The Ascent of Hubbert's Peak.

Back in a snowy and freezing United Kingdom after my trip to Madeira, I was greeted, on arrival at Gatwick Airport, by a posse of Sussex policemen armed to the teeth with Hechler and Koch sub-machine guns, for all the world as if I were Osama bin Laden in person. I thought the pilot of my Easyjet flight had made a mistake, or that perhaps we had been caught up in some weird space-time warp, and transported to a 1970s-style Latin American military dictatorship. I half-expected to see a framed photograph of El Presidente on the wall somewhere.

Getting from Gatwick to St Pancras proved difficult, because of engineering works, but I managed to get as far as London Bridge, and then caught the Tube through to St Pancras. I got held up there because of a fire on an earlier train, but managed to get home eventually. I haven't been out much since, because of all the snow, but at least I'm not completely cut off from civilisation.

In the meantime, my work with APPGOPO (the All Party Parliamentary Group on Peak Oil and Gas) continues. I note with interest that the US Energy Information Administration (EIA), not a body generally associated with the idea of 'peak oil', is now saying that world production is currently running at 85.472 million barrels per day, whereas total world petroleum consumption is 85.534 million barrels per day, a deficit of 62,000 barrels per day, or 22,630,000 barrels per annum. It seems to me entirely possible that the peak of Dr Marion King Hubbert's curve has at last been reached, and that daily crude oil production figures will be declining from now on.

Yesterday, it was announced by the National Grid that demand for natural gas had reached an all-time high in the UK on Thursday, 7th January (see Guardian/Reuters report) of 454 million cubic metres per day (up from the previous record of 449 million cubic metres per day on 7th January, 2003). The National Grid was forced to ask industrial users to consider alternative forms of energy, and put out an alert to its suppliers for additional supplies of gas. Only about 70% of our gas needs are still met from UK domestic sources. I, for one, do not want this country to be as heavily dependent as Germany and Italy are on the dubious beneficence of Mother Russia for its gas supplies.

We should, in any event, not be burning fossil hydrocarbons and continuing to increase the atmospheric concentration of CO2. The current level, 387.75 ppmv, is the highest it has been for 15 million years, as I have said on more than one occasion now, and I really do not think we want the climatic and oceanic conditions of the Pliocene to prevail, as they will do again with certainty, after an appropriate time lag. Rather, we should be using hydrocarbons as raw materials, not sources of energy, and reducing our population size and density to a level that makes sense both for ourselves and for the planet.

It is time for ecological common sense to prevail.

Copenhagen's Dismal Deal.

President Obama needed to get back to Washington, DC, before he was cut off by a snow storm. Perhaps Chinese Premier Wen Jiabao was in a similar hurry. Both men were in a great hurry to reach a shoddy compromise at the Copenhagen Climate Conference - one that 'respected' China's national sovereignty (i.e., meant that China did not have to sign up to any international monitoring of its commitments on CO2 emissions) and, likewise, did not entail the US having to make any commitments to limit (still less reduce) its greenhouse gas output.

The resulting Accord is a non-binding 'political' deal which talks of limiting the increase in mean annual global temperature to below 2 Celsius above the reference datum (the pre-Industrial Revolution level), but does not will the means to achieve that objective, even if that objective were adequate - which it is not.

The politicians have failed - which is no surprise. The present generation of world political leaders do not have the intellectual or administrative ability to sort out this week's laundry, let alone anything of greater import. Barack Obama and Gordon Brown would probably find it difficult to run a bath, let alone their respective countries. The time has come for drastic action, and intervention from below - from, let us say, a socially unexpected quarter: the people, maybe.

Reflections from Madeira (2).

If it is the case that 1.5 billion people could live on this planet with an ecological footprint of 5 GHa/capita, then it is is certainly the case that a larger population could be accommodated with a smaller per capita allocation of bio-capacity.

