# The dimension of the fine structure constant

1. Nov 24, 2007

### Koilon

Everyone "knows" that the fine structure constant (alpha) is a dimensionless number, but I am troubled by the fact that the Planck unit of resistance must be one - if that is true - and that strikes me as suspicious. I wonder if anyone can shed any light on this.

I present the complete calculation here:

In cgs units, Planck's constant, the speed of light and the Newtonian gravitational constant are, respectively:

H = 6.6260755E-27 g.cm^2/sec = M.L^2/T
C = 2.99792458E10 cm/sec = L/T
G = 6.6873E-8 cm^3/g.sec^2=L^3/M.T^2

Solving these equations or the Planck units of mass, length and time, repectively, we get:

M = SQRT(H.C/G) = 5.45020877E-5 g
L = SQRT(H.G/C^3) = 4.055503739E-30 cm
T = SQRT(H.C/C^5) = 1.352770435E-40 sec

Now, the charge on a electron is:

Q = 1.60217733E-20 SQRT(g.cm/mu) = 4.803206799E-10 SQRT(epsilon.g.cm^3)/sec,

where mu and epsilon are magnetic permeability and electrical permittivity, repectively. Multiplying these two values, we get:

Q^2 = 7.695589045E-30 (g.cm^2/sec).SQRT(epsilon/mu)

for the cgs value of the square of the charge on an electron.

Now, (g.cm^2/sec) is the dimension of Planck's constant and SQRT(epsilon/mu) is the dimension of conductivity or the reciprocal of resistance. So, the dimension of the square of charge is the same as the dimension of Planck's constant divided by resistance:

Q^2 = H/R

However

alpha.H/2.Pi = 7.695589115E-30

if

1/alpha = 137.0359895.

So, if the fine structure constant is dimensionless, and

Q^2 = 7.695589E-30 = alpha.M.L^2/2.Pi.T.R,

then R must equal one in Planck units. It makes more sense to me to say that alpha has the dimension of electrical conductivity.

2. Nov 24, 2007

### blechman

You have to be careful what your electrical units are. In Gaussian units (epsilon == 1), electrical conductivity is also dimensionless. This is what people are assuming when they say that $\alpha$ is dimensionless. Jackson's E&M text has a whole appendix dedicated to this.

3. Nov 25, 2007

### Parlyne

There's not really a need to make any assumptions about what is or isn't dimensionless in deriving the fine structure constant. It really comes from the understanding that $\alpha$ is just the strength of the electrostatic force made dimensionless.

Regardless of the units involved, we can write Coulomb's Law as $\vec{F} = k_e \frac{q_1 q_2}{r^2} \hat{r}$. Expressing the charges in terms of e, we can write $q_i = n_i e$, where n is clearly dimensionless. Then, the magnitude of the force will be $F = k_e e^2 \frac{n_1 n_2}{r^2}$. The factor $k_e e^2$ is clearly universal, so it represents the dimensionful strength of the electrostatic force.

Given that $[F] = \frac{ML}{T^2}$, and that $[r^2] = L^2$, it follows that $[k_e e^2] = \frac{ML^3}{T^2}$. As it happens, $[\hbar c] = \frac{ML^3}{T^2}$. Thus, the dimensionless combination must be $\frac{k_e e^2}{\hbar c}$. This will hold generally, no matter what system of units you're using. The only difference will be in the specific form that $k_e$ takes.

4. Nov 25, 2007

### rbj

no, $\alpha$ is dimensionless, whether or not we define the unit of charge as derived from M, L, T so that $\epsilon_0 = 1$ or leave it as some dimensionful value as in the SI system. in the electrostatic cgs system, the unit charge is defined so that the Coulomb force constant (what in SI is: $1/(4 \pi \epsilon_0)$) is the dimensionless one. then the dimension of electric charge is L3/2 M1/2 T -1, it doesn't have its own unique dimension of "stuff", and

$$\alpha = \frac{e^2}{\hbar c}$$

but, in a general system of units, then

$$\alpha = \frac{e^2}{4 \pi \epsilon_0 \hbar c}$$

in either case, $\alpha$ is dimensionless.

5. Nov 25, 2007

### Koilon

Gaussian units

Was I NOT careful about my electrical units?

The Gaussian system of units uses cgs-electrostatic units to measure electrical quantities and, in cgs-electrostatic units, epsilon is, indeed, 1 while mu is 1/C^2. On the other hand, the Gaussian system of units uses cgs-electromagnetic units to measure magnetic quantities and, in cgs-electromagnetic units, epsilon is 1/C^2, while mu is 1.

So, it is not correct to say that in Gaussian units epsilon equals 1. It is also not correct to say that electrical conductivity is dimensionless! The CRC Handbook of Chemistry and Physics, 41st edition, page 3180, gives the dimension of resistance as:

R = time/permittivity.length = permeability.length/time.

Multiplying the two right-hand sides of this equation together gives the dimension of the square of resistance as:

R^2 = permeability/permittivity.

