The Algebraic Positron

Paul Dirac and the Relativistic Electron Equation

Every attempt in 1926 to make Erwin Schrödinger's new quantum wave equation obey Albert Einstein's special theory of relativity produced a physical disaster. Schrödinger had built his famous equation by taking the classical non-relativistic energy formula $E = \frac{p^2}{2m}$ and replacing energy $E$ with a first-order time derivative ($i\hbar \frac{\partial}{\partial t}$) and momentum $\mathbf{p}$ with first-order spatial derivatives ($-i\hbar \nabla$). When physicists Oskar Klein and Walter Gordon tried the same substitution on Einstein's relativistic energy-momentum relation, $E^2 = p^2 c^2 + m^2 c^4$, the resulting Klein-Gordon equation contained a second-order derivative in time ($\frac{\partial^2}{\partial t^2}$). In quantum mechanics, a second-order time derivative allowed the probability density of finding a particle in a box to turn negative—an absurdity—and completely failed to explain why the electron possessed an intrinsic magnetic moment and half-integer angular momentum, or spin $\frac{1}{2}\hbar$, which Wolfgang Pauli had been forced to paste onto non-relativistic waves by hand using two-by-two matrices.

At St. John's College, Cambridge, in the autumn of 1927, Paul Adrien Maurice Dirac refused to accept either ad-hoc patch. Born in Bristol to a Swiss immigrant father who forced young Paul to speak only grammatically flawless French at the dinner table—causing the boy to stay completely silent whenever he could not formulate a sentence with mathematical precision—Dirac had trained as an electrical engineer before moving to Cambridge. Among physicists, his literalmindedness was legendary; colleagues joked that a "Dirac" was the smallest possible unit of human speech, defined as one word per hour.

Staring into the fireplace of the St. John's senior combination room in late 1927, Dirac realized that quantum mechanics and special relativity collided for a structural reason. Quantum transformation theory required the wave equation to be strictly first-order in time ($\frac{\partial}{\partial t}$) so that the state of a wave function at one instant uniquely determines its future with positive probability. Special relativity, meanwhile, treats time $t$ and the three spatial coordinates $(x, y, z)$ on an equal footing. Therefore, a truly relativistic quantum equation had to be strictly first-order in both time and space derivatives simultaneously.

To turn the quadratic expression $p_x^2 c^2 + p_y^2 c^2 + p_z^2 c^2 + m^2 c^4$ into a linear differential operator, Dirac tried to take its algebraic square root:

Squaring the right-hand side to match the left-hand side term by term, Dirac saw that all the cross-terms (such as $p_x p_y$) had to cancel out completely while the squared terms had to equal unity. Ordinary numbers could never satisfy $\alpha_1 \alpha_2 + \alpha_2 \alpha_1 = 0$ and $\alpha_i^2 = 1$ at the same time. Playing with non-commuting algebraic quantities on his desk, Dirac discovered that the two-by-two Pauli spin matrices were too small—there were only three of them, and he needed four mutually anti-commuting objects ($\alpha_1, \alpha_2, \alpha_3, \beta$). The minimal mathematical objects that obeyed the anti-commutation relations were four-by-four matrices, which Dirac later condensed using the four relativistic gamma matrices $\gamma^\mu$ into the covariant Dirac equation published in the Proceedings of the Royal Society in February 1928:

Because the operator was a four-by-four matrix, the electron wave function $\psi$ could no longer be a single number; it had to be a four-component column vector, a bispinor. Without a single extra assumption, placing the Dirac electron in an electromagnetic field automatically yielded the electron's exact spin $\frac{1}{2}\hbar$ and its anomalous magnetic moment $g = 2$.

Yet the four components of $\psi$ posed a riddle: two components described spin-up and spin-down electrons with positive energy $E \ge +mc^2$, while the other two described states with negative energy $E \le -mc^2$. Because any ordinary electron should radiate photons and fall endlessly into negative-energy states, Dirac proposed in 1930 and 1931 that the vacuum is a filled "Dirac sea" of negative-energy electrons, and that an unoccupied hole in this sea would behave in a laboratory as a real particle with the exact same mass as an electron but opposite, positive electric charge. On August 2, 1932, Caltech physicist Carl Anderson photographed a cosmic-ray track inside a vertical cloud chamber that curved the wrong way in a magnetic field, confirming the physical existence of the positron that Dirac's four-component matrix equation had demanded four years earlier.