One of the primary problem solving strategies we use at the Math Forum is “noticing and wondering”. This method reminded me of a certain page in the history of science. In 1928, British mathematician Paul Dirac derived an equation that modeled Einstein’s special relativity in terms of quantum mechanics.
In the plainest English I can use, this equation models the wave-like behavior of subatomic particles while applying Einstein’s famous E=mc² theory, where c is the speed of light. Schrödinger’s ψ(x,t) describes the probabilistic motion of a subatomic particle along a given distance a particular time frame. The constant h is Planck’s constant, which is a fundamental constant in physics which describes proportion between the energy of a particle and the wavelength of its wave-like behavior. Dirac’s contribution in this case is in the Beta, Alpha and P variables. The P variables refer to a 1×3 matrix of a particles momentum in three axes. he Alpha and Beta variables actually relate to a 4×4 matrix, composed of the possible “spin” of an electron and a “negative energy” component of the same electrons.
(Science note: Spin is an inherent property of most particles that works a lot like angular momentum)
At the time, Dirac was satisfied with his alignment of relativity and quantum mechanics, and published his findings and proofs without further delving into the implications of the “extraneous” data created by the matrices involved. He posited that the negative energy component in the matrices were “holes” in a field of electrons, because empty space is technically more positively charged than an electron. Dirac wanted to write this off as a proton, but since momentum was involved, the “other” particle would need to be the same mass. Though, Dirac’s ultimate objective was to unify relativity, Quantum Mechanics and spin and he was unable to explain or prove that the “negative energy” electrons were just holes in space. By answering one question, Dirac opened up another
Perhaps if he “noticed and wondered” he might have come to the conclusion that the specifics of his mathematics implied that an entirely new class of matter exists. It wasn’t until 1932 that the implications of Dirac’s extraneous solutions were characterized in an experiment. Using a device known as a cloud chamber in a strong magnetic field, Carl D. Anderson observed electron-like particles moving counter-intuitively. The only way he could explain this phenomenon was that these were positively charged electron, which he dubbed positrons. (Clever.) This was the first recorded observation of Anti-matter!
(Historical note: Carl D. Anderson wasn’t the first to observe this behavior, Chung-Yao Chao had observed it in a similar experiment three years earlier, but he never followed up on the phenomenon. Though, Anderson mentioned Chao’s findings in his report. How considerate of him!)
Though the equation here isn’t so simple, looking deep into the meanings of mathematical statements and operations can lead to great discoveries. Every time you write an equation or function, you’re writing much more than you think. Behind every rule and formula, there are pages and pages of proofs and expanded fundamentals of mathematics. Even in well known mathematical machinery, there may still be mysteries to unlock. Noticing something strange in one of our PoWs probably won’t turn the world of physics on it’s head, but it might get your work published on the internet. It’s good practice for when you eventually come across something hidden in the math.