Einstein summation notation, as I understand it:
By writing $A_i B^i$ one implicitly means a sum over elements of the rank 1 tensors A and B. The key is the contraction of an "up" and a "down" index. In a formalism where a metric raises/lowers we should see this as an inner product, where the metric encodes the information of how an inner product is taken in such a space.
This notation is convenient tool that I employ on a daily basis. However, the Hamiltonian for the Dirac Equation of QFT fame can be written: $$ H = \alpha_i p_i + \beta m $$ Two down indices? Summed together?
Now, in this case we're considering a flat Minkowski space-time with $i$ summing over just the spacial indices. As such, we can raise and lower the indices for "free" with a Euclidean metric.
Is this not an abuse of notation? This is not Einstein's summation convention but instead a bastardised summation notation in which we just do not write summation symbols?
Surely it would be more explicit to write: $$ H = \alpha_i p^i + \beta m $$
Am I right here? Am I having some kind of mathematical mental breakdown? Both?