In quantum mechanics, the wave function is a way to represent the state of a system. It need not be just one body, which is what your question is tacitly alluding to, but I'll get to that in a moment. If there are $n$ bodies, each with $3$ coordinates, then it's a function of $3n$ coordinate. It all goes down the same kettle as an undistinguished $3n$-dimensional mess. The wave function for the Helium atom - if treating the nucleus as an external source - would include two electrons in it, not just one, and so would have $6$ coordinates, not $3$.
Frequently, however - typically seen in textbook presentations - the wave function is used as a representation for a single body, as if the body, itself, were a wave in ordinary $3$ dimensional space. These are the one-body states.
However, it's not really a case of a body being "smeared out" in space, which is the usual visualization behind the "wave function" - betrayed by the name, itself - but rather the configurations of an entire system that are smeared out, non-locally.
For two bodies, for instance, the states, themselves, may not be factorable into a product of a state for one body, and the state for a second body, although the state spaces where they reside are factorable. Thus, if you have two electrons whose states are described by a Hilbert space $H$, then the two electron-states reside in $H⊗H$. A wave function would have the form $ψ = \sum_{m,n∈ℤ} ψ_{m,n}|m⟩_0|n⟩_1$, with $\{|m⟩_0: m∈ℕ\}$ being the (countable) basis for the first $H$ for the first electron (its basis indexed by the natural numbers $ℕ = \{0,1,2,⋯\}$, and $\{|n⟩_1: n∈ℕ\}$ the basis for the second $H$ for the second electron. The state, itself, can't generally be factored, e.g $ψ = ψ_0 ψ_1$, since the sum may involve entanglement (e.g. $ψ = |0⟩_0|0⟩_1 + |1⟩_0|1⟩_1$).
The quantum theory of many-body systems, technically, still resides in quantum mechanics, rather than quantum field theory, but starts to cross the bridge that your question is asking about.
Many-body state spaces are called Fock Spaces. Quantum field theory uses many-body spaces - with a twist will be described in a moment. The same Hilbert space $H$ can be used for the one-body state, and as the foundation for the many-body state space, as its the "1-body" sector.
The historical reason quantum field theory fell back into a formulation in terms of many-body states is because of vacuum polarization and its reverse: two or more bodies can come into or or out of existence and transform into or from other bodies. The classical case is an electron and positron cancelling each other out and leaving behind two photons. The reversal of the process would correspond to pumping energy into a vacuum (the two photons) and polarizing the vacuum into an electron and positron.
So, out of necessity, you have to use many-body states.
The following notations and conventions - which are not standard, but are very useful in bridging the gap between the two cases - will help explain where the difference lies and what remains of the "one-body" state in the larger setting of the Fock space. The one-body state is what's left of your original wave function, if thinking of the wave-function as a one-body state.
Maxwell-Boltzmann Fock Spaces In Algebraic Form - "Stack Machines"
So, start with a Hilbert space $H$. Create the following spaces:
- The "zero body" states: $H^0 = ℂ$. It's a trivial one dimensional Hilbert space. Apart from a normalization factor, there is only one such state. That's the "vacuum".
- The "one body" states: $H^1 = H$, itself. That's where your wave functions lie. The basis is just $\{|m⟩: m ∈ ℕ\}$.
- The "two body" states: $H^2 = H⊗H$, consisting of the two-body states $\{|m⟩|n⟩ = |mn⟩: m, n ∈ ℕ\}$.
- The "three body" states: $H^3 = H⊗H⊗H$, consisting of the three-body states $\{|m⟩|n⟩|p⟩ = |mnp⟩: m, n, p ∈ ℕ\}$.
- Do the same for $H^n$, for $n > 3$.
Together, we collect all the states to form the Hilbert space that we denote using the Kleene star (just like those seen for regular expressions):
$$H^* ≡ H^0 ⊕ H^1 ⊕ H^2 ⊕ H^3 ⊕ ⋯.$$
It has the property that it can be described recursively as the least solution to the following fixed-point inclusion:
$$H^* ⊇ ℂ ⊕ \left(H⊗H^*\right).$$
Every state in
$ψ ∈ H^*$ has one of the following forms:
- The $0$-body states $ψ = z ∈ ℂ$,
- The $n+1$-body states $ψ = v ψ'$, where $v ∈ H$ is a $1$-body state and $ψ'$ an $n$-body state.
