## The Joy of Natural Units

I’m glad it’s the end of the week. It’s been ridiculously busy. It didn’t help that I was already exhausted before it started, after a hectic three days in Geneva. Part of the reason for being so heavily occupied is that my teaching duties have just doubled. I teach the second half of a module called Nuclear and Particle Physics, and I’ve just taken over  for the second half of the semester to cover the part about particle physics. I started my set of 11 lectures with one about natural units, which is a lot of fun because it usually divides the class into two opposing camps.

About half the students think natural units are crazy, and the other half think they’re great. I’m in the second camp. The motivation is straightforward: particle physics combines quantum theory, which involves Planck’s constant

$\hbar \simeq 1.05 \times 10^{-34}\,\,\,{\rm Js}$

with special relativity, which involves the speed of light

$c\simeq 3 \times 10^{8}\,\,\,{\rm m s}^{-1}$ .

Using everyday SI units (metres, seconds and kilograms) to deal with quantities that are either ridiculously small or ridiculously large doesn’t make any sense but, more importantly, the SI units don’t really reflect the physics very clearly.

In natural units we take these two constants to be equal to unity, so they don’t appear in any formulae:

$\hbar = c =1$

For example, the energy invariant in special relativity is usually written

$E^2=p^2c^2 + m^2c^4$

This is where the most famous equation in physics

$E=mc^2$

comes from. However, the equivalence between mass and energy (and also momentum) is much more clearly expressed in the natural units system:

$E^2=p^2 + m^2$

None of those tiresome factors of $c^2$ to remember! Mass, energy and momentum are all expressed in terms of the same natural unit of energy (usually, in particle physics, the GeV).  You can keep track of which is which by the simple expedient of using different names.

Velocities are, of course, always expressed as a fraction of $c$ in this system so have no units.

In quantum theory we find energy $E=\hbar \omega$ becomes $E=\omega$ so energy is expressed in the same units as frequency. Energy is thus a measure of inverse time.  Momentum $p =\hbar k$ becomes just $p= k$ so momentum is an inverse length.  This is in accord with the various forms of Heisenberg’s Uncertainty Principle too:  $\Delta p \Delta x \sim \hbar$ is $\Delta p \Delta x \sim 1$ and $\Delta E \Delta t \sim \hbar$ becomes $\Delta E \Delta t \sim 1$. A particle with a finite lifetime thus has a finite energy width which is inversely proportional to the lifetime. It makes sense to use energy units for both of these things.

As an extra bonus we can dispense with the clumsy way that electromagnetism is handled in the SI system by noting that

$\frac{e^2}{4\pi \epsilon_0 \hbar c} \equiv \alpha\simeq \frac{1}{137}$

is dimensionless. In the SI system the coulomb force between two electrons is $\frac{e^2}{4\pi \epsilon_0 r^2}$ whereas in natural units it is just $\frac{\alpha}{r^2}$, which is much nicer. Incidentally, the strange quantity $\epsilon_0$ that appears in the SI version is called the permittivity of free space. Nice name, but I wonder what it means?

The dimensionless quantity $\alpha$ on the other hand, has a very clear  physical meaning: it is the fine structure constant,  a coupling constant that measures the strength of the electromagnetic interaction.

Some people – including emeritus professors of observational astronomy – object to natural units because they hide the units that things are expressed in. They don’t actually. What they do is express things in units that are better geared to the physics. In any case, if you want to convert back to SI units you can always do so straightforwardly with a little bit of dimensional analysis. This is necessary if you have to talk to engineers and the like, perhaps so they can build you a particle accelerator, but in the more elevated company of particle physicists you should definitely follow proper etiquette and keep your units natural.

### 22 Responses to “The Joy of Natural Units”

1. Think you missed a + from your E^2=p^2+m^2?

• telescoper Says:

You chaps are fast, but quite right. Sorry the wordpress latex thing is so clunky I missed the “+”…

2. Sure E^2 = p^2 + m^2
Or Pythagoras is turning in his grave somewhere….

3. But a good read anyway 😉
I never quite got my head around “natural” units – they always felt counter intuitive to me. For some reason, having lots of constants made me feel comfortable as an undergraduate.
But I’m now discovering astronomers units are worst of all. Ergs? grams? cm?
I’ll stick to SI, thanks!