For example, 3 billion people could live on Earth if they each had an ecological footprint of 2.5 GHa/capita. It would then only be necessary to reduce the Earth's human population from its present 6.79 billion to 3 billion, a reduction of 3.79 billion. (Let us remind ourselves that the ecological footprint is measured in terms of the area of biologically productive land or water that must be used to provide for the needs of one human being, and to eliminate his or her wastes.)

This implies, however, a substantial reduction in the material standard of living for those of us who live in Europe, for 5 GHa/capita is our current allocation, as indicated in my previous 'blog entry. It is a sacrifice that, IMHO, we can and should be prepared to make, in order to allow more of our fellow human beings the chance to live on this beautiful planet.

Reflections from Madeira.

I greet you, one and all, from the (currently) sunny Island of Madeira, where I am sitting in an Internet Café on the Municipal Square of Funchal, near the Museum of Religious Art, having eaten a good lunch, and feeling reasonably content with life (well, to the extent that that is possible) and ready once again to update this 'blog.

We have had rain. There was torrential - virtually tropical - rain yesterday, and drizzly rain this morning. Otherwise, however, Madeira has (apart from a brief shower on Saturday, just as the Easyjet flight bringing me here from Gatwick was landing) been bathed in warm sunshine, with temperatures in the upper teens Celsius.

The landscape is extremely hilly, not to say mountainous, and much of the vegetation is tropical or Mediterrranean. There are banana, palm and pineapple trees. I have sampled the Madeira (a secco), and found it very palatable. There are all sorts of oddities - teapots that don't pour unless you lift the lid, for example - but then I don't expect the Portuguese to know how to make tea (or 'cha') anyway - and they don't. They don't even know how to make a decent latte (in fact they don't even know what a latte is!).

The news from Copenhagen is bad - but I didn't expect it to be good. The idea that Ban Ki-Moon is apparently trying to float that the developing countries will have to agree to a deal with no definite financial provisions for them is a complete non-starter. Furthermore, I suspect that the White House 'optimism' about a deal is based around limiting atmospheric CO2 to 450 ppmv and the increase in annual mean global temperature to 2 Celsius. The problem is that the current level of CO2 (387.75 ppmv, according to the NOAA) is the highest level the Earth has seen since the Pliocene, 15 million years ago, when there were no Polar ice caps, and sea levels were 75 feet higher than they are today. James Hansen is right: we need to reduce CO2 from current levels, not allow it to increase.

Having watched Sir David Attenborough's very thoughtful Horizon TV documentary about human population on BBC2 the other day, and having read the WWF literature on sustainability, I am increasingly persuaded that, if we are to tackle the twin problems of global warming and resource-deficiency (which includes energy-, water- and food-scarcity), then we need, not merely to reduce human population growth, but the present size of the population. There are simply too many of us for this planet to sustain, and our numbers are such that, if they keep on growing at the predicted rate, so that they reach 9.1 billion by 2050, then Homo sapiens sapiens will join all the other members of genus Homo, and become extinct, along with the many other species that we will take with us into the Sixth Extinction. We have already killed off far too many other species, and are doing so right now.

The environmental feedback systems have a nasty habit of eliminating any population that threatens its local ecology - and there is no reason to suppose that the same logic would not apply on the vastly larger scale of the Earth. There are still those who think that the human population can go on increasing almost without limit on this planet - notably in the Roman Catholic Church, which remains steadfastly opposed to all forms of artificial birth control. The influence of the Vatican is important, given the fact that there are well over 1 billion Catholics in the world - but few, if any, of the Catholic laity in the developed world pay much heed to the teaching of Humanae Vitae; rather, it is the ill-educated poor of the developing world who are more likely to be turned away from practising safe methods of birth control (and prophylaxis against the transmission of STIs) by the intervention of the Church.

Even the most widespread availability and use of birth control will not be sufficient, however. The fact is - to put it as brutally and bluntly as possible - the population needs to be reduced and reduced drastically, if the human race is to have any chance of surviving past the mid-21st Century. That means that the present 6.79 billion figure needs to go down to 1.5 billion, a reduction of of 5.29 billion, or 77.9%. Clearly, the death rate per thousand of population will have to exceed the birth rate by quite a substantial margin to achieve that goal - how long for being determined by how much by.

