Continuity and discontinuity.
Continuity was one of the properties Einstein originally assumed he would have to attribute to space-time in Relativity.
When he based his investigation of the nature of the properties of space-time, which subsequently led him to use the functions of fields to explain gravitation, on the elementary concept of a minimum conceivable distance, Einstein was fully aware of what this implied.
The interval between two events that were as close to each other as possible in space-time was the smallest conceivable distance.
As he had accepted Newton’s heritage concerning the continuity of space and time, Einstein assumed as a basis for his physical investigation that space-time was continuous. In consequence, Einstein’s concept of the minimum interval was adapted by tradition to the Newtonian concept of the “infinitesimal”.
The space between two distinct physical events that occurred in space could theoretically be reduced to an infinitesimal, and the same was true of time.
The concept of the infinitesimal carried with it all the ballast of the paradoxes of continuity. An infinite number of other linear infinitesimals can be conceived to exist within a single linear infinitesimal. And each one of them can contain another infinite number and on and on.
As a result, we lose sight of reasonableness and intelligibility in determining events in space. The same is true also of time. All this is the consequence of the arbitrary assumption of continuity.
Why not change this underlying assumption in physics that has proved so awkward in a world which can be perceived and which operates at the elementary level in terms of discrete and discontinuous quanta ?
Einstein’s own convictions about the necessity for continuity in the description of fields were often shaky. In 1931, he uses the following words:
In electrodynamics, the continuous field appears alongside the material particle (the source) as representing physical reality. This dualism, although it does not satisfy any systematic mind, has not yet been eliminated.
Ever since Maxwell’s day, physical reality has been conceived to be represented by continuous fields governed by differential equations with partial derivatives that cannot be subjected to any mechanical interpretation […].
We must admit that the program contained in this idea has never been carried out completely. The physical systems that have since been constructed and that have been so successful in describing reality represent a kind of compromise between these two programs (Newton’s and Maxwell’s) and it is this very willingness to compromise that make them seem logically provisory and incomplete, even though each has achieved great progress in its own field.
Abraham Pais, a strong advocate of quantum mechanics ( it seems strange that one of the most careful biographies of Einstein was written by one of his adversaries ), provides an altogether misleading commentary on these statements regarding the more than significant doubts Einstein expressed about the validity of the description of fields as continuous.
Pais writes:
This is the clearest expression I know of Einstein’s profound confidence in the validity of the description of the universe based exclusively on fields that are everywhere continuous.
Quite the opposite, Einstein wanted to emphasize how little confidence he had that continuity would remain an appropriate hypothesis for the description of the nature of fields and particles in future.
It is even more difficult to comprehend Pais’ assertion when we read a few lines below his own quotation of an article Einstein wrote along with Jacob Grommer as early as 1927 that states:
All the attempts in recent years to explain the elementary particles of nature using continuous fields have failed. The numerous failed attempts have considerably increased our suspicion that this is not the correct way to conceive of elementary particles.
As we establish the foundations of the new Wave Theory of the Field, we are agreeing with Einstein’s doubts.
These were also shared by one of his toughest adversaries, Werner Heisenberg, the founder of quantum indeterminism. In the final years of his work, Heisenberg admitted several times he was convinced that a discontinuous space-time characterized by discrete units of length and time would in future have to take the place of continuous space-time. He thus denied the validity of the hypothesis of continuity, which up to now has been accepted a priori in physics.
As we construct a new theory of physics, we will change the underlying hypothesis of “continuity”, which physicists have considered fundamental to the present time.
We will replace it with a perfectly legitimate and equivalent working hypothesis that will allow us to consider “discontinuity” in space-time as the basis of a new way of interpreting the world of physical phenomenons.
And we shall see where this takes us.
In late 1986, two years after I had myself published my first book, “Il Campo Unificato” I happened to reread “Storia del Concetto di Massa” Ed. Feltrinelli Milano 1974 (Concept of Mass in Classical and Modern Physics, by Max Jammer –Haward College 1961).
