Paul Adrien Maurice Dirac OM FRS^{[7]} (/dɪˈræk/; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century.
Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions and predicted the existence of antimatter. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger "for the discovery of new productive forms of atomic theory".^{[8]} He also made significant contributions to the reconciliation of general relativity with quantum mechanics.
Dirac was regarded by his friends and colleagues as unusual in character. In a 1926 letter to Paul Ehrenfest, Albert Einstein wrote of Dirac, "This balancing on the dizzying path between genius and madness is awful".^{[9]}
He was the Lucasian Professor of Mathematics at the University of Cambridge, a member of the Center for Theoretical Studies, University of Miami, and spent the last decade of his life at Florida State University.
Paul Adrien Maurice Dirac was born at his parents' home in Bristol, England, on 8 August 1902,^{[10]} and grew up in the Bishopston area of the city.^{[11]} His father, Charles Adrien Ladislas Dirac, was an immigrant from SaintMaurice, Switzerland, who worked in Bristol as a French teacher. His mother, Florence Hannah Dirac, née Holten, the daughter of a ship's captain, was born in Cornwall, England, and worked as a librarian at the Bristol Central Library. Paul had a younger sister, Béatrice Isabelle Marguerite, known as Betty, and an older brother, Reginald Charles Félix, known as Felix,^{[12]}^{[13]} who committed suicide in March 1925.^{[14]} Dirac later recalled: "My parents were terribly distressed. I didn't know they cared so much [...] I never knew that parents were supposed to care for their children, but from then on I knew."^{[15]}
Charles and the children were officially Swiss nationals until they became naturalised on 22 October 1919.^{[16]} Dirac's father was strict and authoritarian, although he disapproved of corporal punishment.^{[17]} Dirac had a strained relationship with his father, so much so that after his father's death, Dirac wrote, "I feel much freer now, and I am my own man." Charles forced his children to speak to him only in French, in order that they learn the language. When Dirac found that he could not express what he wanted to say in French, he chose to remain silent.^{[18]}^{[19]}
Dirac was educated first at Bishop Road Primary School^{[20]} and then at the allboys Merchant Venturers' Technical College (later Cotham School), where his father was a French teacher.^{[21]} The school was an institution attached to the University of Bristol, which shared grounds and staff.^{[22]} It emphasised technical subjects like bricklaying, shoemaking and metal work, and modern languages.^{[23]} This was unusual at a time when secondary education in Britain was still dedicated largely to the classics, and something for which Dirac would later express his gratitude.^{[22]}
Dirac studied electrical engineering on a City of Bristol University Scholarship at the University of Bristol's engineering faculty, which was colocated with the Merchant Venturers' Technical College.^{[24]} Shortly before he completed his degree in 1921, he sat the entrance examination for St John's College, Cambridge. He passed and was awarded a £70 scholarship, but this fell short of the amount of money required to live and study at Cambridge. Despite his having graduated with a first class honours Bachelor of Science degree in engineering, the economic climate of the postwar depression was such that he was unable to find work as an engineer. Instead, he took up an offer to study for a Bachelor of Arts degree in mathematics at the University of Bristol free of charge. He was permitted to skip the first year of the course owing to his engineering degree.^{[25]}
In 1923, Dirac graduated, once again with first class honours, and received a £140 scholarship from the Department of Scientific and Industrial Research.^{[26]} Along with his £70 scholarship from St John's College, this was enough to live at Cambridge. There, Dirac pursued his interests in the theory of general relativity, an interest he had gained earlier as a student in Bristol, and in the nascent field of quantum physics, under the supervision of Ralph Fowler.^{[27]} From 1925 to 1928 he held an 1851 Research Fellowship from the Royal Commission for the Exhibition of 1851.^{[28]} He completed his PhD in June 1926 with the first thesis on quantum mechanics to be submitted anywhere.^{[29]} He then continued his research in Copenhagen and Göttingen.^{[28]}
Dirac married Margit Wigner (Eugene Wigner's sister), in 1937. He adopted Margit's two children, Judith and Gabriel. Paul and Margit Dirac had two children together, both daughters, Mary Elizabeth and Florence Monica.
