In thinking how to ask this question (somewhat) succinctly, I keep coming back to a Microwave Oven.
A Microwave Oven has a grid of holes over the window specifically designed to be smaller in diameter than the wavelength of the microwaves it produces, yet larger than the wavelengths of the visible light spectrum - this is so you can watch your food being heated without getting an eye full of microwaves.
The "realness" of electromagnetic waves seems indisputable - both from the microwave example above, and also because if I want to broadcast a radio wave with a certain wavelength, then I need to make sure I have an antenna of corresponding length to produce the wave I'm looking for. Furthermore, we discuss and treat these waves as real, measurable "objects" that exist and can be manipulated.
Now, if I want to describe the behavior of my Microwave Oven in the framework of QM (let's pretend my oven is going to only produce 1 photon of energy corresponding to a wavelength of the microwave spectrum for simplicity) then I'll describe the behavior of that photon as a wavefunction that evolves over time and gives a probability distribution within my microwave that similarly does not allow the photon to pass through the safety grid and exit the oven cavity giving me a retinal burn.
The difference is, the wavefunction is never treated as something "real" in this description. When the safety grid is described as working to protect you because it has holes smaller than the wavelength of the classical waves it's blocking, this is a useful description that seems to describe "real" objects/be-ables. While it is possible to describe why an individual photon has low probability of passing through the same grid but extended physical properties such as wavelength (in space) are treated as non-real because we're dealing with a point particle and with behavior described by something we also treat as non-real (the wavefunction); it seems unclear to me why we insist this wave function which predicts behavior of physical measurements so well is somehow "non-real."
Put another way, if we have no problem treating EM waves as "Real," then why do we insist on treating the wavefunction that describes the same behavior as "unreal?"
I understand there is recent research (Eric Cavalcanti and his group for one) trying to argue this point, but as every respectable physics professor, I've ever encountered, has treated the wavefunction as an indisputably non-real mathematical tool, I needed to ask this community for an answer.
Answer
When dealing with a single quantum mechanical particle, both the wavefunction and the electric field appear to belong to the familiar class of "fields", both $\mathbf{E}(x)$ or $\psi(x)$. This analogy completely breaks down when you consider multiple particles, in which case the wavefunction depends on all of the particle coordinates, i.e. $\psi(x_1,x_2,x_3,\ldots,x_N)$. This is totally different from the behaviour of the "physical fields" such as the electromagnetic fields, which can be described by a function of a single coordinate, no matter how many particles one has in the system. These physical fields are included in our theories precisely because they enable us to describe physics in a local way. On the other hand, taking the object $\psi(x_1,x_2,x_3,\ldots,x_N)$ to be a physical field, which now propagates in a $3N$-dimensional configuration space, leads to a grossly non-local description that is philosophically abhorrent to many physicists.
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