Order of index in Lorentz transform

Question

I am reading Schwartz's "QFT and the standard model". On pg 13 he gives the Lorentz transform of a rotation around the x-axis:

$ \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos \theta _x & \sin \theta _x \\ 0 & 0 & -\sin \theta _x & \cos \theta _x \\ \end{array} \right) $

For a boost along the x-axis he gives:

$ \left( \begin{array}{cccc} \cosh \beta _x & \sinh \beta _x & 0 & 0 \\ \sinh \beta _x & \cosh \beta _x & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right) $

I believe this corresponds to the Lorentz transform in the form $\Lambda^\alpha{}_\beta$. I believe the first index is the row and the second is the column. But at the bottom of the pg he has $ V^\mu=\Lambda_\nu^\mu V^\nu. $ as though the order of the indices on $\Lambda$ doesn't matter. But surely they do matter as $\Lambda^\alpha{}_\beta\not=\Lambda_\beta{}^\alpha$ in general?

Answer 1

Short answer

His notation is not ambiguous because the expression $$V^{'\mu} \equiv \Lambda^\mu_\nu V^\nu$$ can only mean sum along the $\nu$ component. Since $\Lambda$ is a representation of the Lorentz group, it is a linear operator, hence it can only act on a vector by the usual way that matrices act on vectors. Hence the above is unambiguous.

Longer answer

I'll explain why in general this notation is unambiguous.

The convention is the following: the upper index denotes a vector index and the lower index denotes a dual vector index and. That is, for a general rank $(p, q)$ tensor, one would usually write

$$ T^{\mu_1...\mu_p}_{\nu_1...\nu_q} \partial_{\mu_1} \otimes ... \otimes \partial_{\mu_p} \otimes dx^{\nu_1}... dx^{\nu_q} $$

where $\{\partial_\mu\}$ denotes a basis for your vector space and ${dx^\nu}$ denotes a basis for its dual vector space. (The notation I chose is from differential geometry, which is certainly useful when you define a field theory in over a general spacetime manifold).

As a special case, a linear transformation is a rank $(1,1)$ tensor. Namely $\Lambda^\mu_\nu$ unambiguously denotes components of the tensor:

$$\Lambda^\mu_\nu ~\partial_\mu \otimes dx^\nu$$

Acting on a vector $V \equiv V^\tau \partial_\tau$, we get:

$$\Lambda[V] \\ = \Lambda^\mu_\nu ~\partial_\mu \otimes dx^\nu [V^\tau \partial_\tau] \\ = \Lambda^\mu_\nu V^\tau (\partial_\tau dx^\nu) ~\partial_\mu \\ = \Lambda^\mu_\nu V^\tau \delta_\tau^\nu ~\partial_\mu \\ = \Lambda^\mu_\nu V^\nu ~\partial_\mu $$

Observe no horizontal padding in the indices are needed to make these manipulations unambiguous.