The states, themselves, are represented as "words", e.g.
$u_1u_2u_3Ω$ for a
$3$-body state, where
$u_1, u_2, u_3 ∈ H$, where
$$Ω = 1 ∈ ℂ = H^0,$$
denotes the vacuum state, normalized to
$1$. Specifically, the original "wave functions" are now the one-body states corresponding to
$v ∈ H$ which are now
$|v⟩Ω = vΩ ∈ H^*$.
Technically, the state space is called the Maxwell-Boltzmann Fock space associated with $H$. In it, each body is tagged (by its word position, e.g. $u_1$ is at position $1$, $u_2$ at position $2$, $u_3$ at position $3$ in $u_1u_2u_3$), and is distinguishable from the other bodies.
Here's the non-standard, algebraic, part: instead of treating $|v⟩$ as synonymous with $v ∈ H$, we will treat it as an operator defined by:
$$|v⟩: ψ ∈ H^* ↦ v ψ ∈ H^*.$$
That's analogous to the "push" operation of a stack in a computing machine. Instead of treating $⟨u|$ as the dual of $u ∈ H$ that resides in the dual Hilbert space $\tilde{H}$, we will treat it as an operator, defined recursively by:
$$⟨u|: z ∈ ℂ ↦ 0, \quad ⟨u|: v ψ ↦ ⟨u|v⟩ ψ \quad \left(v ∈ H, \quad ψ ∈ H^*\right).$$
This corresponds to a "pop and test for equality to $u$" operation in a computer stack.
Finally, we will introduce the zero-state projection operator, defined recursively by:
$$π_0: z ∈ ℂ ↦ z, \quad π_0: v ψ ↦ 0 \quad\left(v ∈ H, \quad ψ ∈ H^*\right).$$
This tests for the "empty stack" condition.
The algebra then satisfies the following relations:
$$
{π_0}^2 = π_0, \quad ⟨i| π_0 = 0 = π_0 |j⟩, \quad ⟨i||j⟩ = δ^i_j = \left\{\begin{align}1 \quad (i = j)\\0 \quad (i ≠ j)\end{align}\right\}\quad \left(i, j ∈ ℕ\right),\\
π_0 + \sum_{i ∈ ℕ} |i⟩⟨i| = 1.
$$
The Maxwell-Boltzmann State Space $H^*$ is a de facto Hilbert space implementation of a stack memory, formed of "stack words" or "stack configurations", whose elements are the bodies, themselves, represented via their states in $H$.
The one-body sector, within it, is where you will find your (one-body) "wave functions". But the $ψ$'s in $H^*$ are much more far-reaching in scope, as they also include the vacuum and $n$-body states for $n > 1$.
Einstein-Bose Fock Spaces: For Energy
The quanta that make up an energy field, such as the electromagnetic field, are no more distinguishable from one another than waves of the same shape on an ocean are. Unlike Maxwell-Boltzmann Fock Spaces, there is only one way for two photons of the same type to be in two places, with one in each; while for Maxwell-Boltzmann 2-body states, each body is tagged and there are two ways for them to be like so.
Algebraically, the state space has the same structure as $H^*$, except that the elements of the words commute. Define the following relation:
$$ρ_{+}: u v = v u \quad (u, v ∈ H).$$
Then the Einstein-Bose Fock Space associated with $H$ can be defined as:
$$H^{EB} = H^*/ρ_{+}.$$
The bodies in such a state space are called "bosons" are are typically associated with energy fields.
