Photon Polarization: What are indices i and j?

[Silviu]

Hello! I am a bit confused about the completeness relation for photon polarization. So it says $\sum_{s=1,2}\epsilon_i^s\epsilon_j^{s*}=\delta_{ij}-p_ip_j$, with $p_i, p_j$ - unit vectors. So I am not sure what are the indices i and j. I though that they represent the component of the vector, but in that case the RHS would be always 0. So what are i and j? Thank you!

 

[Orodruin]

Why do you think the RHS would be zero?

 

[Silviu]

Why do you think the RHS would be zero?

if i=j, $\delta_{ij}=0$ and $p_ip_j=0$ as $p_i=(1,0,0)$ and $p_j=(0,1,0)$, for example. If i=j, $\delta_{ii}=1$ and $p_ip_i=1$ so in both cases the difference is 0, right?

 

[Orodruin]

Not right. $p$ is a fixed vector, $p_i$ is its ith component.