Can many-particle wave functions be imagined in 3 dimensions - instead of $3N$ dimensions? It is usually said that the wave function for $N$ particles cannot be imagined in 3 dimensions, but only in $3N$ dimensions. What is the exact argument? Is there a loophole?
Schrödinger, Feynman and many others picture single-particle wave functions in 3 dimensions. So there should be a way to picture many-particle wave functions in 3 dimensions as well. For example, two non-interacting, distant particles can be imagined, each with its own wave function, to "live" in 3d space.
About a possible loophole: I recall reading that many-particle wave functions can be pictured in 3d if one allows them to be multi-valued (in a nice way). But I cannot find any reference on this.   (For example, one could assign, at every point in space, a separate value for the wave function of each particle. The total wave function would then be "multi-valued" at each point in space.)
 A: No, there isn't really a loophole.  Consider two particles which live in 1D, and let them be bosonic.  The wavefunction of the composite system can most generally be written as some symmetric function $\psi(x_1,x_2)=\psi(x_2,x_1)$.
If we make the assumption that
$$\psi(x_1,x_2) = \frac{\psi_A(x_1)\psi_B(x_2) + \psi_B(x_1)\psi_A(x_2)}{\sqrt{2}}$$
and further assume that $\psi_A$ and $\psi_B$ are such that they only have support in regions $R_A$ and $R_B$, $R_A\cap R_B = \emptyset$, then the probability density for the position of the first particle is
$$|\psi_{eff}(x_1)|^2 = \int|\psi(x_1,x_2)|^2 dx_2 = \frac{|\psi_A(x_1)|^2 + |\psi_B(x_1)|^2}{2}$$
and the same holds for $\psi_2$.  Therefore, one could imagine a "reduced" wavefunction
$$\phi(x) = \psi_A(x)+\psi_B(x)$$
which has been projected down into 1D space, and whose norm squared gives the probability density for finding a particle (the identity of which is meaningless) at position $x$.

There were two massively-simplifying assumptions here that do not hold in general.  First, we assumed that the composite wavefunction could be cleanly separated into single-particle wavefunctions $\psi_A$ and $\psi_B$, which is generally not the case.  Consider for example
$$\psi(x_1,x_2) = e^{-(x_1+x_2)^2}$$
For such a wavefunction, the probability of finding the a particle at some $x_1$ is an inextricable function of $x_2$; there is no sense of separate probability distributions for $x_1$ and $x_2$ individually.
Second, we assumed that the single-particle wavefunctions $\psi_A$ and $\psi_B$ were completely separated, in the sense that their respective supports had no overlap.  Were this not the case, then the full probability density would not have cleanly separated into positive-definite parts, and the interference effects would become too important to ignore.
A: This answer is simply to complement the one provided by J. Murray and discuss density functional theory, a much-used theory to study quantum systems with a 3-dimensional function: the electron density.
For all N-electron systems, the kinetic energy and electron-electron interaction terms are the same. Therefore, all you need to specify such a system is the electron number N and the external potential $V_{\mathrm{ext}}$. Hohenberg and Kohn proved that the external potential is uniquely determined by the electron density:
$$
n(\mathbf{r})=N\int d\mathbf{r}_2\cdots d\mathbf{r}_N|\Psi(\mathbf{r},\mathbf{r}_2\,\ldots,\mathbf{r}_N)|^2,
$$
which is a 3-dimensional function, as opposed to the wave function $\Psi$ which is a 3N-dimensional function. The proof, by reductio ad absurdum is pretty simple: you start assuming that there are two external potentials that differ by more than a trivial constant and that have the same electron density $n(\mathbf{r})$. Using the variational principle for both in turn, you arrive at a contradiction.
This simple idea forms the foundation of density functional theory, the workhorse of modern electronic structure calculations of molecular systems and materials. The ground state of any quantum system is completely determined by the electron density $n(\mathbf{r})$. Practical use of this idea is much more involved, and it is a very active area of research in chemistry, condensed matter physics, and materials science.
