One powerful benefit of this was that it gave Dirac his first experience in working in a laboratory, interacting with chemicals, and learning about them on an atomic level, via the recent discoveries of Ernest Rutherford.

In the associated laboratories he would deal with hydraulics, observe the varying strengths of different materials, how they responded to pressure, and learn about the technical machinations of industry: combustion engines, steam turbines, pumps, etc.

I point out that this was his formal introduction to the technique, but in fact, it was the same geometry underlying the technical drawing he'd learned in his earlier schooling.

As far as Heisenberg was concerned, this was a mistake, as he wasn't convinced than any experimenter would ever be able to measure these quantities. However, he knew that would be extremely intensive, perhaps even unwieldy, to describe the movement of an electron in three dimensions, so he settled on one.

While working through the calculations, Heisenberg encountered an oddity: there's no mention of a Real number describing the location of an electron, but rather what is required is a matrix, the elements of which are pairs of numbers that correspond to properties of the electron, namely its energy levels, while also representing the probability that the electron will jump between two of the energy levels.

The standout oddity of this paper, however, for Dirac, was the opportunity to work with non-commutative numbers: numbers that, when multiplied, do not return the same value as they would, had they have been swapped with each-other: position * momentum, is not equivalent to momentum * position here. Dirac had experience with these numbers, having encountered them while learning Grassman algebra, and projective geometry during his education.

Spacetime was a concept a young Dirac spent much time mulling over, and the discovery of relativity was a landmark event in his adolescence, with him taking to the papers with a gusto that would disturb more experienced physicists, who found the complexity of Einsteins mathematics impenetrable.

One of the major hangups of the theory, that Dirac was more comfortable with than Heisenberg, was that it introduced non-commutative numbers (where $a * b =/= b * a$), as representative of certain physical quantities: they were common in the projective geometry and Grassman algebra. To him, these were the keys to making Heisenberg's theory behave appropriately under relativistic conditions.

The Hamiltonian, is a mathematical operator that describes the sum total of energies in a system, both kinetic and potential.

Dirac was confident that by using this construct to root out the answer to his question, he could bring some clarity to Heisenbergs theory, the presence of non-commutative properties in it, and relate this all to the expectation of classical physical theory- and he was correct. Upon researching the mathematical construct, he realized he could apply it to the theory, using mathematical quantities that correspond to classical numerical quantities, including an equation that I would like you to remember for later on in the article:

The next step, for Dirac, however, was to make sure that his new mathematical conception still played by the rules of physics. He did this by showing that his theory had validity, by displaying that it obeyed the law of conservation of energy, and agreed with the recent work of Bohr, who had laid out the nature of atomic electrons, and how they pass between higher and lower energy levels, releasing a quantum of light in the process.

Heisenberg of course, read this paper, and was deeply enthusiastic about it, calling the work beautiful, and ensuring Dirac of the accuracy of his theory. Of course, Heisenberg could only do this, because the results (the relationship between position and momentum) had already been found by none other than Max Born, the man who coined the term 'Quantum Mechanics'.

Dirac had been handed his first defeat in his attempts of theoretic achievement, having been narrowly beat to print by some of the best physicists of the day, working in tandem, Heisenberg, Pascual Jordan, and Max Born, in Göttingen- but he kept on, knowing that quantum theory was nascent, and that there was still plenty of opportunity for scientific glory.

Schrödinger's equation chose to describe quanta of matter, according to the waves that were associated with them (photons and electrons, can behave in a wave-like as well as particle-like manner). His achievement here was a stellar generalization of the work of Louis De Broglie, whose wave theory of matter was only applicable to matter that wasn't under the influence of external forces.

As opposed to Heisenberg's theory, which was largely predicated on non-commutative numbers, Schrödinger's theory was based on the equations of waves, which were much more familiar to physicists, being that it was the foundation of undergraduate physics study at the time.

