Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group

TL;DR

This paper introduces the Instrumental Weyl-Heisenberg Group as a fundamental geometric structure arising from simultaneous position and momentum measurement, challenging traditional energy quantization and expanding the foundation of quantum measurement theory.

Contribution

It establishes a universal measuring instrument modeled by a 7-dimensional group, linking quantum measurement to differential geometry and providing a new perspective beyond unitary transformations.

Findings

Defines the Instrumental Weyl-Heisenberg Group (IWH) as a universal measurement structure.

Shows how SPQM leads to a 7D manifold connecting quantum and classical phase space.

Suggests POVM as an alternative foundation to energy quantization.

Abstract

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.