I have some open-ended questions on the use of staggered indices in writing Lorentz transformations and their inverses and transposes.
What are the respective meanings of $\Lambda^\mu{}_\nu$ as compared to $\Lambda_\mu{}^\nu$? How does one use this staggered index notation to denote transpose or inverse?
If I want to take any of these objects and explicitly write them out as matrices, then is there a rule for knowing which index labels row and which labels column for a matrix element? Is the rule: "(left index, right index) = (row, column)" or is it "(upper index, lower index) = (row, column)" or is there a different rule for $\Lambda^\mu{}_\nu$ as compared to $\Lambda_\mu{}^\nu$?
Are there different conventions for any of this used by different authors?
As a concrete example of my confusion, let me try to show two definitions of a Lorentz transformation are equivalent.
Definition-1 (typical QFT book): $\Lambda^\mu{}_\alpha \Lambda^\nu{}_\beta \eta^{\alpha\beta} = \eta^{\mu\nu}$
Definition-2 ($\Lambda$ matrix must preserve pseudo inner product given by $\eta$ matrix): $(\Lambda x)^T \eta (\Lambda y) = x^T \eta y$, for all $x,y \in \mathbb{R}^4$. This implies, in terms of matrix components (and now I'll switch to linear algebra notation, away from physics-tensor notation): $\sum_{j,k}(\Lambda^T)_{ij} \eta_{jk} \Lambda_{kl} = \eta_{il}$. This last equation is my "Definition-2" of a Lorentz transformation, $\Lambda$, and I can't get it to look like "Definition-1", i.e., I can't manipulate-away the slight difference in the ordering of the indices.
No comments:
Post a Comment