The 1.5 billion number is the number of people the planet can sustain if everyone on Earth has an equal allocation of ~5 GHa (global hectares - i.e., land of average agricultural productivity) to provide for their personal needs and deal with their personal wastes. This is the average allocation for the typical Western European - somewhat smaller than the allocation for a typical American (>9 GHa), but much larger than that of a typical African or Indian (<1>link.

As to the means of achieving this desirable end, I am reminded of the famous debate between Ivan Karamazov and his brother Alyosha, in Dostoyevsky's The Brothers Karamazov, where Ivan asks his (deeply religious) brother if, supposing he could create a perfect world - one where there was no war, no crime, no disease or poverty, no suffering of any kind, only peace, harmony and universal brotherhood - but that, in order to do that he had to murder one tiny baby, would he do it? To which Alyosha replies very firmly in the negative - on the grounds that 'one may not do evil that good may come of it', and that ends can never justify means.

Suppose, however, that one is faced with two courses of action, neither one of which is morally good. Should not one's course then be to choose that action whose outcome is least desirable, not for oneself, but for others (applying the Kantian principle that one should always adopt as a maxim of one's conduct that policy of action which one would wish to be applied as a universal law)?

So, for example, if a policeman is forced to choose between shooting dead a criminal he would prefer to arrest and allowing that criminal to kill innocent bystanders, he will choose the former course.

In this case, we are confronted with the terrifying alternative of reducing the population by 5.29 billion (77.9%!) or having the human race become totally extinct.

Clearly, the former is preferable to species extinction, but does it necessarily entail killing people as opposed to them merely dying, or just failing to be born? Enforced sterilisation was tried in India, under the late Mrs Gandhi, and her son, Sanjay, back in the 1970s. The policy was a disaster for her and her Congress Party politically, and probably led to her eventual assassination - and certainly that of Sanjay. People are living much longer thanks to modern medicine, better food and improved sanitation, and there is even talk of treatments for senescence itself.

The conclusion that I am coming to - ghastly as it undoubtedly is - is that mass murder on a hitherto unprecedented scale may indeed be necessary in order to ensure the long-term survival of our species. That logic does not exactly fill me with great joy, to put it mildly. I hope that I am wrong (oh, completely, absolutely and hopelessly wrong, please!), and I will spend a great deal more time on this, before committing myself to anything (well, such as? My own little murder campaign, perhaps? And what would be the point of that, apart from to cause a great deal of unnecessary misery and suffering and get me - quite rightly - locked up?).

There has to be some alternative - like all of us reducing our ecological and carbon footprints, for example - but unless we do, and do so drastically, in combination with a reduction in population growth, we will most assuredly be doomed - of that I am certain.

A very 'Merry Christmas', indeed!

Parallel worlds.

The Heisenberg Uncertainty Principle states that:

 

∆p_x.∆x  =  ∆E.∆t  ≥  ħ/2 .

 

In other words (as is well known), the uncertainty in a quantum wave-particle’s momentum multiplied by the uncertainty in its position is greater than or equal to half the Dirac constant (Planck’s constant divided by 4π, in effect).

            The Uncertainty Principle allows for – strictly temporary – violations of the Law of the Conservation of Energy, whereby ‘virtual particles’ – the mediators or means of exchange of the four forces of nature – are allowed to ‘borrow’ energy from the ‘bank’ of the ‘quantum vacuum’, and ‘pay the loan back’ after a given period of time – the product of the energy and the time being no greater than ħ/2.

            Thus ‘virtual gravitons’ are emitted by one massive body and absorbed by another – and you have gravitational attraction between the two bodies; ‘virtual photons’ are emitted by an electron and absorbed by a proton and you have electro-static attraction between them, and so on.

            Let me introduce you to my friend, the d’Alembert Operator, named after Jean le Rond d’Alembert (1717-83).  It is represented thus:

 

  =  (1/c²)(∂²/∂t²) – (∂²/∂x² + ∂²/∂y² + ∂²/∂z²) .

