antimatter British physicist Paul Dirac predicted that the electron should have an equivalent with a positive charge, later called a positron. This was discovered to exist; the first example of an antimatter particle; subsequently further examples were found for all matter particles. When energy converts into matter it produces a pair of matter and antimatter equivalents, which can recombine, annihilating to energy.
bosons A particle that obeys Bose-Einstein statistics (as opposed to a fermion). Typically bosons are the particles that carry forces, most notably photons and the famous Higgs boson, but the term also applies to atomic nuclei with an even numbers of particles. Unlike a fermion, many bosons can be in the same state at the same time.
divergent series A series in which the sum is infinite. 1 + 1/2 + 1/3 + 1/4 + 1/5 … (where ‘…’ means ‘continue this series forever’) is divergent. By contrast, a convergent series has a finite sum. The total of 1 + 1/2 + 1/4 + 1/8 … is just 2, even though it contains an infinite set of fractions, because each subsequent item takes the total closer to 2, but never exceeds it.
electron shells Electrons only occupy fixed orbits around an atom, with jumps between these orbits usually involving absorbing or giving off a photon – this is the quantization of the electron. The possible levels of orbit are sometimes called shells, most often in chemistry. Each shell holds a maximum number of electrons (2 in the first shell, 8 in the second, 18 in the third and so on).
fermions One of the two principal types of particle (the other being bosons). Matter particles (electrons, quarks, protons and neutrons) and neutrinos are fermions. Atoms with an odd number of fermions are also fermions, while those with an even number are bosons. Fermions obey Fermi–Dirac statistics, and the Pauli Exclusion Principle (see bhere), which prevents more than one being in an identical state.
fields (quantum fields) A field is a mathematical construct filling all spacetime with a value at every location – a bit like a three-dimensional map of the Earth, where the height above sea-level is the field value. A quantum field is one producing the same effects as quantum objects that can be in a superposition of states, requiring more complex mathematics than a classical field like a map of the Earth.
fine-structure constant One of the fundamental constants of physics, with a value of about 1/137. The fine-structure constant (a) reflects the strength of the electromagnetic attraction (in effect, the probability an electron will emit a photon) and controls the way electrons bind in atoms and molecules.
matrix mechanics An early formulation of quantum theory by Heisenberg that made no attempt to provide an illustrative picture, but merely predicted what was observed as the outcome of the changing values of a set of numbers over time.
neutrons Neutrally charged particles found in the nucleus of atoms, consisting of three quarks. A particular element can have variants called isotopes with different numbers of neutrons in the nucleus.
neutron stars The outcome of the collapse of an old star of between 1.4 and 3.2 times the mass of the Sun, primarily consisting of neutrons crammed together resulting in a huge density. A piece of a neutron star the size of a grape would weigh around 100 million tonnes.
positrons Another name for an anti-electron, the positively charged antimatter equivalent of an electron.
quantum numbers The values of quantum states of a particle that can only have integer or half-integer values. An electron in an atom is described by four quantum numbers corresponding to energy level, angular moment, magnetic moment and spin.
spacetime Relativity treats time as a fourth dimension. In relativity there is no absolute position or absolute time because the way things move influences their position in time, so it is necessary to consider spacetime as a whole, rather than to think of space and time independently.
time-reversed waves Maxwell’s equations describing electricity and magnetism have two solutions allowing for a wave from the transmitter to the receiver, forwards in time (retarded waves) and from the receiver to the transmitter, backwards in time (advanced waves). Traditionally the time-reversed waves were ignored, but they are useful in explaining mathematical problems with the way an electron recoils when emitting a photon.
wave mechanics An early formulation of quantum theory by Schrödinger that treated particles as ‘matter waves’. The wave itself, described by Schrödinger’s equation, was interpreted by Max Born as representing probability rather than location. Shown to be equivalent to matrix mechanics.
In 1913 Niels Bohr explained how atoms emit or absorb photons of specific wavelengths when the electrons circling the nucleus leap from one orbit to another in a set of fixed orbits. These orbits are assigned integer quantum numbers (1, 2, 3 …), called principal quantum numbers. This model worked for hydrogen, the simplest atom known, but in the case of more complex atoms the model did not account for the extra wavelengths that appeared in the spectra of atoms. In 1915 German physicist Arnold Sommerfeld showed that a second quantum number, called the fine structure constant, could account for these wavelengths. In a magnetic field the electrons also behave like tiny magnets, and because they also have spin, physicists had to add a third and a fourth quantum number. The energy of each electron is determined by these four quantum numbers. That same year Wolfgang Pauli discovered that no electrons with the same set of four quantum numbers can orbit the same atom. This concept, the Pauli Exclusion Principle, explains why electrons, even when the atom is in its lowest energy state, distribute themselves over several shells (orbits with the same principal quantum number) and this distribution accounts for the chemical properties of the elements.
