Energy and Context

Warning: This is an opinion piece on what a "fundamental" physical unit is. Like all arguments of this nature, it's as ascetic as it is logical. Hence this article is full of bias.

From modern particle physics by Thomson

“In natural units everything can be expressed as a power of GeV’s” my professor proclaimed. It's my first class on particle physics. After class ended I asked him how one could take a multi-dimensional space of units and make it one dimensional. He explained in natural units physicists simply “set $\hbar = c = 1$". Oh, Obviously. I stopped asking questions. 

I had actually seen the reduction of the number of units before in physics. In introductory physics you usually learn SI units or Système International d'Unités. In SI there are seven fundamental units:

1. Meter (Distance)
2. Kilogram (Mass)
3. Second (Time)
4. Ampere (Charge)
5. Kelvin (Tempature)
6. Mole (Amount)
7. Candela (Luminous intensity)

This system is clearly a bit redundant. Moles and candelas feel out of place. After all, luminous intensity can be defined in terms of power per solid angle. Why should it be fundamental? And a countable amount like the mole is a unit, sure, but is it a physical one? After all, does nature count?

This conjures the question: what is a "physically fundamental" unit? Put simply, a physical unit is a unit that cannot be written in terms of other units. Distance, time, and mass are examples of this.

S.I. units are more about precise measurement in than philosophical realness. The units are based off highly constant and measurable standards. And with this in mind I felt safe rejecting moles and candelas as not physically fundamental units.

I knew temperature to be an emergent property. It’s often described as simply the kinetic energy of the material’s molecules, but this isn’t quite true. However, it really is just energy stored by other fundamental forces in a certain way within matter.

Out of this mess four units remained. The kilogram, ampere, meter, and second. Two were from fundamental forces: gravity and electromagnetism, and the other two: meters and seconds, came straight from space and time.
 

And for a while that was what nature was to me. I accepted these units as fundamental and in a mathmatical sense, independent. That is until I met appendix A of Sakurai’s Modern Quantum Mechanics. Here Sakurai describes another type of unit system, Gaussian units. Gaussian units or CMS units have only three units: centimeters, grams, and seconds. There was no charge! Naturally this blew me away.

Here’s why: When you trash a unit, I feel you reject that the unit is fundamental. It is not just a record keeping trick, it’s philosophical. You assert the existence of a relationship that makes that unit redundant. Sakurai said in Gaussian units the force between two charges $Q_1$ and $Q_2$ is simply:
$$ F = \frac{Q_1Q_2}{r^2} $$

            This implies charge can be expressed in units of $\frac{g^{1/2}cm^{3/2}}{s}$. The same way you can build energy, luminosity, and temperature from other physical real units under certain conditions in SI units, you can build charge in CMS from just mass, distance, and time. This is a break down. Clearly charge is something special, right? At least as special as mass. What’s going on?

Well, maybe not. What we need to ask is: what is special about charge? To answer this question let’s diverge to address another very important, but not fundamental unit, the Newton.

$F = ma$ says that force is simply acceleration scaled to the amount of mass. However to say it’s just $\frac{kg*m}{s^2}$ would be misleading. The operations going from distance and time to acceleration are the tricky part. It’s not distance distributed about some area of the time after all, it’s the rate at which an object increases it’s velocity over time.

Is there really a difference between positive and negative charge? Take a positive charge alone under no electromagnetic forces. Is it distinguishable from a negative charge? No, of course not.

We can only define the two groups of charges by how they interact with each other. Repulsion and attraction, forces.

Suddenly it was clear, charge wasn’t really a fundamental unit, but rather a tool for rationalizing the symmetric behavior of electromagnetism. After all gravity and electromagnetism have very similar looking laws in mechanics, the fundamental difference being that gravity is always attractive. It’s a kind of one sided electromagnetism. And only one unit, in this case mass (but it is arbitrary), is necessary to build their shared physical properties: force and energy.

Of course throwing away units doesn’t necessarily make our life easier. Units keep track of things, and that’s important. For instance, earlier we disregarded temperature being merely emergent from energy stored in materials. However, if you seriously wanted to leave temperature behind in record keeping and not use it as a unit then you would have to use units of energy to record the total thermal energy. Then, in order for your description to be as useful as temperature, you also have to record the amount of the substance and type of material (to specify the specific heat curve).

So maybe temperature isn’t a fundamental unit, but it is a pretty damn useful one. The same with charge. Positive and negative might not be fundamental, but labeling and knowing which tends to collect on cloth when it’s rubbed against a glass rod makes our lives so much easier. Thankfully, even if charge doesn’t have a unit in Gaussian units we can still assign positive or negative values to charges accordingly, avoiding some information loss. However if you simply wrote $15\frac{g* cm^{3}}{s^2}$ I wouldn’t know if you were expressing charge squared, energy times distance, or force times area. So it’s important to carefully explain what kind of physical quantity you’re describing.

Ultimately reducing everything to powers of energy in natural units can be seen as a very similar procedure, exploiting the natural symmetry between time and space expressed by $c$. It makes time and space two sides of the same coin and revokes their units. This reduces all of physics to energy and context. It’s bizarre procedure indeed, but one that is philosophically important. And certainly one that deserves a better explanation than “we simply set $\hbar = c = 1$”.