How Does Tensor Notation Work in Group Theory Calculations?

[christodouloum]

1. While reading notes on group theory there is a step I could not reproduce although it seems to me it should be straightforward. Probably there is something I am missing on tensor indices notation. Since R is an orthogonal matrix you can...

2 ...go from $$\epsilon _{lmn}R_{il}R_{jm}R_{kn}=\epsilon_{ijk}$$

to $$\epsilon_{lmn}R_{jm}R_{kn}=\epsilon_{ijk}R_{il}$$


3. Since R is orthogonal I wrote down $$R_{il}R_{ir}=\delta _{lr}$$. So I multiply both sides by R_il and by relabelling r with l it works. But can I do that?

 

[ystael]

Indices are dummy tokens; they could be pink hearts, orange stars, yellow moons, et cetera. You can always rename one index letter to another as long as you do so consistently throughout the entire expression or equation.