{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T01:09:46Z","timestamp":1785287386906,"version":"3.55.0"},"reference-count":76,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T00:00:00Z","timestamp":1692144000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Center for Quantum Information and Control at the University of New Mexico"},{"name":"U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under the Quantum Computing Application Teams (QCAT) program"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The canonical commutation relation, [Q,P]=i\u210f, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as 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 and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. 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 the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.<\/jats:p>","DOI":"10.3390\/e25081221","type":"journal-article","created":{"date-parts":[[2023,8,17]],"date-time":"2023-08-17T10:15:48Z","timestamp":1692267348000},"page":"1221","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-2631-3932","authenticated-orcid":false,"given":"Christopher S.","family":"Jackson","sequence":"first","affiliation":[{"name":"Independent Researcher, Gold Beach, OR 97444, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Carlton M.","family":"Caves","sequence":"additional","affiliation":[{"name":"Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/0003-4916(60)90131-7","article-title":"The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers","volume":"11","author":"Klauder","year":"1960","journal-title":"Ann. 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