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Torsor Theory of Physical Quantities and their Measurement

Abstract The principal objective of this paper is to provide a torsor theory of physical quantities and basic operations thereon. Torsors are introduced in a bottom-up fashion as actions of scale transformation groups on spaces of unitized quantities. In contrast, the shortcomings of other accounts of quantities that proceed in a top-down axiomatic manner are also discussed. In this paper, quantities are presented as dual counterparts of physical states. States serve as truth-makers of metrological statements about quantity values and are crucial in specifying alternative measurement units for base quantities. For illustration and ease of presentation, the classical notions of length, time, and instantaneous velocity are used as primordial examples. It is shown how torsors provide an effective description of the structure of quantities, systems of quantities, and transformations between them. Using the torsor framework, time-dependent quantities and their unitized derivatives are also investigated. Lastly, the torsor apparatus is applied to deterministic measurement of quantities.