 

The d’Alembert Operator is the equivalent of the Laplace Operator (the part of the above equation after the minus sign), which is represented by ∇², and is named after Pierre-Simon Laplace (1749-1827), in Minkowski space-time^[1].  The d’Alembert Operator appears in the Klein-Gordon equation, a version of the Schrödinger wave equation that is compatible with the Special Theory of Relativity, unlike Schrödinger’s original, but only applies to spinless particles, unlike the more general Dirac equation.

            At its simplest, the Klein-Gordon equation, for a particle with zero charge, but non-zero rest-mass, is:

 

ψ + m²c²ψ/ħ² =  0 ,

 

where ψ is the wave function.

            There are a number of things that should be said here.  In conventional quantum mechanics, the wave function is not a real function – in fact, literally not, because it is, mathematically speaking, a complex number, taking the form

 

a ± ib ,

 

where i = ±√-1, and a and b are real numbers.  This complex number, ψ, is said to be a probability amplitude, whose squared modulus is a probability – the probability that a particle will have a location within a given volume of space, or that it will have a given momentum, or a given spin.  These probability amplitudes can be expressed as complex vectors in an n-dimensional space (Hilbert space, named after the German mathematician, David Hilbert).

In contrast, in an ordinary wave equation, which is a hyperbolic second-order partial differential equation with no imaginary or complex terms, the wave function f is given by

 

∇²f  =  (1/c²)(∂²f/∂t²) ,

 

which is easily re-arranged as:

 

f  =  0 .

 

This is, in fact, the Klein-Gordon equation for a spinless particle with zero charge and zero rest-mass.  Photons and gravitons have zero charge and zero rest-mass, but photons have spin 1 and gravitons (if they exist – none have been detected so far) have spin 2.  There are spinless composite particles, but no elementary ones, as far as we know.

            Here, the wave function has the dimension length-squared, or area.  How can something that can be as simple as that in one equation be a complicated thing like a ‘probability amplitude’ in another?

            I am going to suggest something radical.  I am going to suggest we do away with probability amplitudes, and replace them with lengths-squareds.  Unfortunately, they won’t be straightforward lengths-squareds, but imaginary ones, and they will involve higher dimensions – not of space, but of time.

            We need to go back to Heisenberg, but also to de Broglie.  Count Louis de Broglie suggested in 1924 that sub-atomic particles like electrons were associated with waves which would have a wavelength inversely proportional to their momentum, λ_B = h/mv, where h is Planck’s constant, and m and v the mass and velocity of the particle respectively.  It was this idea that set Erwin Schrödinger on the path to his eponymous equation.  Let us remind ourselves what that says:

 

∇²ψ – (2m/iħ)(∂ψ/∂t) – 4πmUψ/ħ²  =  0 ,

 

where i is ±√-1, again, m and U the mass and potential energy of the particle.  The equation is second-order with respect to the spatial co-ordinates, but only first-order with respect to time, so is not, as we have said, compatible with the Special Theory of Relativity – a fault which was rectified by Oscar Klein, Walter Gordon, Paul Dirac, and Ettore Majorana, among others, after January 1926, when Schrödinger published his paper Quantisierung als Eigenwertproblem^[2] in the Annalen der Physik, the journal in which Einstein had published his seminal papers on Brownian motion, the photo-electric effect and the Special Theory in 1905.

            Schrödinger himself was always convinced that his wave function represented something physically real, and that de Broglie waves themselves were physically real, rather than being ‘waves of probability’ (or rather, of probability amplitude, which makes them even more abstract).  However, Marshall Stone and John von Neumann was able to show that wave mechanics – Schrödinger’s system – was equivalent to the matrix mechanics formulated by Heisenberg, Born and Jordan in 1925.  This represented observables – the position and momentum of particles – as infinite matrices whose elements (all the possible values the positions and momenta can have)  are real and non-commutative^[3].  These matrices are no more than mathematical abstractions, and the physical picture they paint is probabilistic.

