There has been a lot of fuss about the fine-structure
constant (α),
perhaps because it’s a specifically odd value, at very very close to
1/137. And 137 is an odd number, both in
that it’s not even and also in that it’s a prime. And it’s a special prime, being a Pythagorean
prime because 88*88+105*105 = 137*137, and the square root of 137 is the
hypotenuse of a triangle with integer legs (4*4+11*11=137). 1/137 has a palindromic period number.
The value of the fine-structure constant is not, however,
precisely 1/137. It’s closer to 137.036,
which is not as sexy.
This doesn’t stop some people from getting excited about,
including our fine-tuning friends – for example Luke
Barnes. The reason for this
(they argue) is that if the fine-structure constant were even slightly
different then stars would either fail to produce oxygen (which I think we can
all agree is important) or fusion could not occur at all – with a margin of
about 4% either way.
The thing that’s a bit odd is that the discussion is all
about this fine-structure constant, and yet the value of the elementary charge
seems never to be mentioned.
What, you may ask, does the elementary charge have to do
with the fine-structure constant? If so,
that means you didn’t follow the Wikipedia link regarding what the fine-structure
constant is, because the very first two sentences are:
In physics, the fine-structure constant, also known
as Sommerfeld's constant, commonly denoted α (the Greek letter alpha),
is a dimensionless
physical constant characterizing the strength of the electromagnetic
interaction between elementary charged particles. It is related to
the elementary
charge e, which characterizes the strength of the coupling
of an elementary charged particle with the electromagnetic
field, by the formula (4πε0).ħcα = e2.
So the fine-structure constant is proportional to the square
of the elementary charge, because ε0, ħ and
c are all constants (and 4 and π are also constant – note that I added the brackets
above, they aren’t there on the Wikipedia site). What I find interesting is that 4πε0, ħ and
c are not only constant but, in Planck units, they all resolve to 1. Note also that ħ is the reduced Planck
constant, the Planck constant divided by 2π.
We could call 4πε0 “raised permittivity
of free space” or the “raised electric constant”.
This might seem to be a little bit of a cheat, but it should
be noted that µ0 has as similar but
inverse relationship to Planck units, in that µ0/4π (“reduced permeability
of free space” or the “reduced magnetic constant”) resolves to 1 in Planck
units, so that not only does c2=1/ µ0ε0 but that relationship
remains the same when µ0 and ε0 are replaced with
their increased and reduced versions respectively. Note also that the fine-structure constant
can be expressed in terms of permeability, by the formula (ε0/4π).ħcα = e2. And these two constants frequently appear
with a 4π in the appropriate place, almost they are begging someone to normalise
them in a similar way to how the Planck constant is normalised.
Normalisation removes the mystery of why, when all the other
fundamental constants seem to resolve to 1 at the Planck scale, these two (µ0 and ε0) don’t. They do when normalised. What remains outstanding however is the fine
structure constant. It’s a dimensionless
value, so how could we possibly resolve it down to 1?
The answer is hiding in those equations - (4πε0).ħcα = (4π/µ0).ħα/c = e2. Or, once reorganised - α = e2/(4πε0).ħc = (e/√((4πε0).ħc))2. So does √((4πε0).ħc)
have any meaning that we should be aware of?
You bet it does – it’s the Planck charge, or the charge on the surface of a sphere that is one
Planck length in diameter and has a potential energy of one Planck energy.
So, put another way: α = e2/qpl2,
the fine-structure constant is effectively an expression of the ratio of the
elementary charge (e) to the Planck charge (qpl), in
much the same way as the gravitational coupling constant is effectively
an expression of the ratio of the rest mass of an electron (me) to the Planck mass (mpl), or αGe = me
2/mpl2.
(If you look up “electromagnetic coupling constant”, you’ll be redirected
to the fine-structure constant.)
If you read about the gravitational coupling constant, you
will note that there “is some arbitrariness in the choice of which particle’s
mass to use”. It appears less arbitrary to
select the elementary
charge when considering the electromagnetic coupling constant (ie
the fine-structure constant), but it is still a little arbitrary. There is a smaller charge that could be selected,
that associated with quarks, which could be as low as e/3 (positive or
negative depending on the type of quark).
Before I take the next step, I have to point out that while the
gravitational and electromagnetic coupling constants (as commonly understood)
are effectively an expression of the ratio between the relevant characteristic
of an electron to the relevant Planck unit, this isn’t the meaning
of these coupling constants. They are both
defined as “a constant characterizing the attraction between a given pair
of elementary
particles”, electromagnetic attraction in the case of the fine-structure/electromagnetic
coupling constant and gravitational attraction in the case of the gravitational
coupling constant. There is also a
definition based on the interaction of these elementary particles with the
related field.
We could naturalise both of these constants by considering
instead “the attraction between a pair of Planck particles” or the interaction
of Planck particles with the relevant field, considering them to have both Planck
charge and Planck mass. When we do, the
values both resolve to 1.
Another way of saying this is the fact that the coupling
constants don’t have a value of 1 is merely because the electron mass and
charge are both smaller than the Planck equivalents (the mass is much smaller, but
the gravitational coupling constant is also much smaller than the fine-structure
constant). When people are talking about
the range in which the fine-structure constant could be varied without
affecting life in this universe (by preventing stars from doing what they need
to do to create the basic building blocks of life as we know it), they are
really talking about how much higher or lower the charge on the electrons and
protons can be. It’s actually a bit odd
that fine-tuners don’t do this because when they say that the fine-structure
constant can only vary by as much as 4% before we run into trouble, this is equivalent
to saying that the charge on an electron or proton can only vary by as much as
2%. If there is fine-tuning
here, then there’s actually twice as much fine-tuning (on this single measure)
as the fine-tuners are claiming.
Either way, it’s a bit unreasonable to point at the fine-structure
constant as an example of fine-tuning in and of itself. If the fine-tuners want to claim any
fine-tuning here, they need to point to the elementary charge. However, if they can explain why
elementary charge is odd in some way or could be something else than it is,
they are welcome to try. There doesn’t
seem to be anyone else looking into that and when people ask awkward questions
there’s a lot of “we just don’t know”. And, so far, the fine tuners appear to have steered clear of the elementary charge.
No comments:
Post a Comment
Feel free to comment, but play nicely!
Sadly, the unremitting attention of a spambot means you may have to verify your humanity.