A short introduction to the easy (but sometimes unnecessarily confusing) topic of natural units.

Units and Dimensions

Einsteins and Plancks

The basic dimensions in physics are length, mass and time, which are commonly expressed in the SI system of units as the metre, the kilogram and the second.

In particle physics, the typical scale of quantities is lengths of \(10^{-15}\mathrm{m}\), masses of \(10^{-27}\mathrm{kg}\) and durations of \(10^{-7}\mathrm{s}\)

As in any branch of physics, we should adopt a system of units that is appropriate for the scale of quantities involved. We adopt a system called natural units, where the fundamental constants \(c\) and \(\hbar\) are set to unity:

    SI Units Natural Units
\(c\) Speed of light \(3\times 10^8\mathrm{ms}^{-1}\) 1
\(\hbar\) Reduced Planck constant \(1.05\times 10^{-34}\mathrm{Js}\) 1

That is to say, \(c\) and \(\hbar\) are used as the base units of velocity and action (or angular momentum) respectively.

One further quirk is that when we write quantities in natural units, there is no need to write in the units of \(c\) or \(\hbar\), because they are defined to be unity, so they are suppressed.

Natural units are very neat in particle physics, because many of the formulae we write down contain factors of \(\hbar\) and \(c\), and so become neater and more transparent (and easier to remember for the exam!) when we use natural units. Contrast

\[\left(\beta mc^2 + c \sum_{n \mathop =1}^{3}\alpha_n p_n\right) \psi (x,t) = i \hbar \frac{\partial\psi(x,t) }{\partial t}\]

with

\[(i{\cancel{\partial}} - m) \psi = 0.\]

(Ok, the Feynman slash tidies it up a little too… That’s a topic for a future post though)

There’s another, more physical reason for using natural units though. The only reason we need the speed of light in formulae is because we don’t measure space and time in the same units. It is properly seen as the conversion factor between distances and times (seconds and metres, for example). But relativity tells us that space and time are unified into one object – spacetime. If we treat time and space on the same footing, we should measure distances and durations in the same units. What’s the speed of light in metres of distance divided by metres of time? One! (for more on this, check out Spacetime Physics).

What about the energy?

We can still specify one more unit, which we choose to be the unit of energy. We take the electronvolt (eV), which is defined as the kinetic energy an electron picks up falling through a potential of one volt.

\[1\mathrm{eV} = 1.6\times 10^{-19}\mathrm{J}.\]

Different authors may use MeV or even GeV. Don’t panic! It’s just scaling by a power of ten.

In natural units, then, all quantities have dimensions of a power of energy, since the whole system is specified in terms of \(\hbar\), \(c\) and \(\mathrm{eV}\).

Quantities in SI and Natural Units

From known relations, we can show that if we have a quantity of some certain dimensions

\[[\mathrm{quantity}] = \mathbf{M}^\alpha \mathbf{L}^\beta \mathbf{T}^\gamma,\]

then this can be written in terms of \(\hbar\), \(c\) and energy as

\[\left( \frac{\mathbf{E}}{c^2} \right) ^\alpha \left(\frac{c \hbar}{\mathbf{E}} \right)^\beta \left(\frac{\hbar}{\mathbf{E}}\right)^\gamma\]

and since in NU \(\hbar\) and \(c\) are set to unity, we have that in the NU system, the quantity has dimensions of

\[\mathbf{E}^{\alpha - \beta - \gamma}.\]

For example, if I have a duration (dimensions of time), then in NU it will have units of \(\mathrm{eV}^{-1}\).

As an example, let’s convert a length of \(l_{NU} = 1\mathrm{eV}^{-1}\) back to SI units. We need to put back in the factors of \(\hbar\) and \(c\) as follows:

\[\left[ \hbar^\alpha \right] \left[ c^\beta \right] \left[ \mathrm{eV}^{-1} \right] = \mathbf{L}.\]

We end up with a set of equations for \(\alpha\) and \(\beta\), and find that

\[\begin{aligned} l_{SI} &= \hbar c\ l_{NU}\\ &= 1.97\times 10^{-7} \mathrm{m}. \end{aligned}\]

Remember to make sure to use the right units for the numerical values of \(\hbar\) and \(c\), and happy physicsing!

References

Robert Klauber - Student Friendly Quantum Field Theory

Taylor & Wheeler - Spacetime Physics: Introduction to Special Relativity

(though I recommend the \(1^{st}\) edition)