega^{\mu}_{\nu}$$ is constant. Then just vary the x term again. I'm assuming that the variation of the 'a' term is zero and so I wonder about your parantheses. But I can't be sure until I know that these letters represent.

 

[fzero]

It's presumably an infinitesimal Poincare transformation (translation + Lorentz), so

$$\delta_1 \delta_2 x^\mu = \delta_1 (a_2^\mu +{w_2^{\mu}}_{\nu}x^\nu) = a_1^\mu + {w_1^\mu}_\nu a_2^\nu +{w_1^\mu}_\nu {w_2^{\nu}}_{\lambda} x^\lambda .$$

This is a bit different than what's written down in the OP.