How can many body QM as well as Fock space QFT emerge out of first quantization with one extra coordinate

One of the key difficulties in the interpretation of quantum mechanics is the fact that “probability amplitude” is complex valued, while the “probabilities” we know from classical physics are positive and real. However, if we think of wave function as an analogue of classical “classical field” AS OPPOSED TO probability amplitude, then there would no longer be a conceptual problem: after all it doesn't contradict our intuition that classical electromagnetic field can have both signs, nor does its interference contradict our intuition either. One of the commonly cited obstacles in comparing wave function to classical field is the fact that we don't understand its collase. However, there are some collapse models (Bohmian mechanics, GRW, etc). Nor is superluminal signaling a problem either: we can claim that we have quasi-relativity as opposed to true relativity, meaning that there is preferred frame we just don't see one. What I believe to be far bigger obstacle in viewing wave function as classical field is the fact that classical field lives in $R^4$, whereas wave function in configuration space lives in $R^{3n+1},$ where $n$ is the number of particles. This also gives us additional reason to compare wave function to “probability” since classical probability also lives in $R^{3n+1}$; and like stated earlier the comparison to classical probability (as opposed to classical field) is what ultimately creates a problem. Now, since $3*1+1=4$, in single particle case the “probability” and “field” is indistinguishable. It is strictly where the number of particles is more than one where we have a problem. Therefore, my task is to reduce many body QM to single particle one. I do that by introducing one single extra dimension, $y=x^5$ and then picturing strings that are stretched out in $x^5$. Intersection of these strings with $x^5= const$ hyperplane would produce particle configuration. I then make shapes of the strings sophisticated enough to make sure that any configuration we have in mind can be approximated by at least one hyperplane (in reality, more than one). In other words, $x^5$ acts like "space filling curve" in a sense that each y=const hyperplane corresponds to particle configuration. Thus, we can replace wave function in configuration space with a wave function in $x^5$. For the purposes of "realism" we will also make sure said wave function varies in $x^1$, $x^2$ and $x^3$, but we will integrate it away when we define probability amplitudes. Finally, if I have time, I will also sketch how to extend this principle from many body QM to second quantization.