Dirac’s legacy

Paul Dirac, who was born 100 years ago this year, is one of the founding fathers of modern science. He created an equation that underpins much of physics and chemistry and has deeply influenced the way theorists think about the nature of the Universe

  The great theoretical physicist Paul Dirac is not well-known to the public – unlike Albert Einstein – but he certainly deserves to be. He was one of the most gifted mathematical thinkers of the 20th century. Moreover, he was British. Dirac was born in Bristol where he received his early education, later becoming Lucasian Professor of Mathematics at Cambridge University – the position that the much better-known Stephen Hawking now holds.

Dirac's extraordinarily imaginative work not only led to an understanding of elementary particles and the kind of experiments carried out at CERN and other particle physics laboratories, but also laid the foundation for modern chemistry and condensed matter physics – semiconductors, magnetism and lasers. In fact, most of the technological advances that make life comfortable today ultimately owe their existence to Dirac's powerful insights into the nature of matter. The importance of Dirac’s legacy cannot be overemphasised.

 

Dirac's famous equation for the electron

The great equation

Dirac entered the field of physics in the early 1920s. This was an incredibly important time, just as the ideas of quantum mechanics were being formulated. The Danish scientist Neils Bohr had already realised that the electrons in atoms must follow orbits with definite energies. This meant that the electron energies were ‘quantised’ – they could have only certain values. Both Erwin Schrödinger and Werner Heisenberg then set about deriving mathematical descriptions to account for this strange behaviour. Schrödinger produced his ‘wave equation’ while Heisenberg took another approach, less-well known, based on the mathematical formalism of matrices. One important quantum concept to emerge was that of non-commutativity – the idea that for variables x and y, x multiplied by y is not the same as y multiplied by x – or if you like, it depends on the order: putting on socks then shoes, as opposed to the reverse! This underlies a crucial tenet of quantum mechanics, the Heisenberg Uncertainty Principle – the mysterious idea that it is impossible to measure the position and momentum of a particle simultaneously.

  Where does Dirac come in? As a mere research student at Cambridge in 1925, he saw the significance of non-commutative variables and used the idea to show that the Schrödinger and Heisenberg pictures were equivalent, thus building a sound mathematical framework for quantum mechanics.

The pièce de resistance was yet to come, however. Schrödinger's wave equation did not take into account the fact that an electron, being small and light, could move very fast, and that its behaviour was thus more accurately described by Einstein's special theory of relativity than by Newton's laws of motion. Relativity, which explained what happened when objects moved close to the speed of light, was a completely different kind of theory, so incorporating it into quantum mechanics represented quite a challenge.

In 1928, Dirac came up with a mind-bending solution – a wave equation for the electron in which the wave had four independent components (rather as a light wave possesses two independent states of polarisation, or orientation). This equation made some remarkable predictions. First, it showed that the electron had a spin, or magnetic moment, which could be positive or negative and had a value of a half. The idea of spin had already been suggested by other physicists to account for certain features in atomic spectra.

The second prediction was much stranger – it appeared that the electron could have negative energies as well as positive ones. In a leap of the imagination, Dirac proposed the existence of an anti-electron (or positron as it is now called) with the same mass as an electron but opposite charge. At first, the idea was not taken seriously, but then in 1932 Carl Anderson discovered the positron when measuring cosmic rays. Antimatter was born. Dirac went on to show that if an electron collided with a positron they would annihilate each other, releasing their mass-energy as gamma-rays according to Einstein's equation E = mc2. This was the basis for further work for formulating the relationship between matter particles, like the electron, and photons of light in quantum terms.

Another prediction that Dirac made was the existence of isolated magnetic poles, or magnetic monopoles. Physicists have searched for these objects but none has ever been found.

Further information on Paul Dirac is available from:

• www-groups.dcs.st-and.ac.uk/~ history/Mathematicians/Dirac.html
• Michael Berry, 'Paul Dirac: the purest soul in physics', Physics World, February 1998. physicsweb.org/article/world/11/2/9/1
• H. Kragh, Dirac: A Scientific Biography, Cambridge University Press, 1991.

Implications and benefits of Dirac's work

The Dirac equation and its ramifications had huge implications for modern science in many ways. It laid down the foundation for descriptions of matter in physics and in chemistry. Some examples are given below

 

The discovery of the W particle in 1982 in the proton-antiproton collider at CERN

...Particle physics

• We now know that every elementary particle has an antimatter partner; this has deeply influenced our theories of the structure of matter, and how the Universe must have evolved. Particle physicists today are studying mechanisms which explain why there is very little antimatter in the Universe.
• Dirac's quantum formulation of the processes governing matter and energy has directly led to modern particle physics theory – the Standard Model which seeks to relate matter and force particles in a unified way. A crucial ingredient is the nature of the physical vacuum. Dirac was the first to propose 'structure' in the vacuum with his extraordinary idea that it is actually full of particles sitting in negative energy states! If a particle is excited from one of these states into a positive energy level then it leaves behind an empty energy state, or hole, which is the anti-particle. In this way a particle-antiparticle pair can be created from the vacuum. This is what happens in particle colliders at very high energies, allowing particle physicists to make new particles so as to study and refine the predictions of the Standard Model.
• Dirac's equation was the first time that relativity (the special theory) and quantum mechanics were brought together. Unifying general relativity (the theory of gravity) with quantum theory is still a major goal of theorists today.