In addition to this, Schrödinger's theory provided a much needed and appreciated tool in the understanding of quantum mechanics: it allowed for the visualization of quantum objects and quantum phenomena, in a way that had been assumed moot for the theory, due to it's bizarre implications about the nature of observable quantities.

Unfortunately, Dirac was a bit too late: shortly after realizing that he in fact could have derived Schrödinger's equation from his theory, if he hadn't been so focused on generating connections between Quantum and Classical mechanics.

There was reason to believe that Quantum Mechanics, could offer up a solution to this issue, but at the time, in 1926, the theory was not evolved enough to be applicable to atoms with more than one electron. Dirac had learned at Bishop Road, that chemicals were comprised of atoms, and now that physicists had expanded greatly our understanding of it, as well as some of its constituent parts, particularly grappling with the electron, it may have appeared that discovering the rules underlying the relationship between groups of electrons, energy levels of the atom, and subsequently a deeper understanding of chemistry, was within reach.

Dirac, at this point in time, was certainly aware of the implications of this principle, but wasn't satisfied with the level of understanding that it gave, knowing that the theory couldn't describe the chemical processes he observed at Bishop Road, nor could it be applied to atoms with more than one electron (a fact that stymied many physicists, in many problems, as all but the simplest elements have more, with hydrogen having one, and helium having two). He was considering, what the relationship between the waves of Schrödinger's theory and heavy atoms, while mulling over the point Heisenberg had made, that quantum mechanical theories should be architected in terms of the observable quantities that come about as a result of some measurement of a quantum system.

In a few pages of mathematics, Dirac algebraically laid out how groups of electrons share energy as they fill in the available energy states of an atom- this very same mathematical formalism is alive and well to this very day, by researchers who focus on semi-conductors, metals, and the flows heat and electricity in them, which is determined by the electron content of the material- of course, the industrial applications of his work were never of any real significance to Dirac, who was satisfied to have developed the understanding that the wave describing a group of electrons change signs if two electrons move places, but for photons, there's no change in the nature of the Schrödinger wave that describes them.

Roughly a month after the release of his paper, Fermi reached out to him, wanting to notify him, perhaps a bit sourly, that he had already solved the problem, assuming that Dirac probably had not seen it, given that this was an unusually active and productive watershed era in theoretical physics.

At the time, one of the biggest issues presenting itself to physicists of the time, was the meaning of the symbols in their equations. Max Born had found a way to interpret Schrödinger's waves usefully, but this required an appalling giving of quarter in the logical intuition regarding the phenomenon he was studying (particles): that the future state of any particle can always be predicted.

Particularly, Born was considering how an electron would scatter upon impacting a target, realizing that one can not ascertain before hand where the electron will be deflected to, but rather one can calculate the probabilities of the electron deflecting by an arbitrary angle. Going one step further, he insisted that should a Schrödinger wave describe an electron, one can calculate the probability of detecting it in any tiny region, and that the wave itself has no physical existence, but rather solely a mathematical existence that allowed for calculating the likelihood of some phenomenon in the future, a sharp blow to the determinism put forth by Newton.

Truly, Dirac was in high spirits as a result of this work, having been particularly pleased at solving the problem of showing the synthesis of the theories at play, especially given that it was the result of one of his favorite techniques for solving problems in quantum mechanics, by drawing out subtle and sneaky classical analogues to the quantum phenomenon.

As per the nature of continuity, there should be fluid variation between values, no 'jumps', so to speak, which is in direct contradiction of the results of quantum theoretic calculations, that put forth a discrete, comminuted picture of nature at the tiniest scales.

The first thread Dirac pulled on, was an old one, the thread that allowed him to generalize the Schrödinger equation, which was taking quantities that were assumed to be regular numbers, and treating them as if they were non-commutative. From this launch point, Dirac was able to mathematically define the creation and destruction of photons, from their creation throughout the universe in various events, from the large thermonuclear processes in stars, and the ejection of high energy photons, often referred to as gamma rays, from atoms, to the destruction of these particles by the human retina, or plants, during photosynthesis, wherein the energy of photons becomes a source of sustenance (as opposed to a source of information, by the human eye).

