Units

• Intuitive
• Concrete
• Abstract

Intuitive

Measuring something means comparing it with a reference. When we say a table is 2 metres long, we mean it is twice as long as the reference length called a metre. The number 2 alone says nothing; it only makes sense together with the unit. Someone using feet would write a different number for the same table.

Units are conventions. Different communities chose different references, which is why the same physics can look different in different books. Engineers mostly use SI units. Many textbooks on electromagnetism use Gaussian units, in which Coulomb's law has no extra constant in front. Particle physicists measure almost everything in electronvolts.

Some constants of nature can be used as units themselves. Because light always travels at the same speed, a distance can be expressed as the time light needs to cover it, as astronomers do with the light-year. Once we do this, length and time become the same kind of quantity, and the speed of light becomes the number one. In the same way Planck's constant links energy and frequency. Physicists who set both constants to one use natural units, in which every quantity can be expressed as a power of a single unit, usually an energy.

Units also give a quick test of any formula: both sides of an equation must have the same units. If they do not, the formula is wrong.

Concrete

• The International System of Units (SI) - the modern metric system of measurement.
• Gaussian - commonly used in electromagnetism.
• Heaviside-Lorentz (rationalized Gaussian)
• Natural units - commonly used in quantum electrodynamics and particle physics in general.

• A great discussion can be found in Section 2.1 in Klauber's "Student-friendly Quantum Field Theory".
• See also BABEL OF UNITS by Neal J. Carron

In natural units, where $\hbar = c = 1$, any physical quantity can be expressed by using as fundamental unit either energy, mass, length or time.

$$\begin{align} ct&\approx \ell\Longrightarrow {\rm time}\approx{\rm length},\\ E&\approx mc^2\Longrightarrow {\rm energy}\approx{\rm mass},\\ E&\approx pv\Longrightarrow {\rm energy}\approx{\rm momentum},\\ Et&\approx \hbar\Longrightarrow {\rm energy}\approx{\rm (time)^{-1}}\approx {\rm (length)^{-1}}. \end{align}$$

Abstract

The group of dimensions. Choose $k$ base dimensions, for example length $L$, mass $M$ and time $T$. The dimension of a quantity is a monomial $L^{a}M^{b}T^{c}$, which we can identify with the exponent vector $(a,b,c)$. Products of quantities add these vectors, so dimensions form a free abelian group $\mathbb Z^k$ (or a vector space $\mathbb Q^k$ if fractional powers are allowed, as for charge in Gaussian units). A system of units is a choice of one reference quantity for each base dimension. If we replace the unit of length by one that is $\lambda_L$ times larger, and similarly for the others, the numerical value of a quantity of dimension $(a,b,c)$ changes by

$$\{Q\} \mapsto \lambda_L^{-a}\lambda_M^{-b}\lambda_T^{-c}\,\{Q\} .$$

So the group $(\mathbb R_{>0})^k$ acts on numerical values, and each dimension is a character of this group. A physical law written in terms of numerical values must be invariant under this action. This is the statement that each term of an equation has the same dimension.

Buckingham $\pi$ theorem. Suppose a physical relation $f(Q_1, \ldots, Q_n) = 0$ holds in every system of units, and the dimension vectors of the $Q_i$ span a subspace of rank $r$. Then the relation can be written as $F(\Pi_1, \ldots, \Pi_{n-r}) = 0$, where the $\Pi_j$ are independent dimensionless products of powers of the $Q_i$. For a pendulum with period $t$, length $\ell$, mass $m$, gravitational acceleration $g$ and amplitude $\theta_0$, the rank is $3$ and the dimensionless quantities are $t\sqrt{g/\ell}$ and $\theta_0$, so $t = \sqrt{\ell/g}\,F(\theta_0)$ and the period cannot depend on $m$. See dimensional analysis.

Natural units as a quotient. Setting $c = 1$ declares the dimension of $c$, $LT^{-1}$, to be trivial. Formally we pass to the quotient group $\mathbb Z^3/\langle (1,0,-1)\rangle$, in which $L = T$. Setting also $\hbar = 1$ quotients by the dimension of $\hbar$, $ML^2T^{-1}$, and leaves a single dimension, usually taken as mass or energy, with $[L] = [T] = [M]^{-1}$. Adding $G = 1$ leaves no dimension at all, which gives Planck units. Because the dimension vectors of $c$ and $\hbar$ are linearly independent, an expression written in natural units determines uniquely which powers of $c$ and $\hbar$ have to be restored to get back SI dimensions, once the dimension of the result is known.

Electromagnetic units. SI and Gaussian units differ in the number of base dimensions, not only in scale. SI has a separate base unit for current, so charge has its own dimension, and the constant $\varepsilon_0$ is needed to convert. In Gaussian units charge has dimension $M^{1/2}L^{3/2}T^{-1}$. The equations change form accordingly. Coulomb's law reads

$$F = \frac{q_1q_2}{4\pi\varepsilon_0 r^2} \ \text{(SI)}, \qquad F = \frac{q_1q_2}{r^2} \ \text{(Gaussian)}, \qquad F = \frac{q_1q_2}{4\pi r^2} \ \text{(Heaviside-Lorentz)},$$

and correspondingly the fine-structure constant is $\alpha = e^2/(4\pi\varepsilon_0\hbar c) = e^2/(\hbar c) = e^2/(4\pi\hbar c)$ in the three systems, with $e$ measured in the respective units. The dimensionless number $\alpha \approx 1/137$ is the same in all of them.

Leads to: Dimensional Analysis.