How to define the position operator in higher dimensions?

Question

I am familiar with the traditional definition of the position and of the momentum operator in several dimensions. I do not like it, as it uses from the very beginning a concrete Cartesian basis. What I am looking for is a definition in the following form:

Let $V$ be a real vector space. Let $\mu$ be a measure on $V$. Let $L^2(V,{\mathbb C})$ be the Hilbert space of square integrable $L^2$ functions with respect to the measure $\mu$. Denote the Hilbert space scalar product by $\langle \phi | \psi \rangle = \int_V \bar{\phi} \cdot \psi d \mu$. Then the position operator $Q$ is the function $Q\colon L^2 (V, {\mathbb C}) \to L^2 (V, {\mathbb C})$ defined by ... and the momentum operator ...

Question: How can I fill in the dots in above definition? I am interested in an abstract construction from the very beginning.

This means: Using ${\mathbb R}^3$ or a specific basis in $V$ and then abstracting away from the specifics by using morphisms is considered cheating in the context of this question. Using some fundamentally covariant structure on $V$ is fine. Using categorial constructions which later allow for natural isomorphisms between them in the sense of category theory is fine. Using a different Hilbert space is fine.

Goal: I want to understand multidimensionality in QM along the same lines as I understand it in SRT and GRT. I do not want to start with some concrete coordinates, then add Lorentz-transformation or general coordinate transformations - I want to start from the abstract clean mathematical concept of a Minkowski space with a (3,1) index quadratic form or from an abstract differentiable manifold, where the Lorentz-invariance or the general covariance is already encoded into the mathematical object as such.

Answer 1

What you want is impossible - there is no "abstract position operator" because quantization can change with coordinates. Famously, quantization is not a functor, and in fact choosing different coordinates for your classical system can produce different quantized systems or fail altogether, see also this answer of mine.

Furthermore, starting with something like $L^2(V)$ is really the wrong way to go about trying to abstract quantum mechanics: The fundamental structure of quantum mechanics is the algebra of observables and the Hilbert space its represented on, not some measure space $(V,\mu)$. And when we construct the algebra of observables from the algebra of classical functions on phase space, we pick the functions $x_i$ and $p_i$ of some particular Darboux coordinate system as our "position" and "momentum" operators - there is no notion of "coordinate transformation" there that we could preserve, since due to Gronewold-van Howe any non-linear expressions in $x_i$ and $p_i$ will not necessarily be preserved after quantization - that is, different coordinates $Q_i(x,p)$ and $P_i(x,p)$ will not necessarily fulfill $\hat{Q}_i(x,p) = Q_i(\hat{x},\hat{p})$, even after resolving ordering ambiguities in the second expression, so we do not preserve the "freedom of coordinate choice" in the course of quantization.

The most "abstract" definition of the position and momentum operators is via the Stone-von Neumann theorem: We start with the abstract algebra of $n$ position and momentum operators with Lie bracket $[x_i,p_j] = \delta_{ij}$ and then ask what its possible representations are - and the Stone-von Neumann theorem says there is, up to isomorphism, only a single irreducible representation, namely that on $L^2(\mathbb{R}^n)$ with $x_i$ as multiplication in the $i$-th variable and $p_i$ as differentiation. So we really need not worry about some more general construction of the quantum mechanical Hilbert space: The Stone-von Neumann theorem guarantees we can always choose our familiar $L^2(\mathbb{R}^n)$.

Since you mention Minkowski space: quantum mechanics of single relativistic particles is a hack, a set of tools that are useful but ultimately inconsistent, and the correct relativistic quantum theory is quantum field theory, where you do not have position operators in the traditional sense (see Reeh-Schlieder theorem, Newton-Wigner localization).