Margit, known as Manci, visited her brother in 1934 in Princeton, New Jersey, from her native Hungary and, while at dinner at the Annex Restaurant met the "lonelylooking man at the next table." This account from a Korean physicist, Y. S. Kim, who met and was influenced by Dirac, also says: "It is quite fortunate for the physics community that Manci took good care of our respected Paul A. M. Dirac. Dirac published eleven papers during the period 1939–46.... Dirac was able to maintain his normal research productivity only because Manci was in charge of everything else."^{[30]}
Dirac was known among his colleagues for his precise and taciturn nature. His colleagues in Cambridge jokingly defined a unit called a "dirac", which was one word per hour.^{[31]} When Niels Bohr complained that he did not know how to finish a sentence in a scientific article he was writing, Dirac replied, "I was taught at school never to start a sentence without knowing the end of it."^{[32]} He criticised the physicist J. Robert Oppenheimer's interest in poetry: "The aim of science is to make difficult things understandable in a simpler way; the aim of poetry is to state simple things in an incomprehensible way. The two are incompatible."^{[33]}
Dirac himself wrote in his diary during his postgraduate years that he concentrated solely on his research, and stopped only on Sunday when he took long strolls alone.^{[34]}
An anecdote recounted in a review of the 2009 biography tells of Werner Heisenberg and Dirac sailing on an ocean liner to a conference in Japan in August 1929. "Both still in their twenties, and unmarried, they made an odd couple. Heisenberg was a ladies' man who constantly flirted and danced, while Dirac—'an Edwardian geek', as biographer Graham Farmelo puts it—suffered agonies if forced into any kind of socialising or small talk. 'Why do you dance?' Dirac asked his companion. 'When there are nice girls, it is a pleasure,' Heisenberg replied. Dirac pondered this notion, then blurted out: 'But, Heisenberg, how do you know beforehand that the girls are nice?'"^{[35]}
Margit Dirac told both George Gamow and Anton Capri in the 1960s that her husband had said to a house visitor, "Allow me to present Wigner's sister, who is now my wife."^{[36]}^{[37]}
Another story told of Dirac is that when he first met the young Richard Feynman at a conference, he said after a long silence, "I have an equation. Do you have one too?"^{[38]}
After he presented a lecture at a conference, one colleague raised his hand and said: "I don't understand the equation on the toprighthand corner of the blackboard". After a long silence, the moderator asked Dirac if he wanted to answer the question, to which Dirac replied: "That was not a question, it was a comment."^{[39]}^{[40]}
Dirac was also noted for his personal modesty. He called the equation for the time evolution of a quantummechanical operator, which he was the first to write down, the "Heisenberg equation of motion". Most physicists speak of Fermi–Dirac statistics for halfintegerspin particles and Bose–Einstein statistics for integerspin particles. While lecturing later in life, Dirac always insisted on calling the former "Fermi statistics". He referred to the latter as "Einstein statistics" for reasons, he explained, of "symmetry".^{[41]}
Heisenberg recollected a conversation among young participants at the 1927 Solvay Conference about Einstein and Planck's views on religion between Wolfgang Pauli, Heisenberg and Dirac. Dirac's contribution was a criticism of the political purpose of religion, which was much appreciated for its lucidity by Bohr when Heisenberg reported it to him later. Among other things, Dirac said:
I cannot understand why we idle discussing religion. If we are honest—and scientists have to be—we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination. It is quite understandable why primitive people, who were so much more exposed to the overpowering forces of nature than we are today, should have personified these forces in fear and trembling. But nowadays, when we understand so many natural processes, we have no need for such solutions. I can't for the life of me see how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why God allows so much misery and injustice, the exploitation of the poor by the rich and all the other horrors He might have prevented. If religion is still being taught, it is by no means because its ideas still convince us, but simply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also much easier to exploit. Religion is a kind of opium that allows a nation to lull itself into wishful dreams and so forget the injustices that are being perpetrated against the people. Hence the close alliance between those two great political forces, the State and the Church. Both need the illusion that a kindly God rewards—in heaven if not on earth—all those who have not risen up against injustice, who have done their duty quietly and uncomplainingly. That is precisely why the honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins.^{[42]}