Something I had underlined the first time I read the book 15 years earlier reminded me that I had planned then to check the physics department library for an article by a certain Alfred Schild in the Physical Review (of 1948) and I had apparently then forgotten to do so.
That article dealt with the formulation of a mathematical model of discontinuous and discrete space-time that would justify an extremely large number of the normal Lorentz transformations.
At that time, I was constantly preoccupied by the task of adapting a relativistic mathematical structure to Wave Field Theory and always put it off because I imagined it would be extremely difficult.
Mathematical support for the theory would have to be consistent with the demands of special and general Relativity. It would also have to describe in detail the model of discontinuous and discrete space-time that I had only illustrated intuitively in my book Il Campo Unificato (The Unified Field) in 1984.
The fact that Schild’s model of space-time allowed for an extremely large number of Lorentz transformations made me assume it had to be to some degree consistent with Relativity. I thought therefore that it would certainly be productive to familiarize myself with it before undertaking my own mathematical research.
What I found proved to be an unexpected revelation from the very beginning. I instinctively saw the complete description of the waves I wanted to justify in it. Immediately afterward, I found a full discussion of the matter in another article published by Schild in the Canadian Journal of Mathematics (… in 1949). And I confirmed that it appeared to fit my model of discrete space-time like a glove.
After almost 35 years from its building, it was exciting to see the Schild’s model applied so perfectly just when I needed to prove my new theory mathematically. It seemed an almost incredible coincidence.
I was particularly amused to learn that its author had believed the time would never come when his mathematical construction would be given a physical interpretation. I, on the other hand, saw in it the basis for constructing an entirely new field of physics.
How had Alfred Schild come to construct such a model without feeling an underlying need to apply it to something real? Or at least to something that could conceivably be of interest? He may have found himself caught in the wake of the attempts set in motion by Einstein to find a theory to unify gravitation and electromagnetism.
Or, like many others, he may have been seeking a possible alternative relativistic interpretation of the structure of space-time.
In any case, it became fashionable in certain mathematical circles around the 1940s to formulate hypotheses concerning the existence of discontinuous and discrete space-time. In these, “the relativistic interval”, understood in the theory of Relativity as the limit of measurement in continuous space-time, would no longer be an infinitely small number, but would be substituted by a discrete interval with a finite length and duration.
They attempted to use this hypothesis as a basis for developing mathematical structures that would allow for a model of reticular space-time. They imagined that space-time of this type would be made up of discrete modules that could be identified by the new intervals like those described by General Relativity.
But, unlike those intervals, these were not to be considered continuous and infinitesimal but rather discontinuous and discrete.
Alfred Schild created one such model of reticular space-time. His description of space-time appeared more consistent than all those that had preceded it.
Claiming that he believed in Relativity, Schild used mathematics of the tensor and spinors along with an algebra using Gaussian integers to obtain a space-time that was mathematically consistent with itself.
This model of space-time was self-sufficient and possessed all the characteristics needed to prove the initial hypothesis of discontinuity.
He had constructed a model of space-time that, although discrete and discontinuous, nonetheless was mathematically homogeneous, isotropic and dense.
It could be described in keeping with all the dictates of Relativity.
Furthermore, it possessed all the characteristics indispensable to a possible space-time as it had many other positive properties that guaranteed that it would be internally consistent both logically and mathematically.
The model was infinite and yet at the same time discrete. And all of its characteristics could be defined geometrically using diofantine equations (equations with finite differences in which quantities can be enumerated using only integer numbers).
Although satisfactory from a purely mathematical perspective, Schild himself felt his model had one important defect.
It seemed his conception of space-time would have to remain an exclusively mathematical model.
One of its most essential characteristics appeared to conflict with physical reality as it only permitted speeds very close to the speed of light.
In his two articles, Schild abandoned all hope that his creation could ever reflect any physical reality.
Schild’s conception of bodies and particles in motion between the points within the lattice that made up his model of discontinuous space-time had forced him to attribute to them impossible velocities that were too close to the speed of light, “c”.
In his own mind, Schild’s construction was a successful mathematical toy within which everything appeared consistent and rational, but nothing more. It seemed there could be no parallels with reality, if the bodies in motion within discrete space-time could only travel at velocities close to the speed of light.
