Question

Firstly, I wasn't sure exactly where to put this. It's a typesetting query but the scope is greater than TEX; however it's specific also to physics and even more specific to this site.

I've recently been reading a style guide for scientific publications (based on ISO 31-11), however there was no mention of quantum mechanical operators. I've seen them written a few ways and was wondering if there was a decision handed down from "up above" that any particular way is best.

H -- I see this most commonly but I suspect it's mostly due to (mild) laziness to not distinguish it from a variable.
H^ -- This is nicer to me because it makes the distinction between operator and variable. From what I understand of the ISO the italic means it's subject to change, which is true of the form of an operator, but not really its meaning? So I'm not totally sure if that's appropriate here.
H -- Roman lettering is used for functions e.g. sinx, erf(x), and even the differential operator (as in ddx) so this seems to me like the most suitable category to put operators in.
H^ -- Probably the least ambiguous but may also be redundant.

Which would be the best to use? Am I being too pedantic?

Answer 1

My taste, never overload your notation unless its necessary.

Many people in quantum information try to avoid "hats" or further ornaments for operators that are just linear maps. Simple capital letters are fine to write Hamiltonians, channels, unitaries and measurements (italics are not really important, but its a de-facto standard). When people write many-body hamiltonians in terms of smaller k-body interactions, it is common that they use low-case letters for the latter (example, the Hubbard model). Also, mind that in finite dimensional systems linear maps are in one-to-one correspondence to matrices.

On the other hand, thinking of linear maps as matrices forces you to choose a basis. It might be more clear in some contexts to use a symbol-with-hat to denote an operator without mentioning the basis and the same symbol without hat for a matrix representation. However, I find that this practice can make your notation more complicated without earning much, since typically there is a natural default basis in every problem.