Pick a flower on Earth and you move the farthest star. -Paul Dirac

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I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful.

- Albert Einstein1

 

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Paul Adrien Maurice Dirac was born in Bristol, England on August 8th, 1902. The second son of Charles Dirac, a Catholic Swiss citizen of French extraction, and Florence Dirac, née Holten, a British Methodist.

Paul had an older brother, Felix, and a younger sister, Betty.

Dirac's father, Charles, was a French teacher and a staunch disciplinarian; he insisted that young Paul address him only in strictly correct French. Dirac found that he preferred to remain silent. For the rest of his life, he was notoriously shy and notoriously quiet. He would often give one word answers to questions, or no answer at all. 

 

 

 

It was said that this silence was a result of his childhood, when his father would allow him only to speak perfect French at meal times. That may be true, but I suspect he would have been silent even without that. But when he did speak, it was all the more worth hearing.

- Stephen Hawking2

 

Dirac entered the University of Bristol in 1918, and studied electrical engineering. After graduating from Bristol he was unable to find a job. He was accepted by Cambridge but could not afford the tuition, and so he returned to Bristol University, where he studied applied math, for two years.

In 1923 he received a scholarship from Cambridge that allowed him to study there.

Even before receiving his doctorate, Dirac helped to establish the theoretical and mathematical foundations of the new quantum theory. While Dirac preferred to work alone and rarely collaborated on papers, the development of quantum mechanics in the 1920s was a group effort: Heisenberg, Born, Pauli, Jordan, Schrödinger, and a handful of others, were constantly reading each other's papers, trying to make sense of each others's various approaches, and competing against one another to solve the various mathematical, theoretical, and interpretation issues created by the development of the new theory, and competing to get their solutions in print first.

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Description: L-R: Dirac, Landau, Darwin, Leon Rosenkevich (standing), Richardson, Ivanenko, Frenkel, Frank, Debye, Pohl (cut off), on a boat on the Volga, Russia

Date: June 1928  Credit: AIP Emilio Segre Visual Archives, Leon Brillioun Collection

Person(s): Darwin, Charles Galton, Sir; Debye, Peter Josef William; Dirac, Paul Adrien Maurice; Frank, Philipp; Landau, Lev Davidovich; Pohl, Robert Wichard; Richardson, Owen Willams; Ivanenko, Dmitrii Dmitrievich

 

 

Dirac was the first to recognize the relationship between the poisson bracket of classical Hamiltonian mechanics and the anti-commutation relations of operators in quantum mechanics.

The exact nature of this relationship is still an active area in research (now known as ''deformation quantization'').

In 1925, Dirac's older brother Felix, killed himself. Dirac's father Charles was devastated, never fully recovering from the blow.

Dirac parents did not get along. HIs mother spoke only English and his father only French. His father treated his mother like a nurse and a maid, and secretly carried on affairs. Dirac hated going back to his parents house.

In 1927, Dirac began to try to develop a version of quantum mechanics that was consistent with Einstein's special theory of relativity.

When Dirac mentioned to Bohr that he was working on this problem, Bohr responded that Oscar Klein had already solved it. In fact, the equation now known as the Klein-Gordon equation had originally been derived by Schrödinger--who rejected it when it gave predictions inconsistent with experiment--before he derived what came to be known as ''Schrödinger's equation.'' Schrödinger had found an equation that gave the right results for non-relativistic quantum particles, but, because it was linear in time derivative, but quadratic in the space derivative, and hence treated space and time in fundamentally different ways, was not consistent with Einstein's special theory of relativity, in which space and time are related by Lorentz ''boosts'' which mix time and space in a certain algebraically well-defined manner. Any such attempt to mix time and space would turn the Schrödinger equation into gibberish.

The Klein-Gordon equation was quadratic in both time and space derivatives, and so was invariant under Lorentz boosts, but it seemed to violate the tenets of quantum mechanics by leading to negative probability densities. Furthermore, it could not account for electron spin.

Schrödinger had obtained his equation by means of an educated guess: in classical mechanics, for a particle with no potential energy, the total energy E, is related to its mass m, and momentum p, through the equation:

 

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Schrödinger teated energy and momentum as operators, with the correspondence (in natural units, where Plank's constant  = 1):

 

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Substituting these relations into the energy equation:

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gives the (time-dependent) Schrödinger equation (with zero potential energy):

 

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In special relativity, the energy and momentum of a particle (again with zero potential energy) together form a four-vector with length equal to the particle mass. In natural units, with c = the speed of light =1, this gives:

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Using Schrödinger's educated guess about the operators that correspond to energy and momentum in quantum mechanics leads to what is now called the Klein-Gordon equation:

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The Klein-Gordon equation is consistent with relativity in a way that Schrödinger's equation is not, but because it is not linear in the time derivative, it gives can give quantum mechanical results that make no sense.