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The Pauli Exclusion principle explains why electrons in atoms always occupy several orbits instead of simply populating the lowest energy level.
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Because electrons occupy several shells around the atomic nucleus and cannot be confined in one shell, atoms have a minimum size and cannot be squeezed together. This explains why ordinary matter occupies space and is stable. Neutrons, like electrons, are fermions and have half-integer spin. As all fermions are subject to Pauli’s principle, the neutrons in neutron stars cannot merge; this prevents these stars from collapsing further under the enormous gravitational forces in such condensed matter.
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NIELS BOHR
1885–1962
Danish pioneer of quantum theory who introduced a first model of the atom
1900–58
Austrian theoretical physicist who introduced the exclusion principle that took his name
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Alexander Hellemans
The Pauli Exclusion Principle solved the riddle of why the elements with similar chemical properties are arranged in columns in the Periodic Table.
Niels Bohr proposed in 1913 that atomic spectra are created when atoms give off and absorb different light wavelengths as electrons jump from one orbit to another. The difficulty was that measurements of the atomic spectra of hydrogen did not fully agree with Bohr’s theory. So in the summer of 1927 British theoretical physicist Paul Dirac set about trying to solve this puzzle by analysing the behaviour of electrons. To do this, he welded together the wave equations of quantum mechanics developed by Erwin Schrödinger with the mathematical description of particles moving close to the speed of light embodied in the theory of special relativity. Other physicists had tried this approach but had been stumped by the difficulty of incorporating the fact that electrons possess spin into such a relativistic structure. Dirac resolved this issue through the use of some clever algebra and the incorporation of four-by-four matrices into the equation. The result was a relativistic quantum wave equation, now called the Dirac equation, which had solutions for both positive and negative energy electrons, predicting the existence of antimatter. Dirac’s brilliant insight led directly to the development of quantum field theory, the basis of modern particle physics.
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In order to arrive at his equation, Dirac had to combine the physics of the very small with the physics of the very fast.
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The real significance of Dirac’s equation may go even deeper than the development of quantum field theory: for a piece of pure mathematics had correctly predicted the existence of a new fundamental particle – Dirac’s negative energy electron could equally be an equivalent to an electron with a positive charge. The discovery of a real particle that fitted the description, the positron, identified by Carl Anderson in 1932, may indicate that mathematics is intimately connected to the very fabric of our universe.
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1845–79
British mathematician who first developed the algebra Dirac would later use
CARL ANDERSON
1905–91
Pioneering American experimenter who found anti-electrons in cosmic rays
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Leon Clifford
Dirac put relativity into the frame to bring theory into line with the observed atomic spectra.
British physicist Paul Dirac is the greatest scientist to be almost unknown outside his field. Born in Bristol to a Swiss father and a British mother, Dirac had a strict upbringing. It has been suggested that his taciturn nature originated from his father’s insistence that Dirac only speak perfect French to him. According to the story, rather than risk getting it wrong, Dirac would not speak at all. But the evidence seems strong that Dirac was on the autistic spectrum, a more likely cause of his lack of interpersonal skills.
Dirac originally studied electrical engineering at the University of Bristol, but his increasing interest in applied mathematics found him taking a second degree before moving on to Cambridge, where he was soon working on relativity and the flourishing new subject of quantum physics. Here Dirac took Schrödinger’s already powerful equation, describing the probability of finding a particle in any particular location, and expanded it to take in special relativity for some types of particle, allowing for the effects that high-speed movement would have.
Dirac’s equation was symmetrical, allowing for particles that could have either positive or negative energy. This was a serious problem, as an ordinary electron should plunge into the lower negative energy states, pouring out photons. The dramatic solution Dirac proposed was that apparently empty space contained an infinite ‘sea’ of negative energy electrons, filling all possible negative energy states, preventing electrons from decaying into negative energy. He predicted that this sea could have ‘holes’ in it – missing negative energy electrons, which would be the equivalent of positive energy anti-electrons, or positrons. He had foreseen the existence of antimatter before it was discovered.
Dirac also contributed a major breakthrough in the development of quantum theory by proving that Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics, apparently unconnected, were not just consistent but equivalent, pulling the two together to form quantum mechanics.
Dirac shared some personality traits with an earlier holder of his position in Cambridge, Lucasian Professor of Mathematics, Isaac Newton. Like Newton, he had very limited social skills and was infamous for lacking small talk, answering questions as briefly as possible. There are many stories of his attempts at conversation, most notably when meeting the boisterously outgoing American theoretical physicist Richard Feynman. Dirac, after one of his hallmark uncomfortably long pauses, is said to have remarked: ‘I have an equation. Do you have one too?’ His mathematical brilliance, however, was unquestionable.
