There are very compelling, and in my opinion enlightening reasons to call the Weyl transform a noncommutative Fourier transform.
Background
Classical theories can be seen as (classical) probability theories, where observables play the role of random variables, and states the role of probabilities (measures on the phase space). Quantum theories can be naturally interpreted as noncommutative probability theories. The observables are noncommutative random variables, and states are noncommutative probabilities.
The Classical Fourier Transform
Given a measure $\mu$ on the (symplectic linear) phase space $X$, it is possible to define its Fourier transform $\hat{\mu}:X'\to \mathbb{C}$
$$\hat{\mu}(\xi)=\int_{X} e^{i\xi(x)}\mathrm{d}\mu(x)\;,$$
where $X'$ is the continuous dual of the vector space $X$, i.e. the space of continuous linear functionals on the phase space $X$. The phase factor $e^{i\xi(\cdot)}$ is mathematically called character (by a slight abuse of notation, let us identify the character with $\xi$). Therefore, the classical Fourier transform of a measure (state) is a function from the space of characters to the complex numbers, giving the expectation of each character.
A very important theorem of harmonic analysis is the so-called Bochner theorem, relating measures to functions with special properties by means of the Fourier transform. For simplicity, let me state here the version for $\mathrm{dim}\,X<\infty$ (for infinite dimensional vector spaces there is an analogous theorem, but involves cylindrical measures)
Theorem (Bochner). The Fourier transform is a bijection between measures on $X$, and complex-valued functions $G$ on $X'$ satisfying the following properties:
- $G$ is continuous;
- $\sum_{i,j\in F}\lambda_{i}\bar{\lambda}_j G(\xi_i -\xi_j)\geq 0$ for all finite index set $F$, complex numbers $\lambda_i\in \mathbb{C}$, and characters $\xi_i\in X'$.
Therefore the Fourier transform provides a very useful tool to characterize measures: the Fourier transform of a measure is often much easier and explicit to characterize than the measure itself, and by Bochner's theorem given an explicit function satisfying the above properties one knows there is only one measure of which it is the Fourier transform.
The Noncommutative Fourier Transform
Given a noncommutative measure (a quantum state) $\rho_{\hslash}$, it is also possible to define a Fourier transform. However, in order to do that it is necessary that the quantum theory has canonical observables satisfying the Canonical Commutation Relations (CCR). This is indeed the case in quantum mechanics, but also in bosonic quantum field theory. The canonical commutation relations generate an algebra of observables, the so-called CCR or Weyl C*-algebra. The generators are the Weyl observables $W_{\hslash}(\xi)$, $\xi\in X'$, i.e. the complex exponentials of the canonical variables, satisfying the following properties ($\sigma$ is the canonical symplectic form on $X'$, inherited from $X$):
- $W_{\hslash}(\xi)\neq 0$;
- $W_{\hslash}(\xi)^*=W_\hslash (-\xi)$;
- $W_{\hslash}(\xi)W_{\hslash}(\eta)=e^{-i\hslash \sigma(\xi,\eta)}W_{\hslash}(\xi+\eta)$.
Let us clarify where these Weyl observables come from. They are indexed by the space $X'$, that plays the role of the space of test functions. The idea is that the canonical (classical) variables of the theory are elements of the phase space $X$, and the functionals in $X'$ are test functions (usually, one identifies the dual with distributions rather than test functions, but since the canonical variables are often distributions themselves, as it is for fields, the corresponding functionals are test functions). Quantizing the canonical variables, one gets noncommutative objects, but nonetheless having the same test functions.
The Weyl observables are unitary, and therefore noncommutative phases. In fact, they exactly play the role of noncommutative characters. Therefore, it is natural to define the noncommutative Fourier transform of a quantum state $\rho_\hslash$ as follows: the Fourier transform $\hat{\rho}_{\hslash}:X'\to\mathbb{C}$ satisfies
$$\hat{\rho}_\hslash(\xi)=\rho_{\hslash}(W_\hslash(\xi))=\mathrm{Tr}(\rho_{\hslash}W_\hslash(\xi))\;,$$
the latter provided we are representing the state as a density matrix and the Weyl observable as an operator on some Hilbert space. In standard quantum mechanics, this is exactly the Wigner-Weyl transform (since the dual of the finite dimensional phase space is isomorphic to the phase space itself).
The analogy with the classical case does not stop to the fact that the Weyl observable is unitary, and thus a noncommutative character. It is the "correct" character; in fact, a noncommutative version of the Bochner theorem holds for quantum states. Again, for simplicity let us suppose that $\mathrm{dim}\, X<\infty$, and therefore we are considering quantum mechanics (with some caveats, a version for bosonic quantum field theories holds as well; this very general and nice result is not very well-known, and due to Irving Segal, one of the forefathers of mathematical QFT).
Theorem (Noncommutative Bochner). The noncommutative Fourier transform is a bijection between quantum states on the CCR algebra, and complex-valued functions $G_{\hslash}$ on $X'$ satisfying the following properties:
- $G_{\hslash}$ is continuous;
- $\sum_{i,j\in F}\lambda_{i}\bar{\lambda}_j G_{\hslash}(\xi_i -\xi_j)e^{i\hslash \sigma(x_i,x_j)}\geq 0$ for all finite index set $F$, complex numbers $\lambda_i\in \mathbb{C}$, and characters $\xi_i\in X'$.
The difference with the classical case is that the Fourier transform is no more positive definite, but "almost positive definite", up to an $\hslash$-dependent phase factor involving the canonical symplectic form. This describes the well-known fact that the Wigner-Weyl distribution is not a probability distribution.
The classical limit $\hslash\to 0$
Looking at the above properties of the commutative and noncommutative Fourier transforms, a couple of interesting things are evident.
The first is that the Fourier transform, being it classical or noncommutative, puts quantum and classical states on the same grounds: the Fourier transforms $\hat{\mu}$ and $\hat{\rho}_\hslash$ are both complex-valued functions on $X'$.
The second is that, at least heuristically, taking the limit $\hslash\to 0$ of the noncommutative Fourier transform $\hat{\rho}_{\hslash}$ one should get a classical Fourier transform. In fact, in the limit $\hslash\to 0$ the phase factor disrupting the positive-definiteness of the transform disappears. A mathematical caveat is that, unless one is able to prove uniform equicontinuity w.r.t. $\hslash$, the continuity of the function is not necessarily preserved in the limit. Nonetheless, this convergence is one of the key ideas driving semiclassical analysis, and it holds for states having suitable regularity. The classical measure obtained in the limit $\hslash\to 0$ from a quantum state is called a Wigner measure.