Heisenberg's view was tolerant. Pauli, raised as a Catholic, had kept silent after some initial remarks, but when finally he was asked for his opinion, said: "Well, our friend Dirac has got a religion and its guiding principle is 'There is no God and Paul Dirac is His prophet.'" Everybody, including Dirac, burst into laughter.^{[43]}^{[44]}
Later in life, Dirac's views towards the idea of God were less acerbic. As an author of an article appearing in the May 1963 edition of Scientific American, Dirac wrote:
It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.^{[45]}
In 1971, at a conference meeting, Dirac expressed his views on the existence of God.^{[46]} Dirac explained that the existence of God could only be justified if an improbable event were to have taken place in the past:
It could be that it is extremely difficult to start life. It might be that it is so difficult to start life that it has happened only once among all the planets... Let us consider, just as a conjecture, that the chance life starting when we have got suitable physical conditions is 10^{−100}. I don't have any logical reason for proposing this figure, I just want you to consider it as a possibility. Under those conditions ... it is almost certain that life would not have started. And I feel that under those conditions it will be necessary to assume the existence of a god to start off life. I would like, therefore, to set up this connexion between the existence of a god and the physical laws: if physical laws are such that to start off life involves an excessively small chance, so that it will not be reasonable to suppose that life would have started just by blind chance, then there must be a god, and such a god would probably be showing his influence in the quantum jumps which are taking place later on. On the other hand, if life can start very easily and does not need any divine influence, then I will say that there is no god.^{[46]}
Dirac did not commit himself to any definite view, but he described the possibilities for answering the question of God in a scientific manner.^{[46]}
Dirac shared the 1933 Nobel Prize for physics with Erwin Schrödinger "for the discovery of new productive forms of atomic theory".^{[8]} Dirac was also awarded the Royal Medal in 1939 and both the Copley Medal and the Max Planck Medal in 1952. He was elected a Fellow of the Royal Society in 1930,^{[7]} an Honorary Fellow of the American Physical Society in 1948, and an Honorary Fellow of the Institute of Physics, London in 1971. He received the inaugural J. Robert Oppenheimer Memorial Prize in 1969.^{[47]}^{[48]} Dirac became a member of the Order of Merit in 1973, having previously turned down a knighthood as he did not want to be addressed by his first name.^{[35]}^{[49]}
In 1984, Dirac died in Tallahassee, Florida, and was buried at Tallahassee's Roselawn Cemetery.^{[50]}^{[51]} Dirac's childhood home in Bristol is commemorated with a blue plaque and the nearby Dirac Road is named in recognition of his links with the city. A commemorative stone was erected in a garden in SaintMaurice, Switzerland, the town of origin of his father's family, on 1 August 1991. On 13 November 1995 a commemorative marker, made from Burlington green slate and inscribed with the Dirac equation, was unveiled in Westminster Abbey.^{[50]}^{[52]} The Dean of Westminster, Edward Carpenter, had initially refused permission for the memorial, thinking Dirac to be antiChristian, but was eventually (over a fiveyear period) persuaded to relent.^{[53]}
Dirac established the most general theory of quantum mechanics and discovered the relativistic equation for the electron, which now bears his name. The remarkable notion of an antiparticle to each fermion particle – e.g. the positron as antiparticle to the electron – stems from his equation. He was the first to develop quantum field theory, which underlies all theoretical work on subatomic or "elementary" particles today, work that is fundamental to our understanding of the forces of nature. He proposed and investigated the concept of a magnetic monopole, an object not yet known empirically, as a means of bringing even greater symmetry to James Clerk Maxwell's equations of electromagnetism.
He quantised the gravitational field, and developed a general theory of quantum field theories with dynamical constraints, which forms the basis of the gauge theories and superstring theories of today. The influence and importance of his work have increased with the decades, and physicists use the concepts and equations that he developed daily.
Dirac's first step into a new quantum theory was taken late in September 1925. Ralph Fowler, his research supervisor, had received a proof copy of an exploratory paper by Werner Heisenberg in the framework of the old quantum theory of Bohr and Sommerfeld. Heisenberg leaned heavily on Bohr's correspondence principle but changed the equations so that they involved directly observable quantities, leading to the matrix formulation of quantum mechanics. Fowler sent Heisenberg's paper on to Dirac, who was on vacation in Bristol, asking him to look into this paper carefully.
