The role of unit systems in expressing and testing the laws of nature

Abstract

The paper firstly argues from conservation principles that, when dealing with physics aside from elementary particle interactions, the number of naturally independent quantities, and hence the minimum number of base quantities within a unit system, is five. These can be, for example, mass, charge, length, time, and angle. It also highlights the benefits of expressing the laws of physics using equations that are invariant when the size of the chosen unit for any of these base quantities is changed. Following the pioneering work in this area by Buckingham, these are termed "complete" equations, in contrast with equations that require a specific unit to be used. Using complete equations is shown to remove much ambiguity and confusion, especially where angles are involved. As an example, some quantities relating to atomic frequencies are clarified. Also, the reduced Planck constant ħ, as commonly used, is shown to represent two distinct quantities, one an action (energy x time), and the other an angular momentum (action / angle). There would be benefits in giving these two quantities different symbols. Lastly, the freedom to choose how base units are defined is shown to allow, in principle, measurements of changes over time to dimensional fundamental constants like c.

Angles in the SI: treating the radian as an independent, unhidden unit does not require the redefinition of the term ‘frequency’ or the unit hertz

“…When the angle is made explicit in this way, the SI unit for ω is unambiguously rad/s, for h it remains J•s, the complete unit for action, and for h/(2πθN), commonly called 2 ћ, it becomes J•s/rad, the complete unit for angular momentum [6].…”

“…The second cause is the widespread convention, currently included within the foundations of the SI, to treat plane angles as dimensionless quantities with implicit units of radians. This is described in [6] as the Radian Convention. This convention effectively 'crosses out' a constant that would appear in complete equations, those which are valid for any choice of unit.…”

“…The simplification this brings to equations explains why the convention is so widespread. The constant is a plane angle with the value of 1 rad, which in [6] is called θ N . If the constant is reinserted, or at least is acknowledged to have been crossed out, dimensional relationships are clarified, and much potential confusion can be avoided.…”

“…As explained in reference[6], while the quantity of action h/2π is generally called ћ, the quantity of angular momentum h/(2πθN) is also called ћ, because the θN is habitually crossed out, following the Radian Convention.…”

Solid angles in perspective

“…As discussed in Quincey and Burrows (2019), equations that are valid for any coherent set of units are called "complete" or unit-invariant equations. This type of equation must be used if deductions are to be made about the dimensions of the quantities in the equation.…”