How do you suggest to determine the field strength of the quantum electromagnetic field at every point in space? Surely your remarks re a one-particle system apply a fortiori to a field with variable particle number.

@RobinEkman the question was whether the wave-function is a real thing. And if the field strength can or cannot be measured exactly as each point, is this the criterion to admit or not whether the w.f. is a real thing? If you cannot measure exactly the e.m. field at any point in space, you deny the reality of the e.m. field?

@RobinEkman : how many things you measure exactly in the experiments? You produce a theoretical curve and a fitting of the experimental data. Is this the criterion to judge if the w.f. is real? It is a too major question to be judged at a level as you suggest.

@RobinEkman: My initial remarks are for the classical electromagnetic field, and wasn't thinking in QFT terms here. In a full QFT picture, I think there aren't any fields that are real in this sense. It has to be remarked that the notion of reality I use here is not obviously an epistemically coherent notion, but it is the notion which is the reason for saying "the wavefunction is not real, but the EM field is".

@ACuriousMind I fail to see the difference between these 2 scenarios: 1.) I run a number of experiments with single photons in series, measuring each one. I observe a spread of values which I see conform to the predictions of the wavefunction. 2.) I take a number of measurements at different point in space of a classical EM field, thus measuring the field strength in space. Both require multiple measurements to fully describe the object I'm trying to measure. A single measurement of a classical field is no more useful than a single photon fired in a double slit experiment.

@ACuriousMind : would you care to make me also understand what you say, as one which doesn't work with QFT? The wave-function is the solution of the Schrodinger eq., or Dirac, or etc. What has it to do with QFT? Next, how does one decide if something is real? If one sees the effects. Can someone do anything better than take the results of the measurements on the quantum particles? And by what is the EM field real and the wave-packet is not? Because you are used to it from the classical physics? (Ohhh, don't feel offended by my words, it's a legitimate scientific dispute.)

@James:The single measurement of the classical electric field at a point $x$ gives you the value $\vec E(x)$. Repeated measurements give you $\vec E(x)$ on an arbitrarily fine lattice, and hence on the whole space. In this sense the field $E : \mathbb{R}^3 \to \mathbb{R}^3$ is real - it can be constructed out of the empirical data alone. The measurement of the quantum object does not give you $\psi(x)$. It gives you some value that, under the additional assumption that there is a $\psi$ from which this value is deduced by the laws of quantum mechanics, may be used to approximate that $\psi$.

@Sofia: The QFT part was in response to RobinEkman, who asked about the quantum electromagnetic field. The quantized EM field is definitely the domain of QFT. For the question of "reality", that is why I said that the notion of reality I use in this answer is not necessarily epistemically coherent. I here just provide the answer to the question "Why do people think the EM field is real, but the wavefunction is not?". This is not my personal opinion, but I merely recount the arguments I have heard for this. My personal viewpoint on the matters is "Shut up and calculate.".

@ACuriousMind : There are many things that I would like to discuss with you on these matters, as I am one who lives with these problems for long years. It's your free will to choose the position of "Shut up and calculate", but you see, on the ground of this position one cannot deny, neither confirm the reality of the w.f. Though, if you have time, I would have pleasure to chat a bit with you. The problem is deeper than people address here in their answers.

@Sofia: I have to go in a few minutes, so now is a bad time. Just look for me anytime in the h bar when you have time and want to discuss this.

@ACuriousMind I understand what you're saying but it feels arbitrary - given knowledge of a point charge with constant velocity, I can calculate (and thus predict) the static E field strength at a given distance. I can then confirm this with experimental measurement. I can similarly construct an experiment in state wave function psi which gives a complete description of the position x of photons described by the function, and see it confirmed with multiple runs. Both cases seem to make the same assumption - that individual measurements approximate a field. The method of measurement matters?

@ACuriousMind Re: "it can be constructed out of empirical data alone," if one had no knowledge of QM and observed individual electrons build up an interference pattern in a double slit experiment, how would this not qualify as equally convincing empirical evidence for a field, of which individual electron position detections are completely analogous to any other individual field readings? It seems the method of detection is the hangup, which seems strange since the QM effects predicted by the wave function we observe are certainly as "real" as those ascribed to classical field effects.

@ACuriousMind no problem, I also have things to do now. My best wishes!

@JamesPattarini: Note that all we can obtain from the measurement of the positions of a quantum particle is the PDF $|\psi(x)|^2$. Since the wave function is complex, we actually lose information (the phase). It is not possible to measure the wave function directly, just the PDF.

@0celo7 Since we recover the distribution described by the square of the function, only real values are considered ultimately aren't they? The interpretation of the imaginary component of the wave function prior to squaring is something of a source of debate in and of itself, but it seems most are satisfied with ignoring it since the square of the function leaves only real terms. If I've misunderstood you please correct me

@JamesPattarini: Note that a non-complex wave function has vanishing probability current. We must consider complex wave functions in scattering theory.

@ACuriousMind Yes, of course. That's why I think we should add a qualifier: the correct statement is that the classical EM field is realist (has definite values at all points). The EM field as it exists in nature, however, is quantum, and a QFT can't be more realist than QM. So I feel it is misleading to say that the EM field is realist but quantum mechanics is not. Fundamentally the EM field is quantum and classical, realist behavior only emerges in the proper limit.

@ACuriousMind As others have pointed out better than I, a classical EM field can only be measured by an instrument, which reacts in a QM way with the field, and can be described too in QFT as interaction mediated by virtual photons - the interaction of field charge and measuring device can be described by a wave function, making the "object" of measurement in this case (what we thought was a classical field E) seem just as unreal as the wave function we treat as a mathematical fiction. My point is either they're both unreal, or they're both real, but I see no reason to declare one special.

@JamesPattarini: Again, I do not believe the distinction between "real" and "unreal" I present here is useful, or even coherent. I agree that if we treat the world fully quantumly, there are no truly "real" fields in this sense. Yet, this is the distinction that is made. I'm not arguing with your position, I just answered the question "Why do people think the (classical) EM field is real, and the wavefunction is not?" with the arguments I have heard for it.

@ACuriousMind Thanks for the detailed responses, the fantastic feedback here is the reason this community is one of the best on the internet. I understand the basis for why these objects are treated differently now, and while I still object on philosophical grounds, seeing the healthy discussion here makes me feel like I'm not out in left field at least.

If the wavefunction of the ensemble of photons does not determine the electromagnetic field then the wavefunction is necessarily an incomplete description. Assuming it is complete, there is therefore a way to measure an important aspect of it, namely the resulting electromagnetic field. So a main question is, what is this correlation between wavefunction and electromagnetic field, and secondly, what can then measurements of the EM field say about the wavefunction, to what degree do they constrain what the possible wave functions can be? Sadly I don't see this discussed in the answers so far.