Dirac equation

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later.

• Possibly... [Dirac's] biggest achievement was... the equation for the electron... a sort of extension to Schrödinger's equation, which allowed quantum theory to be combined with Einstein's special theory of relativity. ...Dirac's equation is very ...similar to Schrödinger's equation, the difference being that these ...are not numbers, they are matrices, and this is one of the ...great equations of all time ...Farmelo wrote a book about the eighteen greatest formulae of all time, and that's one of them. ...This equation ...explained other things about the electron. These have been experimentally verified to huge precision.$${\displaystyle \left(\beta mc^{2}+c\;(\sum _{n=1}^{3}\alpha _{n}p_{n})\right)\psi (z,t)=i\hbar {\frac {\partial \psi (z,t)}{\partial t}}}$$
  • Christopher Budd, "The Quantum Mathematician" (Mar 13, 2018) from Maths and the Making of the Modern World Lecture Series, Gresham College. A YouTube video source: 39:24-40:17 (Posted Mar 15, 2018).

• The saddest chapter of modem physics is and remains the Dirac theory... In order not to be irritated with Dirac I have decided to do something else for a change...
  • Wolfgang Pauli, as quoted by Graham Farmelo, It Must be Beautiful: Great Equations of Modern Science (2002) p. 103.

• There is one topic I was not sorry to skip: the relativistic wave equation of Dirac. It seems to me that the way this is usually presented in books on quantum mechanics is profoundly misleading. Dirac thought that his equation was a relativistic generalization of the non-relativistic time-dependent Schrödinger equation that governs the probability amplitude for a point particle in an external electromagnetic field. For some time after, it was considered to be a good thing that Dirac’s approach works only for particles of spin one half, in agreement with the known spin of the electron, and that it entails negative energy states, states that when empty can be identified with the electron’s antiparticle. Today we know that there are particles like the W^± that are every bit as elementary as the electron, and that have distinct antiparticles, and yet have spin one, not spin one half. The right way to combine relativity and quantum mechanics is through the quantum theory of fields, in which the Dirac wave function appears as the matrix element of a quantum field between a one-particle state and the vacuum, and not as a probability amplitude.
  • Steven Weinberg, Lectures on Quantum Mechanics (2012), Preface

: Great Equations of Modern Science by Graham Farmelo.

• In... 1928... Dirac... a twenty-five-year-old recent convert from electrical engineering to theoretical physics, produced... the [remarkable] Dirac equation. ...He wanted to ...describe the behaviour of electrons more accurately ...[Previous] equations had either incorporated special relativity or quantum mechanics, but not both. ...Unlike other physicists, and ...Newton and Maxwell, Dirac did not proceed from a minute study of experimental facts. ...By 'playing with equations' he hit upon...[the] uniquely simple, elegant ...Dirac equation ...[which] predicts ...electrons are ...spinning and ...act as ...bar magnets, and [predicts] the rate of the spin and ... strength of ...magnetism.

• [E]quations developed by Heisenberg and Schrodinger did not take... from Einstein's relativistic mechanics, but from the old mechanics of Newton. ...[E]xperimental data on atomic spectra... were so accurate that small deviations from the Heisenberg-Schrodinger predictions could be observed. So there was a strong 'practical' motivation... [A]ncient and fundamental dichotomies were in play: ...light versus matter; ...continuous versus discrete ...tremendous barriers to ...achieving a unified description of nature ...[R]elativity was the child of light and the continuum ...quantum theory the child of matter and the discrete. After Dirac's revolution... all were reconciled, in the... conceptual amalgam... a quantum field. Maxwell's electrodynamics is a continuum theory of electric and magnetic fields, and of light, that makes no mention of mass. Newton's mechanics is a theory of discrete particles, whose only mandatory properties are mass and electric charge.

• Dirac's equation consists of... four separate wave functions to describe electrons. Two components have an... immediately successful interpretation... describing the two possible directions of an electron's spin. ...The extra ...equations contain solutions with negative energy... Assuming Dirac's equation, if you start with an electron in one of the positive-energy solutions, you can calculate the rate for it to emit a photon and move into one of the negative-energy solutions. Energy must be conserved, but that... means... the emitted photon has higher energy than the electron that emitted it! ...Dirac was well aware of this problem. ...He proposed ...'empty' space ...contains electrons obeying all the negative-energy solutions. ... A positive-energy electron can't go to a negative energy solution, because there's always another electron already there, and the Pauli exclusion principle won't allow a second... [T]he idea... the ordinary state of 'empty' space is far from empty... a different word for it... is 'vacuum'... a medium, with dynamical properties... [S]hine light [photons with enough energy] on the vacuum... then a negative-energy electron can absorb... [a] photon... and go into a positive-energy solution... an ordinary electron... But in the final state there is... a hole... originally occupied by the negative-energy electron... [I]f there is a pre-existing hole... a positive-energy electron can emit a photon and occupy the vacant negative-energy solution. ...Dirac's first hole-theory paper was... 'A theory of electrons and protons'.
  • Ref: Paul Adrien Maurice Dirac, A theory of electrons and protons (Jan 1, 1930) The Royal Society Publishing, Proceedings of the Royal Society (Received Dec 6, 1929).

• Dirac's holes, now called positrons, are no longer a marvel, but a tool. A notable use is... PET scans...