 

Novel electronic devices are being developed that depend on the spin of the electron

...Condensed matter physics

• The concept of holes is also important in semiconductor physics. If a negative electron moves from its normal site in the crystal lattice of a material like silicon, then it effectively leaves behind a positively charged hole – which can be treated like a positive electron. The behaviour of electron-hole pairs in semiconductors is the basis of electronic devices used today.
• Dirac's treatment of electron spin underpins theories of magnetism (spin gives the electron a magnetic moment) which is also becoming increasingly important in electronic applications. Physicists are developing exotic magnetic materials in which electron spins can be manipulated to produce unusual effects that can be exploited in sensors and other novel 'spin-electronic' devices.
• Dirac also opened the way for subsequent theories of the interaction of light with matter – quantum electrodynamics. He was the first to provide a theory of spontaneous emission of light, which eventually led to the concept of the laser and use of lasers to manipulate matter.

...Chemistry

• The notion of spin is essential in understanding how electrons are built up and arranged in atoms and then bound together in molecules. Chemists require a detailed understanding of electronic structure to design new materials like drugs, catalysts and plastics, and Dirac's work helped lay down the quantum rules for chemical behaviour. Furthermore, quantum calculations on heavy metals require treating the electrons in the atom relativistically because they move so fast under the influence of a large positive nucleus, and modern computer software used in these calculations is based on the Dirac equation.

...Mathematics

• Dirac's influence on mathematics and theoretical physics has been huge. Many of the problems Dirac tackled introduced new formalisms that were then developed in other contexts. For example, the components of the Dirac equation pointed to a more generalised mathematical approach for any series of linked differential equations. By considering the solutions to Dirac's equation, some of the most profound results known in differential geometry, in general relativity, and in superstring and supergravity theories, have been obtained.

 

	Positron emission tomography was used to create this image of the brain

...Medicine and engineering

• Antimatter is not just something that turns up in high-energy physics. Positron emission tomography (PET) utilises positron-emitting isotopes to pinpoint features in body tissues or in engineering structures. The target areas containing the isotope are revealed by the gamma-rays released when the positrons annihilate with surrounding electrons.
• Could we harness energy from antimatter annihilation? Most people know about antimatter from the Star Trek series, where it is used as a form of space propulsion. In fact, American scientists are studying antimatter as possible fuel for interstellar travel

Large numbers and cosmology

Biography:

Paul Dirac was born on 8 August 1902 in Bristol, one of three children. His father was Swiss and his mother English. Dirac's upbringing was quite strict. He went to school at the Merchant Venturers Secondary School where his father taught French. Apparently his father was so keen for his children to learn French that he insisted that they speak only French at the dinner table. This may account for the fact that Dirac was known as a man of few, carefully chosen words.

Showing an early ability in mathematics, he went on to study electrical engineering at Bristol University obtaining his degree in 1921 at the age of 19. He was then offered a scholarship at Cambridge to study mathematics but could not afford to take it up. Fortunately Bristol University stepped in, waiving the fees, and Dirac obtained a first class degree in 1923. He was then given a grant to do a PhD at St John's College Cambridge. By late 1925 Dirac had written five papers on quantum mechanics. It was at this time that he came into contact with Heisenberg's work, and laid out his ideas for a mathematically consistent theory of quantum mechanics which became his doctoral thesis in 1926.

A year later, Dirac was offered a Fellowship at St John's College where he continued his pioneering work. His book, Principles of Quantum Mechanics, was published in 1930. It has become a classic and is now in its fourth edition. Dirac was appointed Lucasian Professor of Mathematics in 1932, a post he held for 37 years. In 1933 he received the Nobel Prize for Physics along with Schrödinger. Dirac apparently hated the idea of the publicity and wanted to turn it down until it was pointed out he would receive far more publicity if he refused it than if he accepted it! 1934 was an important year for a different reason.

He met his future wife Margit, the sister of another famous physicist, Eugene Wigner. Dirac retired from Cambridge in 1969 and went with his family to Florida where he lived until his death in 1984

 

Dirac thought that quantum electrodynamics developed by Richard Feynman (right) and others was inelegant

The search for pure beauty

Dirac was fundamentally a mathematician and was driven by the search for beauty and elegance in the equations he formulated. He was, for example, very unhappy with quantum electrodynamics, which has infinities in the equations that are then removed by a brushing-under-the-carpet process called renormalisation. "The resulting theory is an ugly and incomplete one," he said. At the same time, Dirac's early training as an engineer meant that he appreciated the importance of approximations in getting to the heart of explaining physical phenomena.

Dirac was just as careful in expressing himself verbally as in expressing himself mathematically, saying "I was taught at school never to start a sentence without knowing the end of it