measure for measure The nature of natural units Nick van Remortel demystifies natural unit systems — and advises what to do when you see a mass expressed in GeV.

everal unit systems for describing the to the natural units is done through the natural world exist. The International relations (rounded to 4 significant digits) System of Units (SI)1 — the most widely 1 s = 1.519 × 1015 ħ eV, 1 m = 5.068 × 106 c ħ eV–1 S 35 –2 accepted system across the sciences — is, and 1 kg = 5.610 × 10 c eV. The large although clever and successful, mainly powers involved in these conversions reflect focused towards serving the scientific the awkwardness of the SI for describing community at large, and relies heavily atomic and subatomic scales. on precision measurements of standard If one chooses the speed of light as the prototypes, objects or systems that define a fundamental scale for velocity, all velocities physical unit. As an alternative, natural units can be expressed as dimensionless fractions are truly ‘natural’ for studying the physical of that constant and c can be omitted from world of the small, in particular relativistic all equations (often referred to as putting and quantum-mechanical systems2. The c = 1). To restore the correct dimensionality, philosophy underlying natural units is one needs to remember to implicitly to have as few base units as possible, and multiply all velocities by the speed of light.

to define them directly through natural 2007 MEDIA, LLC. © SPRINGER SCIENCE+BUSINESS The same applies to the (reduced) Planck physical constants. constant and ultimately, all quantities The first natural units were proposed (E and B, respectively) in the same units, can be simply expressed in powers of eV. by George J. Stoney3 (pictured). The through the introduction of the speed The latter is often a cause for confusion starting point is to adopt a minimal base of light c as a proportionality constant when dealing with natural units — but of fundamental units, from which many in the expression for the Lorentz force this confusion is not at all inherent to the other (non-base) units can be derived via F = qE + q(v/c) × B experienced by a particle natural unit system itself! So, when a particle

dimensional analysis of the physical laws that with charge q moving with velocity v in an physicist says that the mass mH of the Higgs connect them. The choice of the base depends electromagnetic field. The constant needs boson is 125 GeV, this actually means that 9 –2 –25 on conventions, but for the fundamental to have the dimension of a velocity, and c mH = 125 × 10 c eV = 2.228 × 10 kg (ref. 4). laws of physics a base of three units, such as is the natural choice. Using this constant An alternative natural unit system mass, length and time (abbreviated m, l and t, immediately puts into question the choice of follows from considering Newton’s law of respectively) is sufficient. base units adopted in the SI. Indeed, there is gravitation and absorbing the gravitational Troubles arise with the laws of nothing particularly natural about the meter, constant G into a unit for mass — the Planck electromagnetism, in particular with kilogram or second. If one were to adopt mass, leading to the Planck unit system, Coulomb’s law of electrostatics: the velocity as a base quantity, it would have c as which is very appropriate for studying convention to use a specific unit for electric its natural standard. quantum gravity. charge (the coulomb) in the SI implies The same applies to mechanical action Natural unit systems are as natural as it the need for a proportionality constant (energy times time), for which the reduced gets, and lead to valuable insights concerning

1/(4πε0). One can argue whether the vacuum Planck constant ħ is the natural standard. The the scales of energy, time and distance where permittivity ε0 is truly a fundamental constant units for velocity and action can be completed our current descriptions of the forces of of nature, as it is introduced solely to match with a third fundamental constant of nature: nature no longer apply. ❐ a force, expressed in mechanical units, to the rest mass of the electron, which is related electric charges expressed in electrical units. to energy via Einstein’s relativity formula NICK VAN REMORTEL is in the Department A valid alternative is to derive the unit of E = mc2. (The choice of the electron rest mass of Physics, University of Antwerp, 2020 charge from the mechanical units by forcing is more arbitrary than the other fundamental Antwerp, Belgium. the proportionality constant to be unity. constants, as indeed any elementary particle e-mail: [email protected] In this way, the unit of charge acquires the could be chosen to provide a reference mass.) 1/2 3/2 –1 References dimension of m l t . This is how charge As a consequence, the electron volt (eV) 1. SI Brochure: The International System of Units is defined in many unit systems, such as the can be taken as the energy unit — widely (SI) 8th edn (BIPM, 2006); http://go.nature.com/2ehp2XP Gaussian cgs system. used in the fields of atomic, nuclear, particle 2. Jaffe, R. MIT Quantum Theory Notes (2007); When developing the Gaussian system, and astrophysics. One thus arrives at a new http://go.nature.com/2ebYqeU 3. Elliott, I. in Biographical Encyclopedia of Astronomers a solution was found for expressing the base of units expressed in fractions of c, ħ (eds Hockey, T. et al.) 2072–2074 (Springer, 2014). strength of electric and magnetic fields and eV. Going from the mechanical units 4. Olive, K. A. et al. Chinese Phys. C 38, 090001 (2014).

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