Dirac's attention was drawn to a mysterious mathematical relationship, at first sight unintelligible, that Heisenberg had reached. Several weeks later, back in Cambridge, Dirac suddenly recognised that this mathematical form had the same structure as the Poisson brackets that occur in the classical dynamics of particle motion. From this thought, he quickly developed a quantum theory that was based on noncommuting dynamical variables. This led him to a more profound and significant general formulation of quantum mechanics than was achieved by any other worker in this field.^{[54]} Dirac's formulation allowed him to obtain the quantisation rules in a novel and more illuminating manner. For this work,^{[55]} published in 1926, Dirac received a PhD from Cambridge. This formed the basis for FermiDirac statistics that applies to systems consisting of many identical spin 1/2 particles (i.e. that obey the Pauli exclusion principle), e.g. electrons in solids and liquids, and importantly to the field of conduction in semiconductors.
Dirac was famously not bothered by issues of interpretation in quantum theory. In fact, in a paper published in a book in his honour, he wrote: "The interpretation of quantum mechanics has been dealt with by many authors, and I do not want to discuss it here. I want to deal with more fundamental things."^{[56]}
In 1928, building on 2×2 spin matrices which he purported to have discovered independently of Wolfgang Pauli's work on nonrelativistic spin systems (Dirac told Abraham Pais, "I believe I got these [matrices] independently of Pauli and possibly Pauli got these independently of me."),^{[57]} he proposed the Dirac equation as a relativistic equation of motion for the wave function of the electron.^{[58]} This work led Dirac to predict the existence of the positron, the electron's antiparticle, which he interpreted in terms of what came to be called the Dirac sea.^{[59]} The positron was observed by Carl Anderson in 1932. Dirac's equation also contributed to explaining the origin of quantum spin as a relativistic phenomenon.
The necessity of fermions (matter) being created and destroyed in Enrico Fermi's 1934 theory of beta decay led to a reinterpretation of Dirac's equation as a "classical" field equation for any point particle of spin ħ/2, itself subject to quantisation conditions involving anticommutators. Thus reinterpreted, in 1934 by Werner Heisenberg, as a (quantum) field equation accurately describing all elementary matter particles – today quarks and leptons – this Dirac field equation is as central to theoretical physics as the Maxwell, Yang–Mills and Einstein field equations. Dirac is regarded as the founder of quantum electrodynamics, being the first to use that term. He also introduced the idea of vacuum polarisation in the early 1930s. This work was key to the development of quantum mechanics by the next generation of theorists, in particular Schwinger, Feynman, SinItiro Tomonaga and Dyson in their formulation of quantum electrodynamics.
Dirac's The Principles of Quantum Mechanics, published in 1930, is a landmark in the history of science. It quickly became one of the standard textbooks on the subject and is still used today. In that book, Dirac incorporated the previous work of Werner Heisenberg on matrix mechanics and of Erwin Schrödinger on wave mechanics into a single mathematical formalism that associates measurable quantities to operators acting on the Hilbert space of vectors that describe the state of a physical system. The book also introduced the delta function. Following his 1939 article,^{[60]} he also included the bra–ket notation in the third edition of his book,^{[61]} thereby contributing to its universal use nowadays.
In 1931, Dirac proposed that the existence of a single magnetic monopole in the universe would suffice to explain the quantisation of electrical charge.^{[62]} In 1975,^{[63]} 1982,^{[64]} and 2009^{[65]}^{[66]}^{[67]} intriguing results suggested the possible detection of magnetic monopoles, but there is, to date, no direct evidence for their existence (see also Magnetic monopole#Searches for magnetic monopoles).
Dirac was the Lucasian Professor of Mathematics at Cambridge from 1932 to 1969. In 1937, he proposed a speculative cosmological model based on the socalled large numbers hypothesis. During World War II, he conducted important theoretical and experimental research on uranium enrichment by gas centrifuge.
Dirac's quantum electrodynamics (QED) made predictions that were – more often than not – infinite and therefore unacceptable. A workaround known as renormalisation was developed, but Dirac never accepted this. "I must say that I am very dissatisfied with the situation", he said in 1975, "because this socalled 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!"^{[68]} His refusal to accept renormalisation resulted in his work on the subject moving increasingly out of the mainstream.