This was precisely the opposite of the physical reality we know.
In the world of our experience, bodies cannot in fact travel at the speed of light and have great difficulty even coming close to what according to Relativity is the maximum velocity.
Schild passed up the greatest opportunity of his life. He was thinking in terms of a direct correlation between his model of discrete space-time and the normal representation of physics, in which bodies constitute the main actors on the scene of space-time.
It did not occur to him that “only perturbations affecting the state of the lattice” could rationally be in motion within the lattice.
Schild was therefore unable to suggest the hypothesis that the velocities in the lattice so close to the speed of light to which he feared to attribute a physical interpretation might represent velocities at which perturbations in the lattice itself spread out.
Schild had discovered a mathematical representation of a possible and rational aether, but he failed to recognize it as a physical model. He failed to understand that his reticular model of space-time, whose structure was particularly compressible and elastic mathematically, was capable of transmitting perturbations of its own structural properties.
These perturbations in turn could modify all the properties that made up the lattice’s structural “state”.
As they moved from one “event point” (place where something occurs) to any other event point in the lattice, such perturbations disturbed different areas of the lattice work.
Schild failed to see that the only entities that could possess motion within the lattices were these variations in its structure.
He also failed to imagine that these perturbations could move like waves at velocities that, although they could vary locally to a limited degree, were all very close to “c”.
The young Einstein’s ideas come easily to mind in the light of Schild’s failure to make this discovery. Let us interpret loosely what Einstein wrote to Klein:
… Mathematics is clearly not a reliable research tool in physics if one has no physical model in mind to which it can be applied.
What is extraordinary about this story is that the very properties that embarrassed Schild prove very important and indispensable for the interpretation of his discovery as a real model of possible space-time.
Taken together, Schild’s model provided an image of the physical reality of space-time that was consistent with the hypothesis of the existence of a rational and logical aether that can be identified with space-time itself.
The model also possesses one quality that I personally consider absolutely indispensable: despite its mathematical complexity, it is perfectly comprehensible and easy to visualize.
Picture 3. Variations in the geometry of the lattice work of discreet space-time. The lattice is made up of modular elements of a discreet length.
To form an idea of the reality of the lattice, imagine placing a series of identical elastic nets with meshes of identical sizes one on top of the other. Use lengths of the same elastic thread of the same length as the meshes to tie each knot in the upper net to the corresponding knot in the net below it.
You have thus created an ideal three-dimensional elastic net with constant units.
Imagine stretching or compressing one mesh in the elastic net and then letting it go. This would initiate a vibration that would spread out like a wave. The physical parameters of these waves would depend on the elasticity of the net.
Such waves would transmit the structural perturbation to ever more distant sections of the net.
Now imagine that we replace the physical net made of elastic threads with an abstract and immaterial system consisting of discrete integer modular units of a constant basic size.
This gives us what mathematicians call “a discrete metric space”.
Let us set up this lattice so that it characterizes all the “empty” space you can imagine and suppose that nothing else exists in the universe but your consciousness of existing and the existence of this discrete space
On this basis, let us begin our search for our role as observers.
Let us seek to discover how hypotheses and attempts of Popperian falsifications will can be used to “construct” the models of our experience. And we determine whether in turn this single model of “aetheric” space-time can explain the existence of matter, radiation and fields along with the laws which regulate their interactions.
We will need to be able to view our presumptuous-ness as we take a seat at the workbench of the hypothetical Creator with a healthy dose of auto-irony.
We have “only” Schild’s discrete model of space-time and Relativity to give him and wish to use these scanty materials to create the entire Universe with all its radiations, elementary particles, interactions, atoms, molecules, life, stars, galaxies, clusters and super-clusters with the use of a few (and wherever possible simple) immortal laws.s they are put together constitute cells of space which mark the boundaries of a discreet volume of space.
The arrangement of the cell can vary as a function of variations in the time quanta. The perturbation surfaces move about within the lattice passing from the surface of one cell to another in a time quantum.