I've started to study in details quantum optics and I find difficulties in linking the concepts of coherence and correlation among fields, especially because I'm building right now a background on classical optics aven if I already have a strong background in QFT.
As far as I have understood, roughly speaking, coherence is meant to be some kind of measure on how similar the properties of one or more waves are at two different spacetime points. To describe with a quantity this feature, we make use of correlation functions to build the coherence factor (I'm thinking about the second degree of coherence): the closest the factor to one, the greater coherence my wave(s) shows between the two points. Furthermore, correlation functions usually arise in photodetection experiment, stemming for the importance of coherence itself in any process regarding the measurement of EM field (I know there are alternative ways to takle detection, but I'm sticking the question to the very basics of quantum optics). Now, this means that correlation functions give me the "degree" of dependence between the photodetection probabilities at two different points. This means that they have more or less the same role that in any other QFT.
But now, this seems to me conflicting with the nature of coherence itself: more specifically, in (free) QFT we have the so called "miracolous cancellations" where spacelike correlation functions vanish due to commutation relations, but in the optical theory spacelike correlation functions are at the grounds of spatial coherence measurements, and thus are usually non-zero. I don't get what am I missing, probably I'm misunderstanding some basic stuff regarding coherence itself, and I'm hoping to get enlightned about it. Thanks for the help.
