The third in a (possibly infinite) series on quantities, units, and dimensions – see the previous entries here and here.

The SI system defines seven basic units, although not everyone thinks they are all necessary. Two can perhaps be fairly perfunctorily dismissed: the candela, the unit of luminous intensity, and the mole, which is a way of quantifying matter in terms of the number of atoms or molecules in a given sample. Once we have units of mass and energy, surely those two bring nothing new to the table. The third is the kelvin, the unit of temperature, which some would claim is superfluous since it is a measure of average kinetic energy, although there are also arguments for keeping it, which might form the subject of a future blog piece.

That leaves four units, for the four basic dimensions – mass, length, time and electric charge (or current). But there are still plenty of people around who would argue that electricity is a mechanical phenomenon, and that charge does not exist as a fundamental quantity. That would leave us with only three – the good old M, L and T. And as we have seen, there are some who would want to reduce the number of fundamental quantities even further.

But in this piece I want to argue for an increase in this number. (And I’m not just talking about rather specialist quantities such as radiation dose, with its SI unit, the sievert, which seem to have got forgotten along the way somewhere). Firstly, let’s look at a quantity that crops up frequently in science, namely angle.

It is sometimes said that angular measure is dimensionless, because it is defined as the ratio of the arc length subtended to the radius at which it is subtended. A ratio of similar quantities (lengths), and hence dimensionless. Voilà. But is it in fact so defined? And what do we mean, in any case, by dimensionless?

According to Percy Bridgman’s definition of dimension (which was articulated somewhat better by Grigory Barenblatt) there is a 1:1 relationship between units and dimensions. If a quantity has units, it has dimensions, and vice versa. So, since angle certainly has units, shouldn’t it also have dimensions?

If we follow Norman Campbell’s meticulous description of how to make a measurement (or even Bridgman’s more rough-and-ready one), the starting point (once we have satisfied ourselves that angle is the sort of quantity that can be measured, taken as read for the time being) must be to define a unit – and this makes us think of perhaps the angle of a set square, or perhaps one division of a protractor (where the protractor needn’t be calibrated in degrees). Measuring angles using this arbitrary unit, and measuring the arc lengths subtended at various radii, we will notice a relationship between these quantities – in fact the first is proportional to the second and inversely proportional to the third. We can therefore write that θ = ka/r, where a is the arc length, r is the radius and k is a constant of proportionality. If θ is in degrees, then k = (180/pi) degrees.

Problems arise, however, when angles are measured in radians, when k has the numerical value 1, and thus becomes “invisible” in equations. It then becomes common for textbooks to say that angles are just pure numbers. This is clearly an application of “natural units” to angular measure. And the same mistake has been made as was made in the case of orbital motion and particle physics (see my earlier blog on this): angles measured in radians are not just numbers, but consist of a number and a unit, as in the case of any other measurement; in this case the unit is “radians”.1

It is of course understandable why radians are often chosen above any other angular unit. The Small Angle Approximation says that, for small angles, sin θ ~ tan θ ~ θ  when θ is measured in radians; in general terms this would, of course, be expressed as sin θ ~ tan θ ~ θ/k, and this means that, when deriving the expressions for the derivatives of trigonometric functions, we get k appearing in these formulae too; furthermore the Taylor expansions for trigonometric functions will also include factors of k.

The prospect of rewriting many of our equations to include the k factor, just so that they can become units-invariant, probably seems strange, and tedious, to most people who have been used to using the simpified versions of the equations in which it is assumed angles are measured in radians. But that only puts us in the same position as physicists who were brought up under the CGS system of units, in relation to electrical measurements.

Prior to the introduction of SI, a limited form of natural units held sway in electrical theory. Back in the 19th century, the unit of electric charge was defined as that charge which repels a similar charge placed one centimetre away with a force of one dyne. Of course, Coulomb’s Law, which describes the electrostatic forces between charges, gives this as a proportional relationship: F = kq1q2/r2, where q1 and q2 are two charges, r is the distance between them and F is the force, with k as the constant of proportionality. The CGS system then replaces k with the number 1 – a pure number – so that it disappears from equations, and electric charge then appears as a purely mechanical quantity, albeit with dimensions which are strange fractional powers of M, L and T. This formed the basis of electromagnetism and became deeply ingrained into the whole of physics, including quantum electrodynamics – probably a major reason why Feynman and others adopted a similar system of natural units for QED in the 1940s.

