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P. A. M. Dirac

"Of course the most famous geniuses of theoretical physics in the twentieth century were Niels Bohr, Albert Einstein, and P. A. M. Dirac. They went far beyond conceiving models or explaining some experimental facts. Einstein revolutionized our concepts of space, time and gravity. Bohr created the most important concepts necessary for dealing with atomic reality. Dirac succeeded in unifying relativity theory with quantum mechanics, leading both to the concept of antimatter and to quantum field theory, a consistent way of dealing with the interaction of matter with electromagnetic and other fields." Victor Weisskopf

"Physical Laws should have mathematical beauty." This statement was Dirac's response to the question of his philosophy of physics, posed to him in Moscow in 1955. He wrote it on a blackboard that is still preserved today.1

Paul Adrien Maurice Dirac (1902-1984), known as P. A. M. Dirac, was the fifteenth Lucasian Professor of Mathematics at Cambridge. He shared the Nobel Prize for Physics in 1933 with Erwin Schrodinger.2 He is considered to be the founder of quantum mechanics, providing the transition from quantum theory. The Cambridge Philosophical Society awarded him the Hopkins Medal in 1930. He was awarded the Royal Medal by the Royal Society of London in 1939 and the James Scott Prize from the Royal Society of Edinburgh. In 1952 the Max Plank Medal came from the Association of German Physical Societies, as well as the Copley Medal from the Royal Society. The Akademie der Wissenschaften in the German Democratic Republic presented him with the Helmholtz Medal in 1964. In 1969 he received the Oppenheimer Prize from the University of Miami. Lastly in 1973, he received the Order of Merit.3

Dirac was well known for his almost anti--social behavior, but he was a member of many scientific organizations throughout the world. Naturally, he was a member of the Royal Society, but he was also a member of the Deutsche Akademie der Naturforsher (Germany) and the Pontifical Academy of Sciences (Vatican). He was a foreign member of Academie des Sciences Morales et Politiques (France) and the Academie des Sciences (France), the Accademia delle Scienze Torino (Italy) and the Accademia Nazionale dei Lincei (Italy) and the National Academy of Science (United States). He was an honorary member and fellow of the Indian Academy of Science, the Chinese Physical Society, the Royal Irish Academy, the Royal Society of Edinburgh, the National Institute of Sciences in India, the American Physical Society, the Tata Institute for Fundamental Research in India, the Royal Danish Academy, and the Hungarian Academy of Sciences. He was a corresponding member of the USSR Academy of Sciences.4 The world wide respect he earned for his work was well deserved.

A prolific writer, Dirac published over two hundred works between 1924 and 1987, mainly papers in physics journals on topics relating to quantum mechanics. His book Principles of Quantum Mechanics , published in 1930, was the first textbook in the discipline and became the standard.5 Some predictions made by Dirac are still untested because his theoretical work was so far reaching, but many other predictions have been verified, assuring him of a special place in the history of physics.6

Dirac was three years old when Einstein published his famous papers on relativity in 1905 and a year old when his predecessor Joseph Larmor began his tenure as Lucasian professor. Physics had just begun its incredible transformation of the twentieth century when Dirac arrived on the scene.

Dirac came to Cambridge as a graduate student in 1923 after graduating from the University of Bristol. As a student in mathematics in St. John's College, he took his Ph.D. in 1926 and was elected in 1927 as a fellow. His appointment as university lecturer came in 1929.7 He assumed the Lucasian professorship following Joseph Larmor in 1932 and retired from it in 1969. Two years later he accepted a position at Florida State University where he lived out his remaining years. The FSU library now carries his name. 8

While at Cambridge, Dirac did not accept many research students. Those who worked with him generally thought he was a good supervisor, but one who did not spend much time with his students. A student needed to be extremely independent to work under Dirac.9 One such student was Dennis Sciama, who later became the supervisor of Stephen Hawking, the current holder of the Lucasian Chair. Dirac's lectures were attended by Sir M. J. Lighthill while he was a student at Cambridge and Lighthill was Dirac's successor to the Lucasian Chair.

Dirac offered the first course in quantum mechanics in Britain, entitled Quantum Theory (Recent Developments) . Among his students was J. R. Oppenheimer, an American, who later on was in charge of the Manhattan Project, which created the first atomic bomb.10

Dirac's work should be understood in the context of the development of quantum physics. The theoretical work had been underway for several years before his entry into the field. It was plagued with difficulties, in part because of the radical change in the way one thought about the world around us, and in part because it was a difficult problem. The important developments of the beginning of this century were carried out by Max Plank, Max Born, Niels Bohr, Albert Einstein, Werner Heisenberg, Erwin Schrodinger, and Wolfgang Pauli. Quantum mechanics was brought to life during the few short years of 1925 through 1927 by most of these men.11

