units and nature

Even seasoned professionals disagree on this one. Trialogue on the number of fundamental constants by M. J. Duff, L. B. Okun, G. Veneziano, 2002:

This paper consists of three separate articles on the number of fundamental dimensionful constants in physics. We started our debate in summer 1992 on the terrace of the famous CERN cafeteria. In the summer of 2001 we returned to the subject to find that our views still diverged and decided to explain our current positions. LBO develops the traditional approach with three constants, GV argues in favor of at most two (within superstring theory), while MJD advocates zero.

Okun's thesis is that 3 units (e.g. $c$, $\hbar$, $G$) are necessary for measurements to be meaningful. This is in part a semantic argument.

Veneziano says that 2 units are necessary: action $\hbar$ and some mass $m_{fund}$ in QFT+GR; or a length $\lambda_s$ and time $c$ in string theory; and no more than 2 in M-theory although he's not sure.

Finally, Duff says there is no need for units at all, all quantities are fundamentally subject to some symmetry, and units are merely conventions for measurement.

This is a very fun paper and answers your question thoroughly.

The number of SI units is not at all fundamental to nature: the natural Planck units are the fundamental things. But at low energies, large systems, and slow speeds, there are scaling laws that allow you to pick three units arbitrarily.

This is why humans think they get to pick three arbitrary scales. This is false, but it seems true from our experience. An alien civilization will probably choose a similar convention of units, with a different triplet of arbitrary quantities, even though nature doesn't have units at the fundamental level.

The correct natural system of units fixes $\hbar=c=1$, and $G$ to something order 1, determined by the details of the quatum gravity theory. So that there are no units at all in nature. In addition, you fix the constant $\epsilon_0=1$ to give a natural unit of charge, Boltzmann's constant $k_B=1$ to give a natural unit of temperature, and the remaining SI units, the mole and the candela, are just silly. The cgs system goes partway, and eliminates electrostatic units. This is perfectly fine, and cgs will work for any situation.

The reason for using units is that you have a kind of emergent scale invariance in certain regimes. This does not necessarily mean that the theories are exactly scale invariant, but that when you are given one theory with a certain scale, you can always imagine another theory with another scale also knocking around, and the two theories can talk to each other, and there is no reason to prefer one over another.

This type of scale invariance, scale indifference really, requires you to choose an arbitrary unit for convenience, to fix a scale. The fundamental theory is not scale indifferent at all, but we don't work with Planck-length sized black holes on a day-to-day basis, so we are in a scale indifferent regime. This justifies some of the SI unit choices.

For energies less than the Planck mass-energy, but everything relativistic, you have an approximate energy scale indifference, in the sense that there are particles of different masses that behave approximately the same. Choosing a mass scale in this regime becomes arbitrary, so you need a unit, traditionally the eV, for masses, and inverse length/times.

If you go nonrelativistic, time and space no longer scale together, and energies are quadratic in wavenumbers. Now for any fixed mass, you are allowed to scale space by a certain amount, so long as you scale time by the square of that amount. This nonrelativistic scaling law means that you to pick a unit of space, since c is no longer 1 but infinity, while the eV determines a unit of time because hbar is still 1.

If you go to macroscopic systems, the action of anything is enormous compared to hbar, and you lose your natural unit of mass. Then you can choose a unit of mass arbitrarily, and the eV trades in to a macroscopic unit of time, and the unit of length is still around.

In macroscopic systems, the dynamics is generally invariant under separate rescalings of mass, space, and time, in that if you have a system with certain values of any one of these, you can find rescaled systems which go twice as fast, or half as fast, etc. This is a rule of thumb, of course, magnetic effects coupled with electric effects determine c, and so relate time to space, but it is generally valid.

These three units are significant, because they correspond to the $G\rightarrow 0$ (low energy) $c\rightarrow \infty$ (low speed) and $\hbar \rightarrow 0$ limits in which you pick up 3 arbitrary scales. The remaining SI units are not as significant, the mole and the Kelvin, the candela (this one is especially strange), and the Ampere (which is defined naturally anyway). The Kelvin can be replaced by a Joule/mole without any loss, the mole is really a pure number, and the Ampere is defined by the condition that two wires of 1 Ampere at a distance of 1m repel with a force of 2 10^{-7} Newtons, which allows you to define the Ampere as a certain combination of other SI units. The existence of approximate charge scaling can be used to justify a unit of charge, or equivalently a unit of current.

The questioner cannot seem to wrap his head around some of the concepts involved here. I will try to illustrate, with a little story, why one can legitimately consider temperature to not be an independent unit, by comparing it to torque.

Let's talk about my imaginary friend Joe. Joe takes this position:

"The temperature of a system is defined as the average energy of an ideal harmonic oscillator in thermal equilibrium with it."

Joe expresses all his temperatures using only energy units, not kelvins. When I asked Joe what temperature it is outside, he says "the temperature is 26 milli-electron-volts". You or I might say that Joe is "setting the Boltzmann constant to one", but that's not Joe's opinion. Joe has never heard of the Boltzmann constant, and has never even heard of the word "kelvin". Nevertheless, Joe has no problem discussing any nuanced aspect of temperature and thermodynamics with his like-minded friends.

If it's not yet clear what I'm getting at, let's talk about my other friend Moe, the arch-nemesis of Joe. Moe is a mechanical engineer who often is calculating torques, but he thinks it is totally crazy to express torques in newton-meters. He expresses torque in "moe-units". There's a fundamental constant Moe's Constant: $$M_C = 1248 \text{ (moe-units)}/(\text{newton-meter})$$ Moe uses this formula for torque: $$\vec{\tau} = M_C \, (\vec{r} \times \vec{F})$$ I asked Moe how many units there are, and he said "six: length, time, mass, electric charge, temperature, and torque."

The point of this fairy-tale is that you may think Moe is crazy for using moe-units when newton-meters would work just fine...but Joe thinks (with equal justification) that you are crazy for using kelvins when joules would work just fine. You may think Joe is crazy for measuring temperature in joules not kelvins, but Moe thinks (with equal justification) that you are crazy for measuring torque in newton-meters not moe-units.