However, from his once rejected notes he managed to work on putting quantum electrodynamics on "logical foundations" based on Hamiltonian formalism that he formulated. He found a rather novel way of deriving the anomalous magnetic moment "Schwinger term" and also the Lamb shift, afresh in 1963, using the Heisenberg picture and without using the joining method used by Weisskopf and French, and by the two pioneers of modern QED, Schwinger and Feynman. That was two years before the Tomonaga–Schwinger–Feynman QED was given formal recognition by an award of the Nobel Prize for physics.
Weisskopf and French (FW) were the first to obtain the correct result for the Lamb shift and the anomalous magnetic moment of the electron. At first FW results did not agree with the incorrect but independent results of Feynman and Schwinger.^{[69]} The 1963–1964 lectures Dirac gave on quantum field theory at Yeshiva University were published in 1966 as the Belfer Graduate School of Science, Monograph Series Number, 3. After having relocated to Florida to be near his elder daughter, Mary, Dirac spent his last fourteen years (of both life and physics research) at the University of Miami in Coral Gables, Florida, and Florida State University in Tallahassee, Florida.
In the 1950s in his search for a better QED, Paul Dirac developed the Hamiltonian theory of constraints^{[70]} based on lectures that he delivered at the 1949 International Mathematical Congress in Canada. Dirac^{[71]} had also solved the problem of putting the Schwinger–Tomonaga equation into the Schrödinger representation^{[72]} and given explicit expressions for the scalar meson field (spin zero pion or pseudoscalar meson), the vector meson field (spin one rho meson), and the electromagnetic field (spin one massless boson, photon).
The Hamiltonian of constrained systems is one of Dirac's many masterpieces. It is a powerful generalisation of Hamiltonian theory that remains valid for curved spacetime. The equations for the Hamiltonian involve only six degrees of freedom described by , for each point of the surface on which the state is considered. The (m = 0, 1, 2, 3) appear in the theory only through the variables , which occur as arbitrary coefficients in the equations of motion. There are four constraints or weak equations for each point of the surface = constant. Three of them form the four vector density in the surface. The fourth is a 3dimensional scalar density in the surface H_{L} ≈ 0; H_{r} ≈ 0 (r = 1, 2, 3)
In the late 1950s, he applied the Hamiltonian methods he had developed to cast Einstein's general relativity in Hamiltonian form^{[73]} and to bring to a technical completion the quantisation problem of gravitation and bring it also closer to the rest of physics according to Salam and DeWitt. In 1959 he also gave an invited talk on "Energy of the Gravitational Field" at the New York Meeting of the American Physical Society later published in 1959 Phys Rev Lett 2, 368. In 1964 he published his Lectures on Quantum Mechanics (London:Academic) which deals with constrained dynamics of nonlinear dynamical systems including quantisation of curved spacetime. He also published a paper entitled "Quantization of the Gravitational Field" in the 1967 ICTP/IAEA Trieste Symposium on Contemporary Physics.
From September 1970 to January 1971, Dirac was Visiting Professor at Florida State University in Tallahassee. During that time he was offered a permanent position there, which he accepted, becoming a full professor in 1972. Contemporary accounts of his time there describe it as happy except that he apparently found the summer heat oppressive and liked to escape from it to Cambridge.^{[74]}
He would walk about a mile to work each day and was fond of swimming in one of the two nearby lakes (Silver Lake and Lost Lake), and was also more sociable than he had been at Cambridge, where he mostly worked at home apart from giving classes and seminars; at FSU he would usually eat lunch with his colleagues before taking a nap.
Dirac published over 60 papers in those last twelve years of his life, including a short book on general relativity. His last paper (1984), entitled "The inadequacies of quantum field theory," contains his final judgment on quantum field theory;
"These rules of renormalisation give surprisingly, excessively good agreement with experiments. Most physicists say that these working rules are, therefore, correct. I feel that is not an adequate reason. Just because the results happen to be in agreement with observation does not prove that one's theory is correct."
The paper ends with these words;
"I have spent many years searching for a Hamiltonian to bring into the theory and have not yet found it. I shall continue to work on it as long as I can and other people, I hope, will follow along such lines."
(Source: "Paul Dirac: The Man and his Work" by Abraham Pais et al.)