            So, how can we improve on this – assuming we can?  By introducing a second time dimension – which cannot be a straightforward affair, because if it were, there would be all sorts of untold consequences, but which we unveil here as a means to represent sample space – the space that is normally represented by one axis on a graph of a probability distribution function and indicates all the possible values assumed by a given variable.

            This time, however, I am suggesting that ‘sample space’ is something physically real, and not merely a line or space on a piece of graph paper (although that’s real enough, in a way).  I am suggesting that it is the space of all possible times that a particle can occupy, so that all of its possible positions, energies, spins and momenta are realised in each one.

            I think I can express this idea most simply, thus:

 

(1/c²)(∂²f/∂t² + ∂²f/∂σ²) – (∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²) + m²c²f/ħ²  =  0 .

 

This, of course, simplifies to

 

(1/c²)(∂²f/∂t² + ∂²f/∂σ²) –∇²f + m²c²f/ħ²  =  0 ,

 

which is a version of the Klein-Gordon equation for a spinless, non-zero rest-mass, zero charged particle.  In this case, the wave function f is real, and has the dimension length-squared (or area, as above).  The dimension σ is given by

 

σ  =  ±iλ_B/c ,

 

where λ_B is the particle’s de Broglie wavelength.  Consequently, σ² =  –λ_B²/c², and

 

(1/c²)(∂²f/∂t²) – f/λ_B² –∇²f + m²c²f/ħ²  =  0 ;

 

in other words:

 

f – (f/λ_B² + m²c²f/ħ²)  =  0 .

 

Although the equation’s application is very limited, there are – rather more complicated – versions that could be constructed using the same basic principle, namely, that of adding a time dimension.

            The idea of travelling sideways in time is one that has fascinated me ever since I saw the Dr Who story, Inferno, with Jon Pertwee as the Doctor, back in the 1970s, and read CS Lewis’s unfinished tale, The Dark Tower, in the 1980s.  I also remember reading – I think when I was still in primary school – a book by the American science fiction author, Lester del Rey, called The Infinite Worlds of Maybe, which I re-discovered a few years ago, and is still on my bookshelves.

            The Everett de-Witt, or ‘Many Worlds’, Interpretation of quantum mechanics, to my mind, lacks one thing to make it fully plausible, and that is a satisfactory geometry – one that meshes with the metric of space-time of the ΛCDM Model of the Universe.  It is not ‘quantum gravity’ that will provide this – this scientific ‘Holy Grail’ is a mirage – but a recognition that a full account of time itself requires that every event not only has spatial co-ordinates, x, y, and z and a linear time co-ordinate, t, but an alternate time, or possibility, co-ordinate, σ.

            For an electron in a hydrogen atom, with a de Broglie wavelength of 1.66 × 10^-10 m, σ would equal  ± i × ~5.545 × 10^-19 s.  For a macroscopic object, however, say a body weighing 1 kg, travelling at 5 m s^-1, its de Broglie wavelength would be 1.325 × 10^-34 m, and for it, σ would equal ± i × ~4.42 × 10^-43 s!  The degree of uncertainty at the macroscopic level is simply too small, when you are dealing with simple, large objects.  It’s much different, of course, when you are dealing with complex systems, involving a mutlitude of interacting entities – like the weather, or non-linear dynamical systems.  That, however, is a whole different story!

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^[1] Named, of course, after Hermann Minkowski (1864-1909), Albert Einstein’s Mathematics Professor at the Zurich Polytechnic.

^[2] Trans., ‘Quantisation as an Eigenvalue Problem.’  ‘Eigen-’ means ‘quality, property, attribute or adjective.’  ‘Wert’ is ‘worth’, or ‘value’.  Clearly, Schrödinger was not talking about property values!  For the technical meaning of the term in mathematics, see: http://en.wikipedia.org/wiki/Eigenvalue.

^[3] This is because they are vector quantities, with a scalar, or value, part (magnitude) and direction.  The value part of these matrices is an eigenvalue; the set of all the eigenvalues is an eigenspace, and the vectors themselves are eigenvectors.  Determining a particle’s energy, say, or its position, to any degree of accuracy, entails rendering its momentum indeterminate, and vice versa, precisely because of the non-commutativity of these eigenvectors.