Correspondingly, we have the following relation between the $|u⟩$ operators:
$$|u⟩|v⟩ = |v⟩|u⟩ \quad (u, v ∈ H).$$
To accommodate this change, we redefine the operators $⟨u|$ for $u ∈ H$, recursively by:
$$⟨u|: z ∈ ℂ ↦ 0, \quad ⟨u|: v ψ ↦ ⟨u|v⟩ ψ + |v⟩(⟨u|ψ) \quad \left(v ∈ H, \quad ψ ∈ H^*\right).$$
The algebra satisfies the following relations:
$$
{π_0}^2 = π_0, \quad ⟨i| π_0 = 0 = π_0 |j⟩ \quad (i, j ∈ ℕ),\\
\quad [⟨i|,|j⟩] = ⟨i||j⟩ - |j⟩⟨i| = δ^i_j, \quad ⟨i|⟨j| = ⟨j|⟨i|, \quad |i⟩|j⟩ = |j⟩|i⟩ \quad (i, j ∈ ℕ),\\
\sum_{i ∈ ℕ} |i⟩⟨i| = n,
$$
where $n$ denotes the "number" operator that determines how many bodies are in the state.
Since order no longer matters, the bodies are no longer tagged or distinguishable. Each "word" is now just a monomial over the set $\{|0⟩, |1⟩, |2⟩, ⋯\}$, and states are generated by sums of monomials, i.e. polynomials, e.g.
$$|0⟩(|2⟩Ω) + |1⟩(|1⟩Ω) = (|0⟩|2⟩ + |1⟩|1⟩)Ω.$$
The operators $\{⟨0|, ⟨1|, ⟨2|, ⋯\}$ act as derivatives on the polynomials, e.g.
$$\begin{align}
⟨1|(|0⟩|2⟩ + |1⟩|1⟩)Ω
&= ⟨1|(|0⟩|2⟩Ω) + ⟨1|(|1⟩|1⟩Ω)\\
&= |0⟩⟨1|(|2⟩Ω) + |1⟩Ω + |1⟩(⟨1||1⟩Ω)\\
&= |0⟩|2⟩⟨1|Ω + |1⟩Ω + |1⟩(Ω + |1⟩⟨1|Ω)\\
&= |1⟩Ω + |1⟩Ω\\
&= 2|1⟩Ω\\
&= \frac{∂}{∂|1⟩}(|0⟩|2⟩ + |1⟩|1⟩)Ω
\end{align}$$
Fermi-Dirac Fock Spaces: For Matter
In contrast to the Einstein-Bose Fock Spaces, in the Fermi-Dirac Fock Space, only one body may exist in a given state. That's the Pauli Exclusion Principle. These are the states typically used for matter. The difference is that the algebra is now defined by the anti-commuting relation:
$$ρ_{-}: u v = -v u \quad (u, v ∈ H),$$
as:
$$H^{FD} = H^*/ρ_{-}.$$
The relation implements the exclusion principle, since, under it, $|v⟩|v⟩ = -|v⟩|v⟩$, for any $v ∈ H$, thus $|v⟩|v⟩ = 0$.
This time, to accommodate the relation $ρ_{-}$, the operators $⟨u|$, for $u ∈ H$ are defined, recursively by
$$⟨u|: z ∈ ℂ ↦ 0, \quad ⟨u|: v ψ ↦ ⟨u|v⟩ ψ - |v⟩(⟨u|ψ) \quad \left(v ∈ H, \quad ψ ∈ H^*\right).$$
The corresponding identities for the algebra are:
$$
{π_0}^2 = π_0, \quad ⟨i| π_0 = 0 = π_0 |j⟩ \quad (i, j ∈ ℕ),\\
\quad \{⟨i|,|j⟩\} = ⟨i||j⟩ + |j⟩⟨i| = δ^i_j, \quad ⟨i|⟨j| = -⟨j|⟨i|, \quad |i⟩|j⟩ = -|j⟩|i⟩ \quad (i, j ∈ ℕ).
$$
In place of "words" and "monomials", the elements of $H^{FD}$ are essentially sets, since only one body of a given type may appear in a state.
All of this might have connection to Finkelstein's notion of Quantum Sets, but I only know of his work vaguely and haven't had an opportunity to examine it in any detail.
