simplification with undo_average

[ChristophHotter]

The function simplify(x) for operator expressions uses something like undo_average( simplify( average(x) ) ), to use the simplification rules for "normal" numbers of SymbolicUtils. This causes problems if x already includes averages like x = a*average(a), which leads to the wrong result a*average(a) -> a*a. Such terms appear if one deals with measurement backaction. To circumvent this problem, one can simply create the average of the expression as soon as possible. I mention this here since I made this mistake multiple times.

@david-pl Do you think there would be a problem to average terms, in general, as soon as possible after the operator algebra is applied? The only "major" change I see is that we would not have the operator_equations. But I think we only use the operator_equations for the correlation function, so maybe we can just create them there with undo_average()?

[david-pl]

This is probably a duplicate of qojulia/SecondQuantizedAlgebra.jl#70

I think there's a much simpler solution: instead of using averages to simplify, we can create a new symbolic wrapper that is specifically used for simplification only. It has to be a symbolic function that basically works like an average but without any cumulant expansion. Essentially, it has to be symbolic like

@syms simplification_wrapper(::QNumber)::CNumber

Note that this is basically just pseudo-code, not sure if you can use a Sym like the above. But you can just duplicate some code that we have for Average.

Then, we can do simplification by having this apply to QNumber only and leave averages unchanged. Basically, what I mean is

unwrap_qnumbers(simplify(wrap_qnumbers(x)))

So, for your example you'd have

wrap_qnumbers(a * average(a)) -> simplification_wrapper(a) * average(a)

Then you simplify and undo the wrapper.

That said, your solution probably also works, but is a larger change.