The great Swedish mathematician, Leonhard Euler (1707-83), is responsible for the single most beautiful result in all of mathematics, Euler’s identity:
eiπ + 1 = 0 ,
where e is Euler’s number, the base of natural (or Napierian) logarithms, 2.718281828459..., i = √-1, and π is Archimedes’ constant, 3.1415926535897932384...
I have been fascinated by the numbers e and π for a long time, and by the relationship between them, and between them and certain other numbers – both the numbers of mathematics (for example, the golden mean, f, and the Fibonacci sequence[1]) – and those of physics.
The ones that interest us here are the neutron-electron mass ratio, the proton-electron mass ratio, the fine-structure constant (otherwise known as the electromagnetic coupling constant), and the electron’s magnetic moment, expressed in Bohr magnetons. All of them, as we shall see, are related to e and π.
First, the neutron-electron mass ratio:
mn/me = eπ2κ/2α = 1838.68366 ,
where α is the fine-structure constant[2], and κ is a dimensionless constant, equal to 1.000249044275.
Turning to the proton-electron mass ratio, this is:
mp/me = eπ2ζ/2αμe = 1836.152755656 ,
where μe is the electron’s magnetic moment in Bohr magnetons (= 1 + ae, where ae is its magnetic moment anomaly, = 0.001159652193), and ζ is another dimensionless constant, equal to 1.0000305692511.
It turns out that:
α = eξ/12π3μe = 0.00729735308 .
This time, the dimensionless constant, ξ = 1.0000116993595.
It has been known for a long time that
mp/me ≈ 6π5 = 1836.1181 ,
and now we can see why!
It has also be known – since Julian Schwinger’s pioneering work on quantum electrodynamics, in fact – that a first-order approximation of the electron’s magnetic moment anomaly can be obtained from the expression α/2π = 0.0011614. We should not be surprised, therefore, that α can be derived from its magnetic moment in Bohr magnetons, together with e and π.
What is far more surprising is the relationship between α, e, and π and the neutron-electron and proton-electron mass ratios. The masses of the proton and neutron do not depend solely on their electromagnetic fields, but also on the hyper-strong interactions between their component quarks, and – if they are embedded within atomic nuclei – on the strong interactions with the other protons and neutrons surrounding them and (possibly) weak interactions, such as beta decay, which entail neutrons decaying into protons, electrons and anti-electron-neutrinos.
Consequently, quantum electrodynamics (‘QED’) is not sufficient, by itself, to account for these masses. Electro-weak theory and quantum chromodynamics (‘QCD’) are also required – together constituting, with the theory of the Higgs field – needed to account for the masses of all particles – the so-called ‘Standard Model’.
The Higgs field, and its associated boson (a boson obeys Bose-Einstein, as opposed to Fermi-Dirac, statistics, and has integral, as opposed to half-integral, spin[3], and therefore need not obey the Pauli Exclusion Principle) is something of a deus ex machina, conjured up to rescue the situation because without it the Standard Model would otherwise be in a complete mess.
The equations of the theory – in their raw state, prior to renormalisation – give ridiculous answers, riddled with infinities. After renormalisation, they give sensible, finite answers – but the majority of the parameters have to be written in ‘by hand’, as it were, having been measured experimentally. A good scientific theory is supposed to predict what numbers the experimenters will find, so that they can check to see if the theory is right – or rather, wrong, because (if Sir Karl Popper is right) science proceeds by trying to falsify the deductions from hypotheses, not verify them.
The Higgs Field is supposed to fill all of space and is thought to provide all the various particles of the Standard Model with their respective masses by interacting with them rather like a crowd surrounding and mobbing a celebrity, asking for an interview, his autograph, or simply wanting to take a photograph or shake his hand. It exerts drag on each particle, in other words, and gives it its inertia – which is what mass is, or what it measures.
One further modification of the Standard Model may be needed – and that is supersymmetry, which is a set of symmetry operations occurring in fractal dimensions, allowing for fermions to convert to bosons and vice versa, and allowing for the existence of bosonic counterparts of fermions and fermionic counterparts of bosons. Thus quarks, electrons and neutrinos (fermions) have squarks, selectrons and sneutrinos (bosonic counterparts); whereas gravitons, photons, gluons, W+, W– and Z0 particles (the bosons of the gravitational, electromagnetic, strong and weak fields respectively) have gravitinos, photinos, gluinos, winos and zinos (fermionic counterparts). None of these supersymmetric particles has yet been seen experimentally. The Large Hadron Collider (LHC) at CERN in
The relationship between inertial mass and gravitational mass – which is a measure of the extent to which a given body alters the curvature of its local space-time – is supposedly governed by Mach’s Principle. However, that is a matter for another time.
[1] The golden mean, f = [(√5) + 1]/2 = 1.6180339887498948482... It is yet another transcendental number!
[2] The fine-structure constant, α = e2/4πε0ħc = e2μ0c/4πħ = 0.00729735308 (= 1/137.03598949333), where e is the electronic charge, ε0 is the electrical permittivity of the vacuum, μ0 the magnetic permeability of the vacuum, ħ is Dirac’s constant and c the speed of light in vacuo.
[3] Meaning its intrinsic angular momentum is in whole units of ħ, as opposed to fermions, whose spin is in half-units, ħ/2, or 3ħ/2, and so on. It is important not to take the idea of angular momentum too literally here. An electron’s classical radius re = αħ/mec = 2.81794092 × 10-15 m. If you try working out its ‘angular velocity’ – as if it really were a spinning ball with that radius! – then it would have an angular velocity equal to c/2α, or roughly 68½ times the speed of light! (Its angular momentum is ħ/2, so it’s easy enough to work out, thus: ħ/2mere = c/2α.)