Amongst his many students^{[3]}^{[75]} were Homi J. Bhabha,^{[2]} Fred Hoyle and John Polkinghorne.^{[4]} Polkinghorne recalls that Dirac "was once asked what was his fundamental belief. He strode to a blackboard and wrote that the laws of nature should be expressed in beautiful equations."^{[76]}
In 1975, Dirac gave a series of five lectures at the University of New South Wales which were subsequently published as a book, Directions in Physics (1978). He donated the royalties from this book to the university for the establishment of the Dirac Lecture Series. The Silver Dirac Medal for the Advancement of Theoretical Physics is awarded by the University of New South Wales to commemorate the lecture.^{[77]}
Immediately after his death, two organisations of professional physicists established annual awards in Dirac's memory. The Institute of Physics, the United Kingdom's professional body for physicists, awards the Paul Dirac Medal for "outstanding contributions to theoretical (including mathematical and computational) physics".^{[78]} The first three recipients were Stephen Hawking (1987), John Stewart Bell (1988), and Roger Penrose (1989). The International Centre for Theoretical Physics awards the Dirac Medal of the ICTP each year on Dirac's birthday (8 August).^{[79]}
The DiracHellman Award at Florida State University was endowed by Dr Bruce P. Hellman in 1997 to reward outstanding work in theoretical physics by FSU researchers.^{[80]} The Paul A.M. Dirac Science Library at Florida State University, which Manci opened in December 1989,^{[81]} is named in his honour, and his papers are held there.^{[82]} Outside is a statue of him by Gabriella Bollobás.^{[83]} The street on which the National High Magnetic Field Laboratory in Innovation Park of Tallahassee, Florida, is located is named Paul Dirac Drive. As well as in his hometown of Bristol, there is also a road named after him, Dirac Place, in Didcot, Oxfordshire.^{[84]} The BBC named a video codec, Dirac, in his honour. An asteroid discovered in 1983 was named after Dirac.^{[85]} The Distributed Research utilising Advanced Computing (DiRAC) and Dirac software are named in his honour.
Farmelo also discusses, across several chapters, the influences of John Stuart Mill...
In quantum mechanics, a boson (, ) is a particle that follows Bose–Einstein statistics. Bosons make up one of the two classes of particles, the other being fermions. The name boson was coined by Paul Dirac to commemorate the contribution of Indian physicist and professor of physics at University of Calcutta and at University of Dhaka, Satyendra Nath Bose in developing, with Albert Einstein, Bose–Einstein statistics—which theorizes the characteristics of elementary particles.Examples of bosons include fundamental particles such as photons, gluons, and W and Z bosons (the four forcecarrying gauge bosons of the Standard Model), the recently discovered Higgs boson, and the hypothetical graviton of quantum gravity. Some composite particles are also bosons, such as mesons and stable nuclei of even mass number such as deuterium (with one proton and one neutron, atomic mass number = 2), helium4, or lead208; as well as some quasiparticles (e.g. Cooper pairs, plasmons, and phonons).An important characteristic of bosons is that their statistics do not restrict the number of them that occupy the same quantum state. This property is exemplified by helium4 when it is cooled to become a superfluid. Unlike bosons, two identical fermions cannot occupy the same quantum space. Whereas the elementary particles that make up matter (i.e. leptons and quarks) are fermions, the elementary bosons are force carriers that function as the 'glue' holding matter together. This property holds for all particles with integer spin (s = 0, 1, 2, etc.) as a consequence of the spin–statistics theorem.
When a gas of Bose particles is cooled down to temperatures very close to absolute zero, then the kinetic energy of the particles decreases to a negligible amount, and they condense into the lowest energy level state. This state is called a BoseEinstein condensate. It is believed that this property is the explanation of superfluidity.
Bra–ket notationIn quantum mechanics, bra–ket notation is a standard notation for describing quantum states. It can also be used to denote abstract vectors and linear functionals in mathematics. The notation uses angle brackets (the ⟨ and ⟩ symbols) and a vertical bar (the  symbol), to denote the scalar product of vectors or the action of a linear functional on a vector in a complex vector space. The scalar product or action is written as
The right part is called the ket /kɛt/; it is a vector, typically represented as a column vector, and written
The left part is called the bra, /brɑː/; it is the Hermitian conjugate of the ket with the same label, typically represented as a row vector, and written
A combination of bras, kets, and operators is interpreted using matrix multiplication. A bra and a ket with the same label are Hermitian conjugates of each other.
