Necessary conditions for the uncertainty principle

[andresB]

In (1D) quantum mechanics we deal with wavefunctions $\psi(x)$ belonging to $L^{2}(\mathbb{R})$. We then have the position and momentum operators $$\hat{x}=x, \quad\hat{p}=-i\hbar\frac{\partial}{\partial x},$$
with the canonical commutation relation $[\hat{x},\hat{p}]=i\hbar$. The textbooks then use the variance to prove the Uncertainty principle $\sigma_{x}\sigma_{p}\geq\frac{\hbar}{2}.$

So far, so good. However, consider the position-momentum wavefunctions $\psi(x,p)$ belonging to $L^{2}(\mathbb{R}^{2})$. We can define the prequantization operators $$\hat{x}_{pre}=i\hbar\frac{\partial}{\partial p}+x, \quad\hat{p}_{pre}=-i\hbar\frac{\partial}{\partial x},$$
and they also obey the canonical commutation relation $[\hat{x}_{pre},\hat{p}_{pre}]=i\hbar.$ The thing is, there is no position-momentum uncertainty principle at this level, and $\psi(x,p)$ is allowed to be as narrow in phase space as we want.

Why? What necessary condition for the uncertainty principle is missing at the prequantization level?

 

[PeterDonis]

We can define the prequantization operators

Where is this coming from? Do you have a reference?

 

[renormalize]

Where is this coming from? Do you have a reference?

This seems to be relevant: https://en.wikipedia.org/wiki/Phase-space_wavefunctions
Phase-space representation of quantum state vectors is a formulation of quantum mechanics elaborating the phase-space formulation with a Hilbert space. It "is obtained within the framework of the relative-state formulation. For this purpose, the Hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system. Relative-position state and relative-momentum state are defined in the extended Hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators, acting on these states, are obtained." Thus, it is possible to assign a meaning to the wave function in phase space, ψ(x,p,t), as a quasiamplitude, associated to a quasiprobability distribution.

 

[PeterDonis]

This seems to be relevant

This is about the phase space representation of quantum state vectors.

There's nothing at all about "prequantization" operators, which is what the OP was making claims about.