Brian Clegg
8 August 1902
Born in Bristol to Swiss teacher Charles Dirac and Cornish librarian Florence née Holten
1921
Receives engineering degree from University of Bristol
1923
Receives mathematics degree from University of Bristol and begins PhD in Cambridge
1926
Becomes Fellow of St John’s College, Cambridge
1928
Devises the Dirac equation, describing the relativistic motion of an electron
1930
Proposes an infinite ‘sea’ of negative energy electrons and predicts the existence of antimatter
1930
Elected Fellow of the Royal Society
1932
The antimatter electron, or positron, predicted by Dirac, discovered by Carl Anderson at California Institute of Technology
1932–69
Lucasian Professor of Mathematics at Cambridge
1933
Wins Nobel Prize in physics (with Schrödinger) for discoveries in atomic theory
1937
Marries Margit (‘Manci’) Wigner, sister of physicist Eugene Wigner
1969
Retires and takes up honorary post at Florida State University
20 October 1984
Dies in Tallahassee, Florida
1952
Received the Copley Medal and the Max Planck Medal
1995
Memorial unveiled in Westminster Abbey
Quantum Field Theory (QFT) is the bedrock of modern particle physics and the mathematical basis of our understanding of the nature of reality. It builds on quantum mechanics by expanding the area of study from a handful of particles to a system of very many particles. It describes the behaviour of fields – physical quantities that have a value at every point in space, a little like the contours on a map – such as the electromagnetic field responsible for light and radio waves, at the quantum level in a way that is impossible to do in quantum mechanics. And, crucially, it enables physicists to deal with both fields and particles at the quantum level within one coherent set of equations. This theory treats waves and particles as if they are disturbances in an underlying field: so, for example, light is a ripple in an electromagnetic field while the electron is an especially excited state of an electromagnetic field. In this way, the theory neatly explains the wave-particle duality found in Nature by combining the wave and particle aspects of both light and electrons – and other combinations of forces and particles – into a single mathematical description of a field.
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Quantum Field Theory seeks to describe all of the forces and all of the particles found in Nature in terms of the interactions of such fields.
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So far Quantum Field Theory has been unable to provide a fully consistent quantum description of gravity – the force that operates over the vast distances of space. The successful inclusion of gravity would lead to a unified field theory combining all the known forces and particles that make and shape our world. Such a breakthrough would bring us one step closer to an ‘ultimate theory of everything’.
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1931–
Dutch physicist and one of the pioneers of QFT who helped combine the weak nuclear force with QED
1946–
Dutch physicist who worked with Veltman on the weak nuclear force and worked on QCD and quantum gravity
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Leon Clifford
The magnetic field that shields the earth from the solar winds requires quantum field theory to describe its quantum mechanical behaviour.
QED – quantum electrodynamics – takes the classical theory of electromagnetism developed in the 19th century by James Clerk Maxwell into the world of quantum mechanics and special relativity. Classical electromagnetism explains electric currents and electromagnetic waves such as light and radio waves in terms of electromagnetic fields; but the theory was developed before the discoveries of the electron, which carries electric charge, and the photon, which transmits light. Quantum mechanics explains how electrons and photons behave, but is unable to deal effectively with electromagnetic fields. It also has problems with the behaviour of electrons in orbit around atoms where their rapid motion approaches the speed of light and requires the use of the theory of special relativity. QED was inspired by Paul Dirac, who pioneered the successful combination of quantum mechanics and special relativity in his Dirac equation. But the existence of antimatter predicted by Dirac’s equation caused a new problem. It implied the possibility of a particle and an antiparticle annihilating themselves in a burst of energy that could condense into many possible combinations of new particles. Dirac realized that this required the development of a new theory that was capable of handling all these particles – and so QED was born.
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Through the pioneering work of Paul Dirac, QED brought the theory of electromagnetism into the quantum era.
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The history of physics can in many ways be seen as a series of unifications. Maxwell merged electricity, magnetism and light into a theory of electromagnetism, Einstein unified space and time in special relativity, and quantum mechanics brought together waves and particles. Dirac then welded together special relativity and quantum mechanics and, in turn, these were combined with electromagnetism to form QED. And this story of successive unifications continues right up to the present day.
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JAMES CLERK MAXWELL
1831–79
British scientist who unified electricity, magnetism and light into one consistent theory
PAUL DIRAC
1902–84
British physicist whose work was the inspiration for the development of QED
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Leon Clifford
Dirac extended Maxwell’s classical understanding of electromagnetism to encompass quantum particles.
