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I am wondering whether the five$^1$ units of the natural unit system really is dictated by nature, or invented to satisfy the limited mind of man? Is the number of linearly independent units a property of the nature, or can we use any number we like? If it truly is a property of nature, what is the number? -five? can we prove that there not is more?

In the answers please do not consider cgs units, as extensions really is needed to cover all phenomenon. - or atomic units where units only disappear out of convenience. - or SI units where e.g. the mole for substance amount is just as crazy as say some invented unit to measure amount of money.

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$^1$length, time, mass, electric charge, and temperature (or/and other linear independent units spanning the same space).

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Maybe you should say what you mean by "natural unit system", since to me that means $c = \hbar = 1$ like Richard Feynman intended. –  dmckee Nov 28 '11 at 3:19
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By the natural unit system, I mean e.g. units for length, time, mass, electric charge, and temperature(or and other linear independent units spanning the same space). en.wikipedia.org/wiki/Natural_units –  Hans-Peter E. Kristiansen Nov 28 '11 at 14:47
    
Notice that my question is about dimensionallity, and not about the units themselves - I do not care weather they are based on universal physical constants, or man made artifacts. –  Hans-Peter E. Kristiansen Nov 28 '11 at 15:08
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The number of free units is determined by the number of ignorable fundamental constants in your day-to-day life. In our case, it's basically three, because we live in the nonrelativistic non-quantum weak-gravitational regime. The number of units is not a property of nature, but of our position in it. –  Ron Maimon Nov 28 '11 at 16:54
    
Hans-Peter, can you elaborate how you arrive at five physical dimensions? I count three: e.g. length, time and mass. –  Johannes Nov 28 '11 at 19:43
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5 Answers 5

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.

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I think you misunderstood my question. I do not care about the units themselves(e.g. if it is the meter or Planck length unit used for length scale does not matter). –  Hans-Peter E. Kristiansen Nov 28 '11 at 14:59
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@Hans-Peter: I did not misunderstand, and this is a correct answer. There are no "units" in a fundamental theory of nature, or rather there are as many as you like, you can choose the unit of force to be different from the Newton. The question is why do you have about 3 units for macroscropic physics, and this is because there exists a G=0 c=infinity h=0 limit of microscopic physics. –  Ron Maimon Nov 28 '11 at 16:31
    
Again - I do not care about the scale of the units. It makes no difference. - And setting a unit equal to one does not makes it disappear, it is only done because it makes the notation easier in certain fields of physics. –  Hans-Peter E. Kristiansen Nov 30 '11 at 4:55
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@Hans-Peter: Setting the right constant to unity makes a unit disappear into nothing. To see this, imagine that you measure height in inches and length/width in centimeters. Then you have a universal constant of nature, equal to 2.54 cm/inch which is used to do rotation. When you set this unit to 1, you measure height and length with the same unit, and one unit disappears entirely. The same is true of c, and hbar, and G. Setting them to one makes the units vanish into nothing. It is not choosing a natural unit, it is showing that the units are unnatural to begin with. –  Ron Maimon Nov 30 '11 at 7:37
    
What you are saying, @RonMaimon, is that basically all the units are like the candela. They're just there to make things convenient for human experience. –  Mark Beadles Feb 9 '12 at 3:23
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You will find the answer in this paper A self-similar model of the Universe unveils the nature of dark energy , section II - ON UNITS, PHYSICAL LAWS AND SCALING , part A. On quantities and units . Read until you find that Atomic measures are number counts: 'Summing up, as long as the number of particles does not change, the measures of properties of bodies using atomic units can hold invariant in spite of eventual variations in the properties of elementary particles.'

I'will copy/past a few sentences.

in the Abstract : This work presents a critical yet previously unnoticed property of the units of some constants, able of supporting a new, self-similar, model of the universe. This model displays a variation of scale with invariance of dimensionless parameters, a characteristic of self-similar phenomena displayed by cosmic data. The model is deducted from two observational results (expansion of space and invariance of constants) and has just one parameter, the Hubble parameter. Somewhat surprisingly, classic physical laws hold both in standard and comoving units ...

