What is the summation notation

Regarding Dirac Hamiltonian's use of summation notation:

AccidentalFourierTransform

As long as you understand what you mean, you can use any notation you want. For example, if you write piαich you could mean

pichαich≡ p1α1 + p2α2 + p3α3
or
pichαich≡ - p1α1- p2α2- p3α3
or any other convention you want to follow. However, once you are ready to share your results with others, you need to clearly explain what you mean by a term.

For example the combination

a0b0- a1b1- a2b2- a3b3
is very useful. Some people will write aμbμ for that particular combination, while some others will write aμbμ instead.

When calculating something for yourself, write what feels better for you. Whenever you're writing something for someone else, you always need to define your symbols (unless it is really obviously what something means).

For my taste, two indices are summed up exactly when one is an upper and the other is a lower index. However, many people assume that a pair of repeating indexes will always sum regardless of their position.

Polygon

Thanks, I assumed it was. My main gripe is that I've also seen people write an i-bich with i as a free index: crazy people!

frida

As mentioned earlier, we are in Euclidean space, so the metric is the identity matrix I. If you are in Minkowski space, similar things hold true for gμ ν = ± (+, -, -, -) I. (Take care not around the sign as long as you stick to a convention). There may be other metrics in general relativity, but as long as you are not in GR you can ignore the position of the indices.