Planck charge

I have recently done a couple of posts based on Planck units and this is another in the series. This one though is a little different. This is about charge.

When we thing about a Planck mass or Planck length there are a number of different things to compare them against. Take the Planck mass, we can take the ratio of the Planck mass to the mass of the electron, or the proton, or the neutron. Each of these will give a different value. Similarly if we take the ratio of the Planck length with the radius of an electron, or a proton or a neutron, again different values.

When we consider the charge however this is not the case, the charge on the electron is the same magnitude as that of the proton, only different in sign, positive or negative. So the ratio of Planck charge to the elementary charge will be the same, in fact it turns out that

$ \frac {e^2} {q_p^2} $ $ = \alpha  $    ....(1)

e - elementary charge
q_p - Planck charge
$\alpha$ - fine structure constant

I did a post on the fine structure constant a while back, where I quote Richard Feynman describing it as a magic number, a true mystery. I like mysteries. So here it is, the ratio of the elementary charge and the Planck charge.

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Aside : The number that Feynman describes is actually that given by

$ \frac {e} {q_p} $  $= \sqrt \alpha  = 0.08542455 $    ...(2)
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What baffles me a little though is that the Planck charge does not appear to make sense. The Planck charge is 11.70623 times the charge of the electron, the elementary charge.

Apart from quarks that have a charge of 1/3, charges are integer multiples of the elementary charge. At least that is the way it appears at the moment. Quarks do not appear to exist in isolation so you do not get fractional charges.

In other words, it should not be possible to create a charge in isolation of 11.70623 e. So we can never actually create a particle with the charge equal to the Planck charge.

In which case the question remains, what exactly is the Planck charge?

The equation defining the Planck charge is 

$ q_p = $ $ \sqrt {4 \pi \epsilon_0 \hbar c}$ ... (3)

$\hbar$ - Reduced Planck constant - $1.0545717   10^{-34}  J.s$
$4 \pi \epsilon_0$ - inverse of Coulomb's constant - $1.11265   10^{-10} $
$\epsilon_0$ - permittivity of free space - $8.85418782   10^{-12}  m^{-3} kg^{-1} s^4 A^2$
c - speed of light - 299792458 $m s^{-1}$

which seems straight forward enough. Yes, it might look a little odd, but no more so than the other Planck units. Yet, unless it is some Fractional quantum Hall effect (FQHE) value, it is not real. In which case how can it give us the result in (2)? which is real and can be measured exceptionally accurately.

I can only think that it must be a valid value, does this imply it IS a FQHE value? or that there is another phenomena that has not been discovered yet that will allow a non-integer value of the elementary charge?

Let's say the value of the Planck charge could exist because of the FQHE. Does this give us any insights into why (1) is true?

The Fractional quantum Hall effect (FQHE)

the FQHE is a quantum mechanical version of the Hall effect and it is observed in 2 dimensional systems at really low temperatures in strong magnetic fields. This was discovered in the 1980s. I will post on this later. There is a value called the Hall conductance that is given by

$\sigma =$  $ \frac {I_{channel}} {V_{Hall}} = \nu \frac {e^2} {h}$ ....(4)

$\nu$ is called the "filling factor" and provides the fractional value (1/3, 2/5, 2/3 , 12/5 ....)

So is there a Planck version of this?

 $\sigma_p =$  $ \nu \frac {e^2} {h} = \nu_p \frac {q_p^2} {h}$ ...(5)

giving

$\nu$ $\frac {e^2} {h}$ $= \nu_p$  $\frac{q_p^2} {h}$  ...(6)

which becomes

$\alpha = $ $\frac {e^2} {q_p^2} = \frac {\nu_p} {\nu} $    ...(7)

Returning to the Planck charge, does the above imply that the Planck charge may only be valid in situations where FQHE is valid? to consider it anywhere else, while mathematically meaningful, would actually be physically invalid.

Alternatively, is there another theory of physics where fractional charges can exists in other situations besides the FQHE?

Using the Fractional quantum Hall effect (FQHE) it is possible to get fractional elementary charge. If we think that this may explain the problem of the Planck charge not having a integral charge, we can try the following

take $\alpha$ = 137.035999074 (44)

so $\sqrt {\alpha}$ as 11.7062376

The fraction 35/3 = 11.666666666

11.70623 / 11.666666 = 1.003391, which is pretty close. I do not think it is close enough though.

what about 199/17 = 11.70588235,

11.7062376 / 11.70588235 = 1.0000303, closer, can we do better? Yes. Try writing a simple number cruncher, you find hundreds of examples with closer agreement.

Using equation (7) as a starting point we could try this

$\frac {a}{b} e = \frac {c}{d} q_p$     ....(8)

a,b,c and d are integers. Try a=261, b=25, c=404, d= 453, you have integer values that give the number

11.70623762, squared this is 137.035999215

11.7062376/11.70623762 $\approx$ 1 or

137.035999074/137.035999215 = 0.99999999897

given the degree of error in $\alpha$ this value is pretty close to 1!

What does it mean though? Say the value of a,b, c and d are correct, so

$\frac {261}{25} e = \frac {404}{453} q_p$     ....(9)

or

$\frac {261}{404} e = \frac {25}{453} q_p$     ....(9)

why these value? why not others? These are as much of a mystery as 137.03599....

To see if I can get any further with this idea I shall be taking a look at the Hall Effect and the Quantum Hall Effect in more detail in a future post.