Nowadays, of course, the constant of proportionality has been restored, so that the modern formulation of Coulomb’s Law is units-invariant: F = q1q2/4piε0r2, where ε0 has the value 8.8 x 10-12 farad m-1 in the SI system, and 1/4pi statcoulomb2 dyne-1 cm-2 in the CGS system.

Before the SI formulation was introduced, it was necessary for anyone using the electrical equations to remember what units they were couched in. Similarly, when we deal with angles – particularly when they appear on their own, and not as the arguments of trigonometrical functions – we have to remember that the formulae we have memorised will only work in radians. The recognition of a separate dimension for angular measure would have a similarly generalising effect on formulae containing angles, and would remove ambiguities such as the rather absurd fact that, if we limit ourselves to the three mechanical dimensions M,L and T, distinct quantities such as angular momentum and energy appear to have the same dimensions.

Unfortunately, though, increasing the number of dimensions, and introducing a new fundamental constant, would run counter to a popular tendency among physicists, which is to entrench the hegemony of “MLT”, and even seek to reduce the number of dimensions.

Already we live in a world in which – at least for mechanical considerations – M, L and T are considered to be the only “legitimate” dimensions. We note this – and also how absurd it is – when investigating how certain quantities, whose only crime is to be represented by integers, are handled.

An important statistic for particle colliders is the number of particles in the beam passing through unit area in unit time. It is a measure of how intense the beam is, but it is actually called, for some reason, the luminosity (which must confuse astronomers who cross over into particle physics, since that word is also used to describe the intrinsic brightness of a star). More often, though, what is actually wanted is the integrated luminosity, which tells physicists how many particles have passed through unit area since the experiment began. Since the luminosity can vary with time, and indeed can go to zero for long periods when the beam is switched off, the operation required to obtain this figure is indeed integration. What we end up with is just number-of-particles per unit area. The unit area here is of course a tiny area, and a common unit used for such purposes nowadays is the picobarn, equal to 10-40  square metres. But physicists quote integrated luminosities, not in particles per picobarn, but in “inverse picobarns”. The reason for this appears to be that because “number-of-particles” does not have dimensions which can be expressed in terms of M, L or T, it should be regarded as dimensionless. But as we have already seen, dimensionless numbers tell us nothing about physical quantities. We can apply Bridgman’s definition of dimension to the case of “number of particles”, and show that this quantity too has dimension. Bridgman’s method involves reducing the size of the unit by an arbitrary factor, but we can equally well increase it by some factor. In colliders, particles are grouped together in bunches, each of which contain a large number of particles – let’s say there are B particles in each bunch. So we can represent this as reducing the size of our unit by a factor 1/B. And if we do this, the numerical value of the integrated luminosity will reduce by a factor B, or in other words, increase by 1/B. So by Bridgman’s method, we have to conclude that integrated luminosity has dimensions BL-2, and not just L-2.

Interestingly, the comparable statistic used by neutrino physicists is protons on target (the protons produce pions when they hit the target, and the pions decay to muons and neutrinos) so they do at least acknowledge the protons.

One can, of course, take this to extremes. Protons per picobarn is a distinct quantity from neutrinos per picobarn; similarly, cycles per second (hertz) is distinct from radians per second or revolutions per second – something our 1st year lab students often agonise over, sometimes inventing the fictional unit “rad hertz” in their confusion – and meteorites entering the atmosphere per second, and cars per second on the M1, are different again. In reality, there is thus an indeterminate number of quantities, and not just three.

To be continued …

1 Wikipedia attributes the introduction of the radian “in everything but name” to Roger Cotes in 1714, with the name first being used by James Thomson in 1873. But presumably there were angles before 1714 …

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