4. A. Some of the LaTeX stuff still has typos.

B. Astronomical units? What about the AU itself, or the millijansky, or the megaparsec?

5. Sorry I probably wasn’t as clear as i’d have liked. Things such as the AU etc are great because they fit the scale of what astronomers talk about. But the insistence on cgs units by some is just infuriating. Maybe it’s a generational thing more than anything else???

6. telescoper Says:

I recommend you quote all distances in units of the Planck length.

7. Bryn Jones Says:

Natural units can occasionally have a place when teaching fourth-year undergraduates, but only with great caution. That Peter has devoted a lecture to explain natural units in his particle physics half-module indicates that he is approaching the issue with the necessary caution. However, unless very carefully explained, natural units can confuse many students, and for good reason.

I can remember being very confused as an undergraduate when some lecturers chose to use c.g.s. units, often without stating that they were doing so. Familiar equations then lost the permittivity of free space, and gained or lost factors of pi, often without explanation. Consistency of notation is highly desirable and S.I. should be the standard, although special units (such as parsecs, kpc, Mpc and astronomical units) can have a place for simple quantities like distances.

Units matter critically, as students often need to use equations to calculate quantities for themselves.

8. Surely it’s

m^2 = E^2 – p^2

(I like to keep the invariant on one-side, and frame-dependent on the other)

• telescoper Says:

We all like to have something on the side, but it’s best to keep quiet about it.

However I do object to one of your latter remarks ; “if you have to talk to engineers and the like, perhaps so they can build you a particle accelerator”. I don’t believe “engineers build YOU an accelerator”. Perhaps it is better to rephrase this into “engineers/physicists/scientists/etc. build an accelerator”. Engineers don’t work for particle physicists; engineers work, and particle physicists work. Period.

Last objection: “in the more elevated company of particle physicists…” Thinking of yourself as “elevated” immediately implies that you think of others as “degraded” or the like. This attitude is, in my opinion, regrettable; it is the kind of attitude that at present slows down major scientific endeavours, which require tight collaboration of people with many different backgrounds.

• telescoper Says:

Sorry you took offence at my gentle digs at engineers. I’m not a particle physicist, by the way.

10. Hm, interesting. I think I would belong to the “this is crazy”-side of your class. But could you possibly clear one issue up for me? If c = 1, then I’m allowed to write expressions as exp(c). Yet if I then turn back to actual units, my previous expression would suddenly have fallen ill. So is the natural unit-method simply mathematical tickery (Well, okay, everything in physics is mathematical trickery, but here I mean in the sense of sometimes not making logical sense like resulting into ill-defined expressions; like using it as informal shorthand, much like interpreting dx as infinitesimally small but not zero)?

Thank you 🙂

11. The symbol c is not used in natural units; the speed of light is set to unity.. There is no way you would want to write exp(c), therefore. You are certainly allowed to write exp(1) if you wish. If you wish to write exp (v), where v is some other velocity, then that would be interpreted as exp (v/c) in unnatural units.

There’s no “trickery”.

12. Thank you, I appreciate it.

Let v be a certain constant mathematical number between 0 and 1. Now if you say “the velocity is v” when you’re using natural units, wouldn’t that become “the velocity is vc” in unnatural units? The important difference is that in that case, expressions like exp(v) when in natural units will become exp(vc) in unnatural units and thus have a dimensioned exponent and be invalid, i.e. natural units would in that case also have units but just use them implicitly. Please correct me where I am wrong.

• No. In natural units, velocities are fractions of the speed of light and are dimensionless. The Lorentz factor thus includes terms in 1-v^2 in natural units, which become 1-(v/c)^2 in standard systems.

13. Nice post. Just thought I’d mention that energy is not invariant. I think you mean mass.

• It doesn’t say that energy is invariant, but I agree my wording is a bit sloppy. I meant to say the invariant associated with energy/momentum 4-vector or something like that.

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