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The Weyl system $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ and $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ comprise two "presentation" elements of the Heisenberg group.

To the extent $\hat{p}$ is a derivative with respect to position q, Q is the shift amount q in any function of it is translated by the action of $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$; that is, $f(q) \mapsto f(q+Q)$. The action of the group element $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ on such functions is more prosaic: multiplication by $\exp\left[\frac{i}{\hbar}Pq\right]$, which is a bland rephasing.

What is special about the braiding Weyl relations, $\exp\left[\frac{i}{\hbar} Q \hat{p}\right] \exp\left[\frac{i}{\hbar}P\hat{q}\right]= e^{iPQ/\hbar} \exp\left[\frac{i}{\hbar}P\hat{q}\right]\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ is that they are bounded and thus closer to their finite-dimensional analogs, and thus serve to illuminate the Heisenberg group and the Stone-von Neumann theorem much better than $\hat{q}$ and $\hat{p}$.

The Weyl system, $\exp\left[\frac{i}{\hbar} Q \hat{p}~\right]$ and $\exp\left[\frac{i}{\hbar}P\hat{q}~\right]$, comprise two "presentation" elements of the Heisenberg group.

To the extent $\hat{p}$ is a derivative with respect to position q, Q is but the shift amount that q in any function of it is translated by the action of $\exp\left[\frac{i}{\hbar} Q \hat{p}~\right]$; that is, $f(q) \mapsto f(q+Q)$. 

The action of the group element $\exp\left[\frac{i}{\hbar}P\hat{q}~\right]$ on such functions is more prosaic: multiplication by $\exp\left[\frac{i}{\hbar}Pq\right]$, which is a bland rephasing.

What is special about the braiding Weyl relations, $\exp\left[\frac{i}{\hbar} Q \hat{p}~\right] \exp\left[\frac{i}{\hbar}P\hat{q}~\right]$ $= e^{iPQ/\hbar} \exp\left[\frac{i}{\hbar}P\hat{q}~\right]\exp\left[\frac{i}{\hbar} Q \hat{p}~\right]$ is that they are bounded and thus closer to their finite-dimensional analogs, and thus serve to illuminate the Heisenberg group and the Stone-von Neumann theorem much better than $\hat{q}$ and $\hat{p}$. In particular, they are the continuum limits of the substantially more tractable clock and shift matrices system (group).

The Weyl system $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ and $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ comprise two "presentation" elements of the Heisenberg group.

To the extent $\hat{p}$ is a derivative with respect to position q, Q is the shift amount q in any function of it is translated by the action of $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$; that is, $f(q) \mapsto f(q+Q)$. The action of the group element $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ on such functions is more prosaic: multiplication by $\exp\left[\frac{i}{\hbar}Pq\right]$, which is a bland rephasing.

What is special about the braiding Weyl relations, $\exp\left[\frac{i}{\hbar} Q \hat{p}\right] \exp\left[\frac{i}{\hbar}P\hat{q}\right]= e^{iPQ/\hbar} \exp\left[\frac{i}{\hbar}P\hat{q}\right]\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ is that they are closer to their finite-dimensional analogs, and thus serve to illuminate the Heisenberg group and the Stone-von Neumann theorem much better than $\hat{q}$ and $\hat{p}$.

The Weyl system $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ and $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ comprise two "presentation" elements of the Heisenberg group.

To the extent $\hat{p}$ is a derivative with respect to position q, Q is the shift amount q in any function of it is translated by the action of $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$; that is, $f(q) \mapsto f(q+Q)$. The action of the group element $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ on such functions is more prosaic: multiplication by $\exp\left[\frac{i}{\hbar}Pq\right]$, which is a bland rephasing.

What is special about the braiding Weyl relations, $\exp\left[\frac{i}{\hbar} Q \hat{p}\right] \exp\left[\frac{i}{\hbar}P\hat{q}\right]= e^{iPQ/\hbar} \exp\left[\frac{i}{\hbar}P\hat{q}\right]\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ is that they are bounded and thus closer to their finite-dimensional analogs, and thus serve to illuminate the Heisenberg group and the Stone-von Neumann theorem much better than $\hat{q}$ and $\hat{p}$.

The Weyl system $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ and $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ comprise two "presentation" elements of the Heisenberg group.

To the extent $\hat{p}$ is a derivative with respect to position q, Q is the shift amount q in any function of it is translated by the action of $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$; that is, $f(q) \mapsto f(q+Q)$. The action of the group element $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ on such functions is more prosaic: multiplication by $\exp\left[\frac{i}{\hbar}Pq\right]$, which is a bland rephasing.

What is special about the braiding Weyl relations, $\exp\left[\frac{i}{\hbar} Q \hat{p}\right] \exp\left[\frac{i}{\hbar}P\hat{q}\right]= e^\phi \exp\left[\frac{i}{\hbar}P\hat{q}\right]\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ is that they are closer to their finite-dimensional analogs, and serve to illuminate the Heisenberg group and the Stone-von Neumann theorem much better than $\hat{q}$ and $\hat{p}$.

The Weyl system $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ and $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ comprise two "presentation" elements of the Heisenberg group.

To the extent $\hat{p}$ is a derivative with respect to position q, Q is the shift amount q in any function of it is translated by the action of $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$; that is, $f(q) \mapsto f(q+Q)$. The action of the group element $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ on such functions is more prosaic: multiplication by $\exp\left[\frac{i}{\hbar}Pq\right]$, which is a bland rephasing.

What is special about the braiding Weyl relations, $\exp\left[\frac{i}{\hbar} Q \hat{p}\right] \exp\left[\frac{i}{\hbar}P\hat{q}\right]= e^{iPQ/\hbar} \exp\left[\frac{i}{\hbar}P\hat{q}\right]\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ is that they are closer to their finite-dimensional analogs, and thus serve to illuminate the Heisenberg group and the Stone-von Neumann theorem much better than $\hat{q}$ and $\hat{p}$.