What is the summation notation
Regarding Dirac Hamiltonian's use of summation notation:
As long as you understand what you mean, you can use any notation you want. For example, if you write piαich you could mean
For example the combination
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.
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.
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