THE PHYSICS OF CREATION: LECTURE NO. 1
The Physical Significance of Fermat’s Last Theorem
Every physicist is familiar with the problem posed by Fermat’s Last Theorem, even though the question of whether or not there is a solution to the equation:
in which x, y, z and n are whole numbers has contributed nothing to the historical development of physical science.
The reason is that progress in physics is rooted in devising mathematical formulae which help us to describe physical phenomena and relate the physical properties and behaviour of matter in its mutual interactions with other matter. Physics is not about numerical relationships, unless those relationships say something about the connection of physical properties. Physicists seek to understand and interpret what they observe in their experimental pursuits so that they can formulate a theory using their mathematical skills and then apply such formulations in practical applications which give purpose to their efforts. Other physicists find contentment in theorizing about a realm of science far removed from the realities of the present time, given that there are many of the most fundamental questions that are still outside the scope of accepted theory. Yet even after the passage of some 360 or so years since Fermat introduced the problem posed by his ‘Last Theorem’, and given that every physicist is aware of the unsolved riddle posed by that theorem, none, it seems, has seen what is there to be seen, namely something pertaining to the relationships which govern particle creation.
Once physicists had discovered that electric charge acts on electric charge to set up an energy potential inversely proportional to charge separation, it was but a small onward step from there to realize that charge acting upon itself is an embodiment of energy in the form of a fundamental electric particle. Thus, if that charge is confined in a sphere of radius a, one can express its energy as proportional to:
where e represents that charge.
Now Nature could have allowed charge to exist at even the most fundamental level with random values. If that were the case physical phenomena could well have been so diverse that we would not have been able to formulate laws of physics, owing to the random character of the matter form. Eventually, the discovery of the electron and its the unitary charge feature, Nature’s revelation that e is a universal constant, provided us with the essential clue needed to explore how particles of matter are created.
The advance of physics is a blend of discovery mixed with hypothesis which can be sustained by onward discovery. One hypothesis that emerges, based on charge having that unitary form e in association with the energy e given by equation (2), is inspired by the procreation of the human life form. Male and female association nucleates the seeds of further life and creates male and female offspring which are nurtured by inflow of energy.
It was discovered that electric charge exists in complementary forms as ‘positive’ charge or ‘negative’ charge, which attract one another, whereas like polarity charges repel one another. Guided by that analogy with the male and female genders, the action by which the ‘offspring’ charges of positive and negative polarity are created, could be expected to depend upon the coming together of the positive and negative charges belonging to the same family of particles. This is depicted in Fig. 1, where the positive charges are shown as red spheres and the negative charges are shown as green spheres.
The hypothesis which we will now introduce will seem to be arbitrary but there is an underlying physical basis. Suppose that space, meaning the volume of space that is available for occupation by these spherical charges is at a premium and cannot just be provided as necessary as and when pairs of charges choose to merge and engage in the creative process. That hypothesis can be put in mathematical terms by saying that the combined volume of the two charge spheres on the left hand side of the arrow in Fig. 1 is equal to the combined volume of the four charge spheres on the right hand side of the arrow. It can be put in physical terms by saying that the enveloping space is itself a seat of energy having a uniform distribution and, if that space medium is not to shed energy and become involved in the particle creation process, other than by a transient fluctuation, its own volume must be conserved. The corollary to that is that the volume occupied by those charge spheres has to be conserved as well.
So one can imagine a scenario where the fundamental particles of which matter is composed are charged spheres capable of transformation by processes of charge-pair creation or annihilation, conserving energy if all that happens is that they decay by mutual annihilation and reappear by mutual creation, to fill the mould set by the volume of space vacated. Alternatively if energy can be exchanged as between separate groups of charge pairs then creation of particle pairs of different species becomes a possibility.
The governing physical criteria involved here are:
(1) energy conservation,
(2) space conservation and
(3) charge parity conservation.
The governing mathematical criterion set by that space conservation requirement is that:
where a is the radius of the parent charge form and b and c are the respective radii of the separate charge form of the two species of charge generated by mutual annihilation of the parent charge pair.
