Objection to interpretation of quantum mechanical wave function being wave I am currently studying Quantum mechanics (first time) from Ballentine's book. In Chapter 4, he objects the idea of associating wave functions, which are solutions of Schrödinger equation in position representation, with waves, which is propagating in a real 3D space.
His argument was: If it was a "real wave", then for a $N$-particle hamiltonian there should be $N$-interacting waves in the 3D space. Which is not the case.
My understanding is: If we pick a general interacting hamiltonian of $N$ particles, the equation cannot completely be decomposed as interacting wave functions in 3D. In the end, the time derivative wave function part would still be a function of $N$ coordinates, hence only one wave function in $3N$-Dimensional space.
He further claimed: to associate wave functions with waves is just a introductory analogy, where single particle's configuration space is isomorphic to the real space.
My understanding is: for example a free particle would have configuration space of the 3D real space, which then coincide with the real space, of course then they are isomorphic.
Is there any misconception of my understanding, or do you all agree on the concept stated by Ballentine?
 A: Let me just comment on Ballentine's argument. As @Cosmas Zachos says in his comment, it is indeed standard, but I am not sure it is waterproof.
I have doubts about this argument because, as nightlight noticed, an off-the-shelf mathematical result by Kowalski (see references in the nightlight's post), a generalization of the Carleman linearization, generates for a system of nonlinear partial differential equations (describing, for example, evolution of a wave in 3D) a system of linear equations in the Fock space, which looks like a second-quantized theory and is equivalent to the original nonlinear system on the set of solutions of the latter.
Let us consider a nonlinear differential equation in an (s+1)-dimensional space-time ${\partial_t}\boldsymbol{\xi}(x,t) = \boldsymbol{F}(\boldsymbol{\xi},{D^\alpha}\boldsymbol{\xi};x,t)$, $\boldsymbol{\xi}(x,0)=\boldsymbol{\xi}_0(x)$, where $\boldsymbol{\xi}:\mathbf{R}^s\times\mathbf{R}\rightarrow\mathbf{C}^k$ (function $\boldsymbol{\xi}$ is defined in an (s+1)-dimensional space-time and takes values in a $k$-dimensional complex space; for example, the conventional space-time is (3+1)-dimensional; $D^\alpha\boldsymbol{\xi}=\left(D^{\alpha_1}\xi_1,\ldots ,D^{\alpha_k}\xi_k\right)$, $\alpha_i$ are multiindices, ${D^\beta}={\partial^{|\beta|}}/\partial x_1^{\beta_1}\ldots\partial x_s^{\beta_s}$, with $ |\beta|=\sum\limits_{i=1}^{s}\beta_i$, is a generalized derivative, $\boldsymbol{F}$ is analytic in $\boldsymbol{\xi}$, $D^\alpha\boldsymbol{\xi}$. It is also assumed that $\boldsymbol{\xi_0}$ and $\boldsymbol{\xi}$ are square integrable. Then Bose operators $\boldsymbol{a^\dagger(x)}=\left(a^\dagger_1(x),\ldots,a^\dagger_k(x)\right)$ and $\boldsymbol{a(x)}=\left(a_1(x),\ldots,a_k(x)\right)$ are introduced with the canonical commutation relations:
$$
\left[a_i(x),a^\dagger_j(x')\right]=\delta_{ij}\delta(x-x')I,\\
\left[a_i(x),a_j(x')\right]=\left[a^\dagger_i(x),a^\dagger_j(x')\right]=0,$$
where $x,x'\in\mathbf{R}^s$, $i,j=1,\ldots,k$. Normalized functional coherent states in the Fock space are defined as $|\boldsymbol{\xi}\rangle =\exp\left(-\frac{1}{2}\int d^sx|\boldsymbol{\xi}|^2\right)\exp\left(\int d^sx\boldsymbol{\xi}(x)\cdot\boldsymbol{a}^\dagger(x)\right)|\boldsymbol{0}\rangle$. They have the following property:
$$
\boldsymbol{a}(x)|\boldsymbol{\xi}\rangle =\boldsymbol{\xi}(x)|\boldsymbol{\xi}\rangle. (*)
$$
Then the following vectors in the Fock space can be  introduced:
$$
\nonumber
|\xi,t\rangle = \exp\left[\frac{1}{2}\left(\int {d^s}x|\boldsymbol{\xi}|^2-\int {d^s}x|\boldsymbol{\xi}_0|^2\right)\right]|\boldsymbol{\xi}\rangle=\\
\exp\left(-\frac{1}{2}\int d^sx|\boldsymbol{\xi}_0|^2\right)
\exp\left(\int d^sx\boldsymbol{\xi}(x)\cdot\boldsymbol{a}^\dagger(x)\right)|\boldsymbol{0}\rangle.(**)
$$
Differentiation of equation $(**)$ with respect to time $t$ yields, together with equation $(*)$, a linear Schrödinger-like evolution equation in the Fock space:
$$
\frac{d}{dt}|\xi,t\rangle = M(t)|\xi,t\rangle,
|\xi,0\rangle=|\boldsymbol{\xi}_0\rangle,
$$
where the boson "Hamiltonian"
$$M(t) = \int {d^s}x{\boldsymbol{a}^\dagger}(x)\cdot F(\boldsymbol{a}(x),{D^\alpha}\boldsymbol{a}(x)).$$ Let us note that the states of equation $(**)$ are, in general, multi-particle states.
So it seems that, at least for bosons, Ballentine's argument can be circumvented. This consideration was used, e.g., in my article.
