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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.

The goals of this paper fall into two closely related areas. First, we develop a formal framework for deterministic unital quantities in which measurement unitization is understood to be a built-in feature of quantities rather than a mere annotation of their numerical values with convenient units. We introduce this idea within the setting of certain ordered semigroups of physical-geometric states of classical physical systems. States are assumed to serve as truth makers of metrological statements about quantity values. A unital quantity is presented as an isomorphism from the target system’s ordered semigroup of states to that of positive reals. This framework allows us to include various derived and variable quantities, encountered in engineering and the natural sciences. For illustration and ease of presentation, we use the classical notions of length, time, electric current and mean velocity as primordial examples. The most important application of the resulting unital quantity calculus is in dimensional analysis. Second, in evaluating measurement uncertainty due to the analog-to-digital conversion of the measured quantity’s value into its measuring instrument’s pointer quantity value, we employ an ordered semigroup framework of pointer states. Pointer states encode the measuring instrument’s indiscernibility relation, manifested by not being able to distinguish the measured system’s topologically proximal states. Once again, we focus mainly on the measurement of length and electric current quantities as our motivating examples. Our approach to quantities and their measurement is strictly state-based and algebraic in flavor, rather than that of a representationalist-style structure-preserving numerical assignment.

The goals of this paper fall into three related areas: (i) we present an overview of a universal algebraic paradigm in which measurement specialists can construct formal models of measurement in a unified manner and systematically reason about a large class classical measurement operations, (ii) we construct convenient von Neumann quantity algebras and quantity-channels between them to represent measurements, and introduce the dual framework of state spaces and state-channels between them to investigate the statistical structure of measurements, and (iii) we provide several detailed examples that illustrate the power and versatility of algebraic approaches to measurement procedures.