Braket notation was introduced in 1939 by Paul Dirac and is also known as the Dirac notation. However the braket notation has a precursor in Hermann Grassmann's use of the notation for his inner products nearly 100 years earlier.
Dirac (software)Dirac (named after Paul Dirac) is a relativistic ab initio quantum chemistry program. The full name is "Program for Atomic and Molecular Direct Iterative Relativistic Allelectron Calculations", in short PAM Dirac. It is capable of calculating various molecular properties using the Hartree–Fock, MP2, density functional theory, configuration interaction and coupled cluster electronic structure theories. Dirac is one of the most successful generalpurpose quantum chemistry packages that provides accurate description of relativistic effects in molecules, using the Dirac equation as its starting point. The program is available in source code form, at no cost, to the academic community.
The most recent version,
DIRAC15, was released on December 17, 2015.
Dirac MedalThe Dirac Medal is the name of four awards in the field of theoretical physics, computational chemistry, and mathematics, awarded by different organizations, named in honour of Professor Paul Dirac, one of the great theoretical physicists of the 20th century.
Dirac algebraIn mathematical physics, the Dirac algebra is the Clifford algebra Cℓ_{4}(C), which may be thought of as Cℓ_{1,3}(C). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin½ particles with a matrix representation with the Dirac gamma matrices, which represent the generators of the algebra.
The gamma elements have the defining relation
where are the components of the Minkowski metric with signature (+ − − −) and is the identity element of the algebra (the identity matrix in the case of a matrix representation). This allows the definition of a scalar product
where
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 spin1/2 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. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the nonrelativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation.
Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin1/2 particles.
Dirac large numbers hypothesisThe Dirac large numbers hypothesis (LNH) is an observation made by Paul Dirac in 1937 relating ratios of size scales in the Universe to that of force scales. The ratios constitute very large, dimensionless numbers: some 40 orders of magnitude in the present cosmological epoch. According to Dirac's hypothesis, the apparent similarity of these ratios might not be a mere coincidence but instead could imply a cosmology with these unusual features:
Dirac operatorIn mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or halfiterate, of a secondorder operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of firstorder operators he introduced spinors.
Dirac seaThe Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negativeenergy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, well before its experimental discovery in 1932.
Upon solving the free Dirac equation,
one finds
where
for plane wave solutions with 3momentum p. This is a direct consequence of the relativistic energymomentum relation
upon which the Dirac equation is built. The quantity U is a constant 2 × 1 column vector and N is a normalization constant. The quantity ε is called the time evolution factor, and its interpretation in similar roles in, for example, the plane wave solutions of the Schrödinger equation, is the energy of the wave (particle). This interpretation is not immediately available here since it may acquire negative values. A similar situation prevails for the Klein–Gordon equation. In that case, the absolute value of ε can be interpreted as the energy of the wave since in the canonical formalism, waves with negative ε actually have positive energy E_{p}. But this is not the case with the Dirac equation. The energy in the canonical formalism associated with negative ε is –E_{p}.
In hole theory, the solutions with negative time evolution factors are reinterpreted as representing the positron, discovered by Carl Anderson. The interpretation of this result requires a Dirac sea, showing that the Dirac equation is not merely a combination of special relativity and quantum mechanics, but it also implies that the number of particles cannot be conserved.
Dirac spinorIn quantum field theory, the Dirac spinor is the bispinor in the planewave solution
of the free Dirac equation,
where (in the units )
The Dirac spinor for the positivefrequency solution can be written as
where
In mathematical physics, the Dirac–von Neumann axioms give a mathematical formulation of quantum mechanics in terms of operators on a Hilbert space. They were introduced by Dirac (1930) and von Neumann (1932).
FermionIn particle physics, a fermion is a particle that follows Fermi–Dirac statistics. These particles obey the Pauli exclusion principle. Fermions include all quarks and leptons, as well as all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.
A fermion can be an elementary particle, such as the electron, or it can be a composite particle, such as the proton. According to the spinstatistics theorem in any reasonable relativistic quantum field theory, particles with integer spin are bosons, while particles with halfinteger spin are fermions.
