arXiv subjects

William R. Nugent

Publications and source records attributed to William R. Nugent.

2 recordsLinked to original sources

arXiv2026

Measurement Symmetry and Heisenberg Geometry: Embedding Classical Test Theory in a Noncommutative Representation

Classical measurement theory is traditionally formulated in an algebraic framework. However, it is fundamentally commutative and does not naturally represent noncommutative measurement phenomena identified in the social sciences. This study investigates whether the transformation structure of classical measurement theory preserves a canonical noncommutative geometry. A measurement state vector is introduced, and transformations relating various forms of equivalence (parallelism) between measures are expressed as a Lie matrix group. A faithful matrix representation of the Heisenberg group is introduced, and the conjugation of Heisenberg elements by a general measurement transformation is derived. Results show this conjugation defines an automorphism of the Heisenberg group, preserving its commutator structure. However, if the elements of the measurement state vector are equally scaled the Heisenberg geometry is preserved exactly. The findings establish a symmetry linking classical measurement transformations with the Heisenberg group providing a mathematical foundation for extending classical measurement theory to phenomena exhibiting noncommutative structures.

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arXiv2025

Extending a Matrix Lie Group Model of Measurement Symmetries

Symmetry principles underlie and guide scientific theory and research, from Curie's invariance formulation to modern applications across physics, chemistry, and mathematics. Building on a recent matrix Lie group measurement model, this paper extends the framework to identify additional measurement symmetries implied by Lie group theory. Lie groups provide the mathematics of continuous symmetries, while Lie algebras serve as their infinitesimal generators. Within applied measurement theory, the preservation of symmetries in transformation groups acting on score frequency distributions ensure invariance in transformed distributions, with implications for validity, comparability, and conservation of information. A simulation study demonstrates how breaks in measurement symmetry affect score distribution symmetry and break effect size comparability. Practical applications are considered, particularly in meta analysis, where the standardized mean difference (SMD) is shown to remain invariant across measures only under specific symmetry conditions derived from the Lie group model. These results underscore symmetry as a unifying principle in measurement theory and its role in evidence based research.

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