Of Mayans and Neutrinos.

Having now seen (and been highly amused by – such is my perverse sense of humour) Mr Roland Emmerich’s latest blockbuster disaster movie^[1], 2012, in which civilisation as we know it – and billions of people – succumb to, inter alia, super-volcano eruptions, earthquakes, tsunamis and mega-floods, I find that – although my admiration for the ancient Mayan’s knowledge of astronomy, and their calendrical achievements, is undiminished, the film did stretch one elastic of my credulity well beyond breaking point.

            For, at the very beginning of the film, one of the leading characters (whose name I forget – and it scarcely matters), played by the fine British Shakespearean actor, Chiwetel Ejiofor (a very good Othello, if memory serves), discovers, through his friend, an Indian particle physicist, working in a laboratory deep below ground in a copper mine somewhere in the sub-continent, that the Sun’s neutrino flux has increased, and that the neutrinos are changing into different sub-atomic particles when they reach Earth (what kind we’re not told), causing the Earth’s core to heat up.

            Now, that the Mayans could have worked out the world would end on Friday, the 21^st December, 2012 (at precisely 11:11 AM, GMT) is implausible enough – but by that means?

            This would require a knowledge of quantum flavour dynamics (QFD, or electro-weak theory) – and of the existence of neutrinos in the first place – and of solar neutrinos in particular, ergo of the thermonuclear fusion process that powers the Sun.  It would require that knew that neutrinos had mass, and that there were three kinds of them – electron neutrinos, muon neutrinos and tauon neutrinos, with their respective anti-particles.

            All of this is, of course, completely impossible.  The Mayans may have been smart, but they were not that smart.  (And, what’s more, they, like the Aztecs of Mexico and the Incas of Peru, had a nasty habit of indulging in human sacrifices to their gods.)  So, could solar neutrinos actually heat up the Earth’s core?

            Well, there are an awful lot of them.  The solar neutrino flux incident upon the Earth’s surface is approximately 6.2965 × 10¹⁴ m^-2 s^-1, which means that every square metre of the Earth’s surface is being bombarded with 629.65 trillion neutrinos every second!  See: http://www.cosmicrays.org/muon-solar-neutrinos.php.

            The energy of one of these solar neutrinos could be as high as 18.8 MeV (= 3.012 × 10^-12 J), so, in theory, at least, the irradiance from that 629.65 trillion neutrinos per second could work out at ~1.9 kW m^-2.  In practice, however, neutrinos tend to be far less energetic than that – between 425 keV, for the proton-proton I reaction, up to 15 MeV for the ‘boron 8’ neutrinos and only 18.8 MeV for the rare ‘hep’ neutrinos (an energetic side-chain of the proton-proton, or ‘pp’ reaction, called the pp IV reaction, in which He-3 is converted into He-4 with the addition of one proton, or hydrogen-1 nucleus, and the release of a positron and an electron neutrino)^[2].

Furthermore, neutrinos only interact very weakly with other matter.  They have no electric charge, only a very small mass and only take part in gravitational and weak force interactions.  They pass through us on their way to the Earth’s core, so if they were going to heat that up, they would certainly heat us up first!

Of far greater significance, in terms of the Sun’s energy output, is its photon flux, across the entire electromagnetic spectrum.  The average amount of electromagnetic radiation from the Sun distributed over the entire surface of the Earth is approximately 342 W m^-2, which works out at ~10.785 GWh m^-2 yr^-1.  It is this that keeps us warm, although were it not for the beneficial impact of the greenhouse effect (and there is one!) we would still find our climate very chilly indeed^[3].  The trouble is, we are now having too much of a good thing, by putting too much warming CO₂ (and other greenhouse gases) into our atmosphere – a bit like leaving the electric blanket on for too long.  Not good for the electricity bills – and a fire risk!

