  The canonical commutation relation, $[Q,P] = i\hbar$, stands at the
foundation of quantum theory and the original Hilbert space. The interpretation
of $P$ & $Q$ as observables has always relied on the analogies that exist
between the unitary transformations of Hilbert space and the canonical (a.k.a.
contact) transformations of classical phase space. Now that the theory of
quantum measurement is essentially complete (this took a while), it is possible
to revisit the canonical commutation relation in a way that sets the foundation
of quantum theory not on unitary transformations, but on positive
transformations. This paper shows how the concept of simultaneous measurement
leads to a fundamental differential geometric problem whose solution shows us
the following: The simultaneous $P$ & $Q$ measurement (SPQM) defines a
universal measuring instrument, which takes the shape of a 7-dimensional
manifold, a universal covering group we call the Instrumental Weyl-Heisenberg
Group, IWH. The group IWH connects the identity to classical phase space in
unexpected ways that are significant enough that the positive-operator-valued
measure (POVM) offers a complete alternative to energy quantization. Five of
the dimensions define processes that can be easily recognized and understood.
The other two dimensions, the normalization and phase in the center of IWH, are
less familiar. The normalization, in particular, requires special handling in
order to describe and understand the SPQM instrument.
