I A, a† etc. within integrals: does it have to be so hard?

[Swamp Thing]

I have been trying to teach myself quantum optics for some time.

Up to now, I have often looked at certain types of integrals -- the ones that have operators within them -- without going into too much detail, just trying to get the general purport and moving ahead, only to get mired in some perplexity or other, a bit further on.

Recently I thought I'd re-read certain things in a bid to understand them more rigorously, and I found these quite useful:
Milburn & Basiri-Esfahani : https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4550010/
Bennett, Barlow, Beige: https://arxiv.org/abs/1506.03305

In the first reference we have something similar to this:

$\int d\omega~f(\omega)~a^\dagger(\omega) ~|\psi\rangle$

After literally days of pondering, I either began to realize, or began to suffer the delusion that (please tell me which is true) -- began to think that this integral is not really an integral as most would think of it. It is a mathematician's way of specifying a FOR loop in a computer program. Only, you imagine this loop running to a very, very large terminating value for the iterating index, while you keep shrinking the physical slice that you are dealing with.

Of course, this is true of any integral, but with the operator inside, things can get more interesting. The operator is just a kind of flag that keeps track of things and guides how things are done when certain indexes match or don't match.

Moreover, this integral differs from your common or garden one in that whatever you are adding up, each increment often goes into a different "bin" or "axis" or "degree of freedom" or "direction" as directed by the operator. Each tiny slice of the integral may actually be flying off into different, mutually orthogonal "places".

Why am I posting this?
Firstly, to request someone who is an expert to confirm the above interpretation.
Secondly, to ask (assuming that the interpretation is generally correct) -- to ask :

"Why, why, why, why? Why does it have to be MADE so complicated?" I can see how rigorous formal notation has its own place, and a very important place it is, too. But are there not some good online resources that explain stuff in the spirit that I have indicated above, but with more accuracy and authority than that? If there aren't, there ought to be.

Finally, a request -- if someone, an expert, a Magus, a Wizard, can break his/her Vow of Secrecy, why not write a PF Insight Post about this kind of thing? Generations of ordinary men, women and children will thank you for it.

 

[strangerep]

OK,... there's no vow of secrecy here -- this is just basic quantum superposition. It only seems a hard concept because (apparently) you haven't yet studied the right level of textbook.

Suppose the frequencies were discrete, and you had a sum like $$a^\dagger(1) + a^\dagger(2) + a^\dagger(3) + \dots$$ which you could rewrite as $$\sum_i a^\dagger(i)$$Your integral is just a version of that, but with a continuous index $\omega$ instead of the discrete index $i$ in my sum.

Once you see that it's just a way of writing a linear combination, it suddenly becomes far less puzzling.

If you haven't yet studied the early chapters of Ballentine, which explain much of this kind of math in a quantum context, then that should probably be your next stop.