So, planetary alignments with the Sun (or the Earth) notwithstanding, I think we are quite safe on the Mayan Long Count end date.  The Mayans did not, in any event, believe that the world would end then – their conception of time was, like that of a lot of other ancient peoples, cyclic, so it wouldn’t have made sense to them to speak of the world ‘ending’.  There is nothing unusual about the alignment of the planets on the 21^st December, 2012, as this picture of the inner solar system on that date (and at the required time of the Winter Solstice in the Northern Hemisphere, 11:11 GMT) shows:

 

 

–Venus and Mercury are aligned, but Earth and Mars are not.  In case anyone should think I am indulging in selected reporting of the evidence,  here is the entire Solar System at that day/time:

 

 

 – the size of the inner planets is greatly exaggerated, as is the size of Pluto (which no longer counts as a planet, alas!).  To obtain the information for yourselves, see the website at: http://www.fourmilab.ch/cgi-bin/Solar/action?sys=-Sf.

            However, do go and see the film.  The apocalypse that may be heading our way may not be a Mayan one, and it may be somewhat later than 2012 (if not by much – say, 2030, or thereabouts), but we may at least be entertained by the spectacle of John Cusack escaping near-certain death while we wait for it!

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^[1] http://www.whowillsurvive2012.com/; http://www.imdb.com/title/tt1190080/.

^[2]See: http://en.wikipedia.org/wiki/Standard_Solar_Model;

http://en.wikipedia.org/wiki/Proton-proton_chain_reaction;

http://en.wikipedia.org/wiki/CNO_cycle.

^[3] Possibly as low as –18°C (–0.4°F).  See: http://en.wikipedia.org/wiki/Greenhouse_effect.

A follow-up to 'e and pi'.

One more curious fact, before I leave the subject of those marvellous numbers e and π, at least for now.

            Quantum electrodynamics employs a theoretical coupling constant, instead of the actual, empirical one, which as we know is α = 0.00729735308.  The QED version is called j² and is exactly equal to 0.01; j = 0.1 is the QED equivalent of electric charge.  (The reason being that α^½ = e/√(4πε₀ħc) = 0.085424546 is a way of expressing electric charge in dimensionless terms, with ε₀ = 1/4π, and ħ and c both equal to 1 in ‘natural units’.)

            It so happens that:

 

m_p/m_e  =  8παφμ_e/j⁴  =  1836.152755656 ,

 

where φ is a dimensionless constant, equal to 1.0000005765078.

            We have already seen that

 

m_p/m_e  =  eπ²ζ/2αμ_e ,

 

and

 

α  =  eξ/12π³μ_e ,

 

where ζ = 1.0000305692511 and ξ = 1.0000116993595, respectively.

            It is easy to see from this that:

 

4αξφμ_e/3ζπ⁴j⁴  =  1 ,

 

which relates all of the (apparently arbitrary) dimensionless constants to the values of α, μ_e, j and π.  (Interestingly, e disappears.)

            The one constant we have left is the largest of them, κ = 1.000249044275, which appeared, it will be recalled, in the equation:

 

m_n/m_e  =  eπ²κ/2α .

 

Simple re-arranging gives us

 

2α  =  eπ²κm_e/m_n ,

 

and substituting in the above, we have

 

2eκξφμ_em_e/3ζπ²j⁴m_n  =  1 ,

 

and so e makes its re-appearance here.

            The expression κξφ/ζ = 1.0002307469, which constant I shall label η.  Ergo, the above may be re-written:

 

2eημ_em_e/3π²j⁴m_n  =  1 ,

 

or:

 

m_n/m_e  =  2eημ_e/3π²j⁴  =  1838.68366066 .

These results regarding the proton-electron and neutron-electron mass ratios are quite remarkable.  I have known about them for some time, but I have been unable to interest the scientific community in either of them, because they do not fit into any of the existing paradigms.  That is unfortunate for me, perhaps, but doubly unfortunate for the scientific community, which is yet again showing how it is apt to be blinded by its own prejudices.

            I may well have more to say about this on another occasion.

 

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