  1. Vocabulary : explains "base and derived quantities", Measurement : "Quantities have to be expressed by numbers; this operation is called measurement and consists in the comparison between the quantity to be measured and a scalar quantity of the same kind defined and adopted by convention, called measurement unit" Natural units - Note that the use of constants for defining base units, as done in Natural units, does not fit in the above definitions of metrology because constants are not quantities;
  2. Base units are not independent - One must not confuse quantity with its measure, i.e., the number we attribute to the quantity; this number depends both on the measuring method and on the characteristics of the units. A common confusion is the one between the quantity “speed of the light” and its measure. ... Base quantities, as concepts, are independent one another, but the respective base units are not. The classical approach is to choose length, time, mass and charge. Theoretically, these four quantities are enough; however, physical laws are expressed as a function of temperature as if it was another independent quantity; to consider it a base quantity greatly simplifies the description of physical systems. ... length is a geometrical, static, concept; time is a concept linked to the flow of occurrences, the contrary of static; they are, clearly, distinct concepts. Now, let us look at the SI units of time and length. The unit of time, the second, is defined as the duration of a number of periods of the radiation produced in a transition between two specific energy levels of an atom; the length unit, the meter, is defined as the length of the path traveled by light in the vacuum in a certain time interval. As it is obvious, if by some reason the time unit changes, the length unit will also change, as long as the speed of light does not changes accordingly; or, ...Therefore, while the concepts of length and time are independent, their units are not. This has consequences in the description of the universe; for instance, relativistic space-time is a property of the description of the universe using such units and a reference frame calibrated by the method described by Einstein. Consider now mass and charge. ... The last base quantity is temperature;... Therefore, length, time, mass and charge units are deeply linked through the properties of atoms and speed of light. Note also that, because the atomic structure depends on fields, which propagate at the speed of light and with characteristics defined by field constants, the atomic properties will vary in case of a variation of the speed of light or field constants, implying a change in units. That is to say, not only units are linked one another but they are also linked with field constants
  3. All accepted systems are equivalent - All the differently defined base units of the different accepted systems have shown so far to be invariant in SI units, being not known any system of units able of supporting physical laws that is not invariant in SI units. This indicates that all these units may be just proportional and that no different description of the universe arises from using one system or another. ... Cosmic data allows the definition of a special length unit, known as the comoving length unit, which increases with time in relation to the atomic unit. There is no system of units based on the comoving length unit as it is not known how physical laws could hold in a system of units whose length unit is not invariant in atomic units.
  4. Atomic measures are number counts. ... a measure of the mass of a body using atomic units is basically a baryon count... The above reasoning shows that the measures of mass, charge and length of bodies are independent of the mass and charge of elementary particles and of atoms’ radii, tracing only the number of particles or atoms. In what concerns atomic time unit, it is such that holds invariant the measure of the average speed of light in a closed path in vacuum

The above paper answers a more vast question: It shows that the physical laws stay the same if the atom is not an invariant ! If matter shrinks then we are doomed to measure the 'space expansion'. Why should the matter evanesce? Because particles and fields (grav and electromagnetic) are hand in hand one to the others. No one will deny that fields propagate in space at c speed, and fields have the energy that makes the world evolve. Then, necessarily the energy in the fields are originated (sourced) from the energy in the particles. To keep energy balance (there are no free lunches) the particles must decrease in energy content as long as the field spread away (i.e. always). We can not measure this in the lab but we see it in the universe at large. The cosmological reddening of light is due to bigger atoms of the past, and Dark Energy is not needed to explain the measures.

The paper was born outside of academia and is not peer-reviewed. It is poison. DE quest is like a Snipe hunt (in PT 'caça de gambuzinos').

Altogheter Dark Eenergy , space expansion, inflation, are misconceptions, magical thinking, 'free lunches'. Thats the correct wording when we have 'Consequences without a probable Cause', isn't it?.

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I am ignoring the mole-like and candela-like dimensions here, they really are worthless. Add one or two to my answer if you want to consider these

Yes. Any measurement will have to involve 5 parameters. We really can't prove the absence of more, though. If a new phenomenon is discovered, we may have to assign a new dimension/unit to it.

In SI units(minus mol and cd), we have five independent dimensions. No closed relations between quantities of different dimensions can be written (By 'closed' I mean without introducing new variables. eg length and time are related, but the relation introduces the speed as a new variable). Thus, in natural units, we also must have five. Any group of $n$ variables with no closed relationships can only be written in terms of $n$ or more different variables. Otherwise, one would be able to solve an equation of $n$ variables, $n-1$ equations by transforming the $n$ variables into $n-1$ new variables. So, in planck units, we start with five variables (m,l,t,q,T), and we manage to write them in terms of five new variables. This is the same for any other complete unit system.

About the mole and candela

Actually, the mole is a pretty important unit. I'd personally prefer if one mole=one unit of substance, instead of the cumbersome $N_A$ (Makes more sense, too as long as we don't want to do any chemistry). As you said in a comment above, "And setting a unit equal to one does not makes it disappear". Well, it's not obvious, but we DO need a unit for "amount of substance". In normal day-to-day calculations, while tallying up stuff, we do need to take into account amount of substance. Fortunately, we use a unit where 1 amount of substance=1 unit, so to us the mol looks superfluous. But for analysing the amounts of units required to describe the universe, it is essential. The mol manages to condense an infinitude of other units into one. These units are $, apples, oranges, bananas, houses, chairs, broken tables, etc., i.e. anything you can count. If you want to treat everything dimensionally correctly, you should use mol (or your own unit where 1 unit=1amount of substance) to count these. Normally, I myself would consider the mol superfluous, but as we are talking about fundamental units, I think it deserves a place here.

The candela, on the other hand, is invented to describe human perception of light, so as a fundamental unit it has no meaning. No humans $\implies$ no need for the cd.

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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.

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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 temerature in joules not kelvins, but Moe thinks (with equal justification) that you are crazy for measuring torque in newton-meters not moe-units.

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