Dirac therefore sought a relativistic equation for the electron that was linear in the time derivative.

Consider the equation:

 

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Where i is the square root of minus one, the the "gammas" are some set of not necessarily commutative operators, and "psi" is a wave function.

As an operator equation this says:

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If you square both sides of this equation you obtain:

 

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This will satisfy the Klein-Gordon equation if:

 

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In other words, if the set of gamma operators satisfy the commutation relations:

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where alpha  and beta range from zero to three, and eta is the Minkowski metric, i.e.:

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This turns out to be the defining relation for a Clifford algebra. Dirac had learned about Hamilton's quaternions during his mathematical studies. Quaternions are one, particularly simple example of a Clifford algebra, (in fact they are simple in a strict mathematical sense) but he did not seem to have known about the general theory of Clifford algebras. He essentially rediscovered it.

If the set of gamma operators  do indeed satisfy the above commutation relations, then the equation:

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is a relativistic equation of the electron that is consistent with quantum mechanics. It is the Dirac equation, discovered and published in 1928.

Solutions of the Dirac equation split into two parts: a spin up part, and a spin down part. This is exactly what you want for a spin one-half particle, such as an electron--the only spin one-half particle known at the time--and is not true, in general, for a solution to the Klein-Gordon equation.

 

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So Dirac had found an equation that was both (special) relativistic, and quantum mechanical, and which accounted for electron spin.

The solution to the Dirac equation is the periodic table. That is, if you ask the Dirac equation what energy levels electrons can have, the answer it gives you correspond to the energy levels of electrons in the various shells of atoms in the periodic table. This was a major triumph, one of the greatest in the history of science. There was, however, a problem. While the Dirac equation only gave positive probabilities, it also gave positive probabilities to states where electrons had negative energy.

In classical mechanics this is not a problem-stars, for example, have negative total energy when considered classically--but in quantum mechanics this is a problem. An electron can jump to a lower energy level, and if negative energy levels are allowed, an electron can keep jumping down the ''energy well'' forever. If this were possible, the entire universe would vanish in a great explosion of light.

Dirac attempted to resolve this paradox by offering a novel interpretation. In 1927, he had sought to develop a theory of multiple-particle (i.e. multiple electron and photon) quantum interactions--a predecessor to modern quantum field theory--by considering the quantum vacuum to be full of an infinity of unobservable photons. In this view, when an electron appears to emit a photon, that photon, which had previously been in an unobservable state, moves into an observable state and so seems to ''appear'' out of nowhere. Similarly, when an electron appears to absorb a photon, a photon that is observable shifts into an unobservable state.

 

The light-quantum has the peculiarity that it apparently ceases to exist when it is in one of its stationary states, namely, the zero state, in which its momentum, and therefore also its energy, are zero. When a light-quantum is absorbed it can be considered to jump into this zero state, and when one is emitted it can be considered to jump from the zero state to one in which it is physically in evidence, so that it appears to have been created. Since there is no limit to the number of light-quanta that may be created in this way, we must suppose that there are and infinite number of light-quanta in the zero state3. . . 

 

Dirac now proposed something similar for negative energy electrons. Because electrons obey the Pauli exclusion principle, no two electrons can be in the same state--or put differently--no more than two electrons (one spin up and one spin down) can be at the same energy level.

Dirac therefore proposed that there is an unobservable negative energy electron sea that is already filled. Positive energy electrons cannot jump to negative energy states, because all those energy states are already filled. Occasionally, however, a negative energy state might ''open up.'' Dirac argued that this hole in the negative-energy electron sea would look like particle with the opposite charge as the electron.

Dirac originally hoped that these ''holes'' in the negative-energy electron sea might be protons--electrons and protons and photons were the only known ''fundamental'' particles known at the time (it is now known that the proton is not fundamental). But Pauli and Hermann Weyl both demonstrated that Dirac's holes would have to have the same mass as the electron, whereas protons were known to be much more massive.

Dirac was nervous about this result, and cautious about actually predicting a new, as-yet-unseen particle. But in fact, that's what his equation did. In particular, it predicted the existence of anti-matter.