In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin statistics relation is in fact a spin statisticsquantum number relation.As a consequence of the Pauli exclusion principle, only one fermion can occupy a particular quantum state at any given time. If multiple fermions have the same spatial probability distribution, then at least one property of each fermion, such as its spin, must be different. Fermions are usually associated with matter, whereas bosons are generally force carrier particles, although in the current state of particle physics the distinction between the two concepts is unclear. Weakly interacting fermions can also display bosonic behavior under extreme conditions. At low temperature fermions show superfluidity for uncharged particles and superconductivity for charged particles.
Composite fermions, such as protons and neutrons, are the key building blocks of everyday matter.
The name fermion was coined by English theoretical physicist Paul Dirac from the surname of Italian physicist Enrico Fermi.
Gabriel Andrew DiracGabriel Andrew Dirac (13 March 1925 – 20 July 1984) was a mathematician who mainly worked in graph theory. He stated a sufficient condition for a graph to contain a Hamiltonian circuit. In 1951 he conjectured that n points in the plane, not all collinear, must span at least [n/2] twopoint lines, where [x] is the largest integer not exceeding x. This conjecture was proven true when n is sufficiently large by Green and Tao in 2012.
Kapitsa–Dirac effectThe Kapitza–Dirac effect is a quantum mechanical effect consisting of the diffraction of matter by a standing wave of light.
The effect was first predicted as the diffraction of electrons from a standing wave of light by Paul Dirac and Pyotr Kapitsa (or Peter Kapitza) in 1933. The effect relies on the wave–particle duality of matter as stated by the de Broglie hypothesis in 1924.
List of things named after Paul DiracBelow is a list of things, primarily in the fields of mathematics and physics, named in honour of Paul Adrien Maurice Dirac.
Lucasian Professor of MathematicsThe Lucasian Chair of Mathematics () is a mathematics professorship in the University of Cambridge, England; its holder is known as the Lucasian Professor. The post was founded in 1663 by Henry Lucas, who was Cambridge University's Member of Parliament from 1639–1640, and it was officially established by King Charles II on 18 January 1664. It was described by The Daily Telegraph as one of the most prestigious academic posts in the world and its former holders include Isaac Newton, Charles Babbage, George Stokes, Joseph Larmor, Paul Dirac and Stephen Hawking.
Principle of least actionThis article discusses the history of the principle of least action. For the application, please refer to action (physics).The principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. In relativity, a different action must be minimized or maximized. The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action). The physicist Paul Dirac, and after him Julian Schwinger and Richard Feynman, demonstrated how this principle can also be used in quantum calculations.
It was historically called "least" because its solution requires finding the path that has the least value. Its classical mechanics and electromagnetic expressions are a consequence of quantum mechanics, but the stationary action method helped in the development of quantum mechanics.The principle remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory and is a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action.
The action principle is preceded by earlier ideas in optics. In ancient Greece, Euclid wrote in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection. Hero of Alexandria later showed that this path was the shortest length and least time.Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744 and 1746. However, Leonhard Euler discussed the principle in 1744, and evidence shows that Gottfried Leibniz preceded both by 39 years.In 1933, Paul Dirac discerned the quantum mechanical underpinning of the principle in the quantum interference of amplitudes.
The Principles of Quantum MechanicsThe Principles of Quantum Mechanics is an influential monograph on quantum mechanics written by Paul Dirac and first published by Oxford University Press in 1930.
Dirac gives an account of quantum mechanics by "demonstrating how to construct a completely new theoretical framework from scratch"; "problems were tackled topdown, by working on the great principles, with the details left to look after themselves". It leaves classical physics behind after the first chapter, presenting the subject with a logical structure. Its 82 sections contain 785 equations with no diagrams.Dirac is credited with developing the subject "particularly in Cambridge and Göttingen between 1925–1927" (Farmelo).
The Strangest ManThe Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius is a 2009 biography of quantum physicist Paul Dirac written by British physicist and author, Graham Farmelo, and published by Faber and Faber. The book won the Biography Award at the 2009 Costa Book Awards, and the 2009 Los Angeles Times Book Prize for Science and Technology.The title is based on a comment by physicist Niels Bohr four years before his death that of all the scientists who had visited his institute, Dirac was "the strangest man".

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