The creativity of the Dirac equation really struck me when I first learned it. Basically, it uses a mathematical trick to write sqrt(a^2 + b^2) = a + b, which doesn't work for normal math, but does work if a and b anticommute: ab = -ba. You can implement this with matrices, and in the resulting solution quantum spin just pops out. Astounding.
Dirac was a legend and a man who actually believed in the principle that the universe itself was a mathematical structure, rather than making mathematics the tool which we use to describe the universe. His amazing prediction of anti-matter from a pure theoretical / mathematical basis was astounding. He also believed in the principle of mathematical beauty and that one should use this principle in attempting to find the laws of the universe, and I very much agree.
In the James Scott Prize Lecture that he gave on 6 February 1939, Dirac spelled out his new ideas of the relationship between mathematics and physics which focused on the concept of mathematical beauty. Many years before his brother in law Eugene Wigner famously problematized the unreasonable effectiveness of mathematics in the natural sciences, Dirac discussed the same topic. How is it that the mathematical-deductive method is so remarkably successful in physics? According to Dirac:
“This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature’s scheme. One might describe the mathematical quality in Nature by saying that the universe is so constituted that mathematics is a useful tool in its description. However, recent advances in physical science show that this statement of the case is too trivial. The connection between mathematics and the description of the universe goes far deeper than this.”
In his James Scott Lecture and at numerous later occasions, Dirac asserted that the modern history of theoretical physics provides convincing evidence that there is a perfect marriage between the rules that mathematicians find interesting by their own standards and the rules that govern natural phenomena. He thought this was not accidental, but that it might reflect some deep identity between mathematics and physics:
“Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. It is difficult to predict what the result of all this will be. Possibly, the two subjects will ultimately unify, every branch of pure mathematics then having its physical application, its importance in physics being proportional to its interest in mathematics.”
Dirac suggested that future developments in theoretical physics would lead to the “existence of a scheme in which the whole of the description of the universe has its mathematical counterpart.” In accordance with this philosophy, he advised physicists to “begin by choosing that branch of mathematics which one thinks will form the basis of the new theory. One should be influenced very much in this choice by considerations of mathematical beauty. . . . Having decided on the branch of mathematics, one should proceed to develop it along suitable lines, at the same time looking for that way in which it appears to lend itself naturally to physical interpretation.”