In 1932, Carl Anderson at Caltech discovered the positron, a fundamental particle with the same mass and spin states as the electron, but with opposite electric charge. Dirac's theory had been vindicated. In 1933, Dirac and Erwin Schrödinger were jointly awarded the Nobel Prize in physics. (Anderson was awarded the Nobel Prize in 1936.)

The theory of the negative-energy electron sea, however, has now been replaced by the more general notion of the quantum field. From the perspective of field theory, both the electron and the positron are merely specific excitations of the quantum field. The negative-energy electron sea has been replaced by the (perhaps equally extravagant) infinite sea of ''virtual'' particles.

While in Stockholm to receive the Nobel Prize, a journalist asked Dirac whether his theory had any practical ramifications, he answered:

''My work has no practical significance1.''

That may have been true in 1932, but today anti-matter is utilized for medical technology in the form of PET scans (the ''P'' stands for positron), it plays an important part in observational astronomy, may hold the key to propulsion for interstellar travel, and is related to one of the most fundamental mysteries in the standard model of cosmology, namely: ''why is there more matter in the observable universe than anti-matter?''

When a particle of matter meets a particle of anti-matter, the two particles annihilate each other, leaving only photons. (This is exactly what would have happened in Dirac's old infinite negative-energy electron sea model, when an electron filled a ''hole.'')

The current theory is that there was a slight over-density of matter over anti-matter in the primordial synthesis of particles following the big bang, amounting to about one part in ten thousand. The matter and anti-matter then mutually annihilated, leaving us with the matter-dominated universe we see today, and the light which is now the 2.73K cosmic background microwave radiation.

But why the over-density of matter over anti-matter? And is this a generic feature of the entire universe, or only of the part that we can observe?

Dirac spent the 1934-1935 on sabbatical in Princeton. There he mat Manci Wigner, sister of the physicist Eugene Wigner. The two became friendly and continued to meet and exchange letters after Dirac returned to Cambridge. In 1937 they were married. This was Dirac's first serious relationship, but Manci's second marriage. She had two children from her previous marriage, and she and Dirac eventually had two more.

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Dirac was elected a Fellow of the Royal Society in 1930, and in 1932, was made Lucasian Professor of Mathematics at Cambridge, a post once occupied by Newton, and today occupied by Stephen Hawking.

Dirac held this position until 1969, when he retired from Cambridge and moved to the United States. He spent the rest of his life in Florida, at the University of Miami, and at Florida State University in Tallahassee.

Dirac died on October 20th 1984, in Tallahassee Florida.

 

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Dirac Heisenberg

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Dirac & Feynman

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Quotes:

The steady progress of physics requires for its theoretical formulation a mathematics that gets continually more advanced. That is only natural and to be expected. What, however, was not expected by the scientific workers of the last century was the particular form that the line of advancement of the mathematics would take, namely, it was expected that the mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundations and gets more abstract.4

 

The physicist, in his study of natural phenomena, has two methods of making progress: (1) the method of experiment and observation, and (2) the method of mathematical reasoning. The former is just the collection of selected data; the latter enables one to infer results about experiments that have not been performed. There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with reasonable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature's scheme.5

As time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.

In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite.

 

 

[1] Farmelo, Graham (2009). The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom. Basic Books, New York.

[2] Goddard, Peter, et al, (1998) Paul Dirac, The Man and His Work. Cambridge University Press, Cambridge.

[3] Dirac, P. A. M. (1927) The Quantum Theory of Emission and Absorption of Radiation. Reprinted in [6].

[4] Dirac, P.A.M., (1931). Quantised Singularities of the Electromagnetic Field. Reprinted in [6].

[5] Dirac, P.A.M., (1939). The Relation between Mathematics and Physics.

[6] Dirac, P. A. M., edited by R. H. Dalitz (1995 The collected Works of P.A.M. Dirac 1924-1948. Cambridge University Press, Cambridge.

[7] Gamow, George (1966). Thirty Years That Shook Physics. Dover, New York.

[8] http://www.dirac.ch/PaulDirac.html

[9] http://biogrph.wordpress.com/article/paul-dirac-1x4qvbqoz9orn-47/

[10] http://cerncourier.com/cws/article/cern/28693

[11] http://www.physics.fsu.edu/awards/nobel/dirac/RememberingDirac

Additional Info

  • Birthday Date: Friday, 08 August 2014
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