Since, electrical conductivity is the reciprocal of resistance, the dimension of electrical conductivity is SQRT(permittivity/permeability) is ANY units.

"Jackson's E&M text" is not familiar to me. A search on the Web leads me to believe the you are referring to "Electromagnetic Theory" by J.D. Jackson. I will be sure to look up the appendix you mentioned at the earliest opportunity.

6. Nov 25, 2007

### blechman

you're right, i was being stupid. sorry about that post.

Koilon: I'm a particle physicist, where we use "natural units" in which c=1. But rbj's post is correct - the fine structure constant is dimensionless in all unit systems.

jackson's textbook is the standard textbook for graduate-level E&M.

7. Nov 25, 2007

### Parlyne

Conductivity is not the reciprocal of resistance. The reciprocal of resistance is conductance.

8. Nov 25, 2007

### Koilon

I feel that I need to back up and clarify a couple things.

First, I understand that practicing physicists and mathematicians are accustomed to "simplify" their notation by any and all sorts of conventions and omissions but, just as Alfred Tarski once decried the sloppy notational practices of mathematicians and propounded the maxim: NO NOTATION WITHOUT DENOTATION to rectify it, I decry it among practicing physicists and go further to add the maxim: NO DENOTATION WITHOUT NOTATION.

In the present context, the practice of particle physicists of setting Planck's constant, the speed of light, and the gravitational constant to one and leaving them out of the notation is a case of DENOTATION WITHOUT NOTATION. Another case is the practice of leaving out the dimensions. That is like leaving dx or $$\partial$$y out of a differential equation.

I also understand the PRACTICAL advantage of SI units for a PRACTICING physicist, but I am more interested in physical reality and the notations used to model it. There are no conventions in reality and I hold that conventional "simplified" notations complicate, rather than simplify, by removing symmetries from notation that are inherent in the physics.

For example, not to get into a lengthy exposition, simply witness the symmetry of physical reality brought out in the notation simply by using arbitrary units of mass, length, time, permittivity, and permeability as our base units, instead of mass, length, time and current:

$$Q^2 = \frac{ML^2}{T}\sqrt{\frac{\epsilon}{\mu}}\ \ \ \ \ \ \ \Phi^2 = \frac{ML^2}{T}\sqrt{\frac{\mu}{\epsilon}}$$

$$I^2 = \frac{ML^2}{T^3}\sqrt{\frac{\epsilon}{\mu}}\ \ \ \ \ \ \ \ P^2 = \frac{ML^2}{T^3}\sqrt{\frac{\mu}{\epsilon}}$$

$$D^2 = \frac{M}{TL^2}\sqrt{\frac{\epsilon}{\mu}}\ \ \ \ \ \ \ \ B^2 = \frac{M}{TL^2}\sqrt{\frac{\mu}{\epsilon}}$$

$$H^2 = \frac{M}{T^3}\sqrt{\frac{\epsilon}{\mu}}\ \ \ \ \ \ \ \ \ \ E^2 = \frac{M}{T^3}\sqrt{\frac{\mu}{\epsilon}}$$

$$C^2 = T^2}\frac{\epsilon}{\mu}\ \ \ \ \ \ \ \ \ \ \ \ \ \ N^2 = T^2}\frac{\mu}{\epsilon}$$

$$G^2 = \frac{\epsilon}{\mu}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R^2 = \frac{\mu}{\epsilon}$$

where

Q = electric charge
$$\Phi$$ = magnetic flux
I = electric current (magnetic potential, )
P = electric potential (magnetic current)
D = electric flux density
B = magnetic flux density
H = magnetic field intensity
E = electric field intensity
C = capacitance
N = inductance
G = conductance (thank you, Parlyne)
R = resistance

Surley, this has pedagogic advantage if nothing else.

Second, in my original post I probably should have made it more clear that I was thinking in terms of ARBITRARY units. Physics does not depend on which units you use. If you leave out all of the numerical values down to "However" and change "g", "cm", "sec", "epsilon" and "mu" to "mass", "length", "time", "permittivity" and "permeability" - it is just algebra (or operational calculus) in terms of arbitrary units.

So far, I see nothing but what amounts to a tired litany of dogmatic assertions that the fine structure constant is dimensionless and a complete failure to address the evidence presented in my original post that this would imply that the ratio of permittivity to permeability is always unity!

rbj: How did you get the superscripts in $$T^3/2M^1/2Y^-1$$ without using Latex?

9. Nov 25, 2007

### blechman

Koilon: you've gotta cool it, and think about what it is you're saying!

No, this is just wrong. There is absolutely nothing wrong with setting $\hbar=c=1$ and expressing all quantities in units of energy. I leave it to you to figure out exactly what UNIQUE factors of Planck's constant and c you need to include to get any unit you require. Since these factors are ALWAYS UNIQUE!! I see no reason to include them one way or anther.

10. Nov 25, 2007

No comment.