Renormalization is a mathematical technique for solving a problem that arises within quantum field theories such as QED and quantum chromodynamics. This problem involves the appearance of awkward infinities which, without this technique, would make it impossible to tease meaningful solutions from the equations of the theory. Infinities can arise because particle and antiparticle pairs pop into and out of existence for infinitesimally short periods of time within a quantum system. Any attempt to add together the effects of all these different particles quickly leads to infinity, in what mathematicians call a divergent series. In simple terms, renormalization works by parcelling together some of the elements that diverge to infinity and then offsetting them against each other; the remaining balance is replaced in the equation by an arbitrary constant whose value can be determined by experiment. Provided that only a finite number of constants is required and that the values for each such constant can be determined, a theory is said to be renormalizable. For quantum field theories to be accepted by the physics community, they must be shown to be renormalizable. So far, quantum gravity has failed this test.
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Renormalization is a neat mathematical trick that solves a major problem but was described as ‘a shell game’ by its most famous inventor, Richard Feynman.
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Renormalization undoubtedly works, but Richard Feynman was never completely comfortable with the technique he helped to develop. The great physicist Paul Dirac also cautioned against the approach. It is worth remembering that Dirac predicted the existence of antimatter by refusing to ignore some seemingly strange solutions to his own equation. So are these infinities really just mathematical irritations or are they actually telling us something profound about the underlying nature of reality?
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RICHARD FEYNMAN
1918–88
American physicist and a co-discoverer of renormalization
1918–94
American physicist credited with renormalizing QED
1906–79
Japanese physicist who independently developed renormalization
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Leon Clifford
Particle/antiparticle pairs popping briefly into existence contribute to the infinities needing renormalization.
A way of visualizing what occurs when a change occurs in the quantum world is to draw a Feynman diagram, a simple graphic that shows how subatomic particles interact. Feynman diagrams were developed in the 1940s by American physicist Richard Feynman as a means of understanding the processes involved in the theory of photons and electrons known as quantum electrodynamics (QED). These diagrams remain relevant to all areas of quantum field theory and have proved hugely successful in helping scientists gain insights into some of the most complex calculations in high-energy particle physics. Every Feynman diagram has to obey a set of specific rules to ensure consistency and usefulness. All Feynman diagrams represent particles by a combination of wavy and straight lines and interactions take place where the lines meet. Feynman diagrams can capture one or more interactions: one axis represents space and another represents time, and lines representing particles move through both space and time diagonally across the diagram. Interestingly, Feynman diagrams show particles of antimatter moving along the time axis in the opposite direction to particles of matter. This can be interpreted as saying that an antiparticle is the equivalent of a particle of normal matter that is going backwards in time.
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Feynman diagrams reduce the world of quantum physics to a more immediate graphic that represents the interaction of particles in time and space.
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Does the way we view physics affect the way we think about it? Feynman diagrams offer an easily comprehensible visual shorthand that implies a particle-based view of the world. This is at odds with the continuous fields described in quantum field theory (QFT). This reflects the fact that all scientific theories and methods are just models that predict what we observe rather than true descriptions of reality.
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RICHARD FEYNMAN
1918–88
American physicist, inventor of the eponymous diagrams
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Leon Clifford
Feynman’s elegant diagrams proved an essential tool in understanding quantum electrodynamics.
Waves that travel backwards in time are predicted in the famous equations of electrodynamics developed by James Clerk Maxwell and these predictions are carried into quantum mechanics. According to the mathematics, an event that creates a wave that travels forwards in time – an electron emitting a photon as an electromagnetic wave, say – simultaneously creates a different kind of wave, one that travels backwards in time. These time-reversed waves are known as advanced waves – advanced, since they arrive in advance of their creation. Usually, the bizarre mathe- matical solutions that allow for advanced waves are ignored, but that does not mean they do not exist. Indeed, one interpretation of quantum mechanics describes quantum events in terms of the interactions of waves going backwards in time with waves that travel forwards in time. However, no one has ever seen an advanced wave. One suggestion is that this is due to the workings of the second law of thermodynamics and that both kinds of waves are actually produced in equal number, as predicted by the mathematics. The action of the second law means that the forward-in-time wave will be absorbed at some point in the future and that this inevitably results in the erasure of all evidence of the advanced waves.
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In quantum mechanics time is a two-way street, where waves travel both backwards and forwards in time, though only the wave travelling forwards is detectable.
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If advanced waves ever became detectable then it would, in principle, be possible to send a message back in time using an advanced wave by pointing the transmitter at the place in space where the Earth was at some point in the past. What would be the implications of this?
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JAMES CLERK MAXWELL
1831–79
British scientist who unified electricity, magnetism and light in one consistent theory
1911–2008
American physicist who, with Richard Feynman, explained why it is that we do not see advanced waves
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Leon Clifford