Dirac was aware he had done something magnificent, having put an end to the consternation over the contradictions between the wave and particle descriptions of particles. The next step for Dirac, was to test his theory against the facts presented by earlier, tried and true theories, and he chose the work of Einstein- a bold stick to pick from the pile of things by which to measure one's work.

For Dirac, the next step, was to extend his work here, from not just photons, to electrons, and of course, stemming from his love of relativity theory, he desired to put forth a relativistic, quantum field theoretic description of electrons.

In his dedication, and persistence, trying out equation after equation, Dirac hit upon the one that would revolutionize physics in a way that would require all of his peers to reconsider what they thought about nature at the fundamental level. He had found an equation that eschewed using Schrödinger waves, and presented an entirely new type of wave, that had four sections, coupled together.

Usually, such absurdity could be ignored, or hand waived away, but in the context of quantum theory, it begged explanation, as it allowed for a positively energetic electron to jump into the negative energy state- but no such behavior had been observed experimentally. This particularly irked Heisenberg, who found the idea of negative energy levels to be truly absurd, and would go on to be one of the main critics of the equation, pushing Dirac to provide a solution, or relegate it to the pile of "almost were" equations in physics.

Despite the growing concern, and the oddity of the theoretic predictions, Dirac was not swayed, and marched forward sternly, right into the answer he needed. It struck him that if one was to consider all of the electrons in the universe, filling in energy states, common sense dictated that the negative energy states would be occupied first, meaning that there were no available negative energy states for a positive energy electron to jump into, a subtle application of Pauli's exclusion principle, which indicated a negative energy state could only hold one electron.

From this, he extrapolated the idea that the universe is awash with these negative energy electrons, a sea, if you will, of constant density, and unfaltering uniformity, meaning that only a departure from this consistency could be observed experimentally. It became clear to him that there would, however, be occasionally vacant energy states of the negative energy electrons, which would be filled by ordinary electrons, resulting in the emission of a small bit of radiation, meaning one could be detected if a positive energy electron jumped into it.

As confident as Dirac was in his ideas, there was a growing choir of critique regarding the negative energy electrons, and the sheer audacity of assuming that an unobservable type of electron was not only extremely abundant, but was more abundant than normal electrons. Dirac found himself against the ropes, with nothing left to do but the seemingly impossible. A particularly intense moment in this situation was a talk entitled 'The Proton', attended by Delbrück and Landau, who were looking to see whether or not he had made any progress regarding the 'hole theory', and upon realizing that he as of yet, had not, Landau sent a single word telegram to Bohr regarding the talk and it's contents, referring to it simply as 'crap'.

Here, Dirac made a grand leap into the area of theoretical predictions, being lambasted by Oppenheimer and Weyl that the hole must necessarily have the same mass as the electron. Having no other option, he posited that the hole might actually be representative of the existence of a heretofore undiscovered particle: identical to electron except for in the value of it's charge, he presented the idea of the anti-electron. He also mapped this idea onto the proton, moving away from the more familiar idea that the proton and electron had a deeper relationship than being constituents of the atom, he suggested the proton has its own own negative energy states. In a display of theoretical prescience, boldness and creativity, he'd laid eyes on the the antiproton, and positron.

There's something to be said, for me, about the fact that, despite being dealt heavy handed defeats, the likes of which he dealt to Heisenberg, Born, and Pascual Jordan (who didn't weather the one as well as Dirac weathered the many), by people who, while certainly having been lionized greatly by time, where absolutely mountainous in their stature when they were his colleagues. I would not venture so far as to say nature rewarded Dirac for his patience, and fortitude, but I do think the denouement is poetic, cinematic even. Having faltered certainly in his beliefs, ideas, and perspectives, at times, he never wavered in his focus, on his path forward into the unknowns of science, aided by precision and reason, toward his eponymous equation.