You will now begin to see how Fermat’s Last Theorem may creep into this question of particle creation.
We could begin to contemplate the question as to whether space itself can be quantized in the sense that there is some very small unit of volume that Nature admits, so endowing the universal expanse of all space, the extent of the ‘void’, something we can express as a whole number. However, we shall instead consider the related question of whether the whole energy content of the universe, including what we call ‘space’ and whatever there is in that ‘space’, is conceivably made up of a whole number of a minute energy quantity.
In that case, taking the three charge forms, each comprising a pair of charges of opposite polarity, and assigning radii a, b and c to those charge forms, we can, using equation (2), formulate from equation (3), an energy equation:
where Ea, for example is the energy value we could assign to that charge e of radius a as represented by equation (2).
This equation has the form of equation (1), with n equal to 3, and so, if those energies Ea, Eb and Ec were to comprise whole numbers of that primary fundamental energy quantum we have envisaged above, then Fermat’s equation must have an integer solution for n equal to 3.
From this one can see that, towards the end of the 19th century, when physicists believed in the aether, the discovery of the electron by J J Thomson could have stimulated the above thoughts and aroused interest in Fermat’s Last Theorem for reasons connected with physics. At that time, the aether was seen by some even as a plenum, rather than a void. It somehow contains electrical charge in a neutral composition that eludes detection other than by exhibiting its capacity to store energy and propagate electromagnetic waves. The proof or otherwise of Fermat’s Last Theorem could well have become a task for the theoretical physicist searching for that ultimate fundamental energy unit, if such were to exist.
In the event another century was to pass before Fermat’s Last Theorem yielded to proof. Fermat bequeathed to the world of mathematicians the greatest conundrum of all time, when he wrote in the margin of a book:
“I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.”
Mathematicians today tend not to believe that Fermat really had the proof he claimed, namely that equation (1) above has no integer solution.
In this Lecture No. 1, the first I am presenting on this Physics section of my website, I have chosen to discuss a problem that has been around unsolved for some 360 or so years. By that I mean the task of finding that solution to a mathematical conundrum that could well have had some impact on physics and our understanding of particle creation. Professor Andrew Wiles of Princeton has finally brought that problem to a conclusion, though his solution is by no means simple. There are other major problems, physics problems, of very long standing that remain unsolved, at least as seen from the perspective of the orthodox modern day scientist. Notorious amongst these is the question of gravitation and the task of linking G, the constant of gravitation, with electrodynamics. I shall address these problems as I add to these Web pages.
I will end here by summarizing my case as presented above. If minute standard units of energy were to be assembled to create the particles which comprise our universe, then the existence of particles produced by the transmutations discussed by reference to Fig. 1 is not possible unless Fermat’s equation has an integer solution. Therefore, since we are able to prove Fermat’s Last Theorem, we can infer that Nature does not build fundamental particles from standard units of energy. The onward march of experimental science is therefore unlikely to lead to such a discovery. We must, accordingly, build our picture of matter accepting that electrons are its smallest particle form and that, if there is a smaller charge form that features in the structure of the aether, that cannot bear an integer relationship with the rest-energy of the electron, meaning the energy of the electron at its moment of creation. However, we might well find that the charge volume of such an aether charge is an integer multiple of that of the electron. Such an aether particle cannot be a photon or a neutrino, because, although the aether may seem to be populated by minute energy quanta, which physicists term ‘photons’ and ‘neutrinos’, we should not regard these as being ‘particles’. They have no electric charge and they must only be seen as ‘events’ in which energy is exchanged between aether and matter. Photons, for example, are not quantized, in the sense that their energy is a multiple of a discrete fundamental energy unit. It is the atom which determines the energy shed as a photon event by electrons in their mutual interactions and interaction with the atomic nucleus, so determining the photon frequency. How that frequency is generated and how it determines the constant of action governing photon events, namely Planck’s constant, is one of the many problems to be addressed as we advance in our quest to understand ‘The Physics of Creation’.
