Quantum Relativity

We're used to measuring things with a lot of different types of units. In the US, we measure small things in inches, medium things in feet, and large things in miles. We weigh things in pounds. Time is measured in seconds, minutes, and hours. Some things are measured in combinations, like miles per hour. And, there's all sorts of units that don't play into our daily lives very often for things like forces, power, torque, etc. In physics, we deal with all of these units on a routine basis, and more.

Most of these units were chosen for historical reasons, many of which have since been forgotten. Feet were originally the length of a man's foot. Seconds were probably chosen by the average heart rate of a man. A minute is the time it takes for the sun or the moon to rise or set. Miles were originally 1000 paces of a Roman centurion. So, this is all very interesting for walking around Europe or watching sunsets, but it really doesn't have much to do with the fundamental laws of physics.

It's purely an accident of chemistry and biology that we see time and space as so different. Using the speed of light, c, as a conversion constant, we can measure distance in seconds or time in feet. One light second is about 186,000 miles. One foot of time is about one nanosecond. So, we agree to set c equal to 1, and we agree that we will use the same measure for time and space.

Einstein's special theory of relativity tells us that E = Mc². But, we have just agreed that c is one, so energy and mass have the same units, pounds or kilograms or horsepower-hours, we can choose whatever we wish.

From quantum mechanics we learn that everything oscillates with a frequency η = E / h, where h is Planck's constant. This is completely fundamental - all energy oscillates, whether it's a photon or a bowling ball. It's just an accident of history and human perception that we choose to measure energy or mass with a different scale than time or distance. So, we'll agree to measure energy and mass using units of frequency, meaning "per second" or "per meter." Now, having made this agreement, Planck's constant is 1. We still haven't chosen a basic unit, but we have reduced most everything to this one unit.

Distance is now measured in meters. Velocity, distance per time, is meters per meter, so velocity has no dimensions. In our system of measurement, a velocity of one is the speed of light. The speed limit on most freeways is 65 miles per hour which equals about c / 10,000,000. If physicists were running the highways, apparently highway signs would say "Speed limit 10^-7." That's it, no units.

Acceleration is velocity per second, so acceleration has dimensions of "per meter." Mass also has dimensions of "per meter," so F = Ma tells us that force has dimensions of "per meter²." Gauss' law of static electricity tells us that F = e² / r², so e, the electric charge, is dimensionless. The fine structure constant α = e ²/4π is also dimensionless.

Finally, we have one more fundamental unit in nature: G, Newton's gravitational constant. The units of G can be deduced from Newton's equation of gravity, F = GmM / r². Since the force has dimensions of "per meter²," and the 1/r ² term has dimensions of "per meter²," we see that GmM has no dimension. Therefore, since Mm has dimensions of "per meter²," G must have dimensions of "meter²." Now, our big leap: we'll set G to one, and therefore start using the same dimensions that the universe naturally uses - we'll call them "natural dimensions," sometimes referred to as "God's units."

In cgs (centimeter-gram-seconds) units, G = 2/3*10^-7 cm³ / g-s². Thus, we see that G / c³ = 1/4 * 10^-38 s/g. Now, we multiply by h = 6.6*10^-27 erg-sec = 6.6*10^-27 g-cm² /sec and we get 1.63 * 10^-65 cm². Finally, the square root of this number is √(hG/c³) = 4.04*10^-33 cm. This will be our fundamental unit of length and time, which we will call the Planck, abbreviated as P. We can live with just one fundamental unit, but for convenience sake we will define one additional unit. Our mass and energy unit will be h / (c * Planck) = √(hc/G) = 5.45*10^-5 g, which will call the Stone, abbreviated as E. Note that the Stone is simply 1 / Planck. Wherever we use Stone, we could write Planck^-1.

Now we're left with just a few factors of 2π and such in various places. For example, our Lagrangian is now in units of mass, as we expect for an energy term, so the time integral of the Lagrangian, the action, is dimensionless, as we expect. We'll use the action to find the phase of a particle as phase = exp( i 2π S t ). We could have scaled our units to eliminate this 2π, but we prefer to leave it in as an explicit reminder of the difference between time and radians.

Below is a conversion table and a list of constants. This is enough in most cases to work real problems in natural units and get answers in MKS.