Different Hilbert spaces in quantum mechanics

The single-particle Hilbert space is the space of functions $psi: , mathbb{R} to mathbb{C}$ [with value $psi(vec{r})$] with finite $L^2$ norm (normalizable functions). Furthermore boundary conditions may limit this set. In any case it is indeed not the space of all continuous functions. Fock space is the many-particle generalization, the space of normalizable $N$-body wavefunctions, $psi(vec{r}_1,dots vec{r}_N)$ which is now a function from $mathbb{R}^N to mathbb{C}$. Furthermore, for identical bosons or fermions we require that this function is symmetric or antisymmetric in the arguments. I'm not sure if this answers your questions but hopefully it clarifies some things.

As was mentioned in the other answer, a Hilbert space is not necessarily a space of continuous functions. This is just one example of a Hilbert space. The term Hilbert space applies to any set of mathematical objects that satisfy the axioms of a vector space over either $mathbb{R}$ or $mathbb{C}$, that have an inner product defined on the space, and that form a complete metric space with the inner product serving as the distance function (this last requirement is often not explicitly mentioned in physics textbooks).

This means that "normal" vector spaces, like $mathbb{R}^n$, are also Hilbert spaces. As another example, we can form a Hilbert space from continuous, normalizable functions by defining the inner product to be $psicdotphiequivint dx psi^*(x)phi(x)$. We can also form a Hilbert space using continuous, normalizable functions of two variables by defining the inner product to be $psicdotphiequivintint dxdy psi^*(x,y)phi(x,y)$. The possibilities are endless. We could even take two different Hilbert spaces and form another Hilbert space by taking their outer product.

I'll start with a generic perspective, and then I'll apply it to the question about Hilbert space. Here's the generic perspective:

• Sometimes we use one mathematical thing to represent another mathematical thing. Here, a representation is a mapping from $A$ to $B$ that preserves the essential structure of $A$, where $A$ and $B$ are both mathematical things.

• In physics, we also use another type of representation: a mapping from physical things to mathematical things. To define a mathematical model of a physical system, we need to provide this second type of representation.

Hilbert spaces and their (math-to-math) representations

A Hilbert space is a vector space over the complex numbers $mathbb{C}$, equipped with a positive-definite inner product and satisfying a completeness condition. That's all. In fact:

• For any given finite $N$, all $N$-dimensional Hilbert spaces over $mathbb{C}$ are isomorphic to each other: they're all the same as far as their abstract Hilbert-space-ness is concerned.

• When the number of dimensions is not finite, quantum theory requires the Hilbert space to be separable, meaning that it has a countable orthonormal bases. Once again, all infinite-dimensional separable Hilbert spaces over $mathbb{C}$ are isomorphic to each other: they're all the same as far as their abstract Hilbert-space-ness is concerned.

We can represent (math-to-math) a Hilbert space using matrices, or using Fock spaces, or using single-variable functions, or using seventeen-variable functions, or whatever. Those representations introduce extra structure that is superfluous as far as the Hilbert-space-ness is concerned, but such representations can still be useful. In particular, different representations can simplify the task of describing different linear operators on the Hilbert space. In quantum theory, the linear operators on a Hilbert space are more important than the Hilbert space itself.

The physics-to-math representation

The thing that makes a given quantum model interesting is how it represents measurable things in terms of linear operators on a Hilbert space. The word observable is used for both sides of this physics-to-math mapping.

Consider these two models:

• The usual quantum mechanics of a single non-relativistic spinless particle, with a Hamiltonian of the form $H=P^2/2m + V(X)$, where $P$ is the momentum observable and $X$ is the position observable.

• Quantum chromodynamics (QCD). By the way, QCD can be rigorously well-defined by treating space as a discrete lattice, so it's mathematically legit.

Both of these models use the same abstract Hilbert space, namely the one-and-only infinite-dimensional separable Hilbert space over the complex numbers. The two models are different, though, because they describe different (simplified) worlds that have different types of measurable things. QCD does not have position observables, and single-particle quantum mechanics does not have Wilson-loop observables. Even if we only consider the association between observables and spacetime, QCD and single-particle QM are still very different: the pattern of observables in QCD is Lorentz-symmetric (to a good approximation at resolutions much coarser than the lattice spacing) and the pattern of observables in single-particle QM is not.

Sometimes physicists use the term "Hilbert space" to mean a particular representation (math-to-math) of the Hilbert space along with a particular set of observables (physics-to-math) suggested by that representation. I personally prefer to reserve the term "Hilbert space" for the abstract mathematical thing (neither type of representation), because I think that's more clear. Preferences aside, the important message is that models are not distinguished from each other by their abstract Hilbert spaces or by how those Hilbert spaces are represented in the math-to-math sense. Instead, different models are distinguished from each other by their observables — by the physics-to-math mapping from measurable things to linear operators on the Hilbert space.