# symmetry-ccr-loop-complex.mp4

Chapter section: The Heisenberg Symmetry Group

Caption (verbatim):

Phase as the $[\hat X,\hat K]$ commutator

Image description (verbatim):

A complex spiral follows an x–k loop and returns to its original magnitude envelope with a quarter-turn of phase remaining

## Before the animation

Chapter lines 986–990:

```math
\langle T_k(b)\psi,T_k(b)\chi\rangle
=
\langle\psi,\chi\rangle.
```

Chapter lines 992–996:

```math
\langle T_\phi(\beta)\psi,T_\phi(\beta)\chi\rangle
=
\langle\psi,\chi\rangle.
```

## After the animation

Chapter lines 1004–1004:

The infinitesimal closed $x$-$k$ loop is generated by $[\hat X,\hat K]$. Because the finite loop leaves a phase translation, and phase translations are generated by $iI$, we have:

Chapter lines 1006–1010:

```math
[\hat X,\hat K]
=
iI.
```