Dirac was the first to apply quantum mechanics to an electromagnetic field, using the method of second quantization. This work contained the basis for quantum field theory,12 which Dirac called quantum electrodynamics.13 The singular delta function was invented by Dirac in order to prove two problems were equivalent. He was working with the problems of "diagnolizing the energy matrix in the Born--Heisenberg-Jordan theory" and "finding the energy eigenvalues of Schrodinger's wave equation."14 The delta function is now used in many different areas of mathematics and physics and is considered basic. In 1926 he derived Balmer-spectrum energy levels of the hydrogen atom. He was the first to derive the Lorentzian shape of spectral lines using quantum mechanics. He introduced the terms bra and ket from the word bracket to denote the use of parts of the bracket. The half brackets were for state vectors and their eigenvalues. One of his major breakthroughs was the use of an algebraic version of quantum mechanics based on Poisson brackets.

Heisenberg had developed equations that were not working exactly the way he wanted. Dirac realized that he had seen a particular expression of the problem earlier. When he applied the brackets, he overcame the difficulties and gained insight into the problem. He saw that the equations had implications that Heisenberg did not acknowledge, thereby holding back the necessary development of the general theory. The problem now appears somewhat simple, with the hindsight of tremendous development and insight over more than half a century.

The problem was that of communitivity in multiplication. It is elementary that two numbers, say 2 and 3, when multiplied equal 6 whether you write 2 times 3 or 3 times 2. The order of multiplication is not relevant, but the order of matrix multiplication, in fact, does make a difference. So one may write the variables x times y or y times x and expect the answer to be xy or yx and still be the same value. The problem however changes when x and y are matrices. This is significant because the values obtained from the resulting matrices have meaning. They are used as dynamical variables in equations.

In the terminology of physics, A and B in this case are represented by the terms p and q. What Heisenberg had done was to show pq - qp = 0 by virtue of a trivial example that did not depend on the communitivity property. The p represents momentum of a moving object, and the q represents position. One would expect that we could calculate the motion of an object by knowing where it was and how fast it was moving. In the everyday world, this is certainly true, but in the world of the atom, it is not true. The problem is that we are unable to know both the position and momentum of something, which, in this particular case, the something is the electron. In fact, the more certain you are of the electron's position, the less certain you are of its momentum, and vice versa.15 This is a basic statement of Heisenberg's Uncertainty Principle. There are some important ideas to understand to fully appreciate this rather nonintuitive concept, but the end result was that Dirac found the way to connect classical mechanics with quantum mechanics by this non-communitivity property.

In the particular equations being used, when you subtracted pq-qp the result was bar i, so that the difference was not zero, but was predictable. This implied a relationship between the position and momentum of the electron, and the relationship was determined by Planck's constant.16 Dirac made the distinction between q-numbers and c-numbers. One set of numbers, the atomic level numbers, do not commute and the other set, that which we see in a laboratory do commute. Dirac used the bracket notion invented by Simeon-Denis Poisson in 1809 to express this. This breakthrough helped set the stage for Werner Heisenberg's most important contribution to physics, the Uncertainty Principle. Dirac found that Schrodinger's wave equation and Heisenberg's matrix equations were special cases of a generalized formulation.17 He founded what is known now as transformational theory.

In 1927 Dirac united quantum mechanics and relativity by deriving a relativistically invariant form of Schrodinger's wave equation, predicting positively charged electrons, which were seen later in 1932. He also predicted that antimatter will annihilate when in contact with matter. He is best recognized for pulling together all of the disparate theories of quantum mechanics. He was able to justify the spin of the electron on a theoretical level which had eluded physicists up to this point. Spin was considered just another number that needed to be included for things to make sense. It was this insight that separated him from others of his time and won him the Nobel Prize in 1933.18

By 1928, Dirac had produced great steps forward along with other physicists of the time. However all together the steps caused confusion, since not all of what was known was consistent and some was contradictory. With characteristic insight, Dirac produced an equation, now called the Dirac equation, with four parts that described the motion of an electron. Many things suddenly fell into place and new "equations followed describing electron collisions, interaction of photons and electrons, and the correct formulas for the spectral lines of the hydrogen atom."19 These equations were not possible until Dirac produced his equation. It was one of those moments when everything that is in confusion suddenly becomes clear. It was so momentous that there was talk of everything being figured out and there would be nothing for physicists to do. Clearly that thought was premature.

The Dirac equation rejected the idea that an electron was a point in space that orbits around the nucleus of the atom. Instead he saw a sort of cloud that orbited that was made up of probable locations of where the electron may be. These probable locations spun around a point of maximum probability appearing as the spin of the electron. Dirac was able to produce accurate values for the strength the magnetic field around the electron, as well as other important characteristics that had eluded everyone up to that time.20

The Dirac equation, as is often the case with a great step forward, produced some questions and new predictions that were unexpected. For example, it was not just a mathematical artifact that second order equations have two roots, one positive and one negative. The reality was that matter was properly represented. The possibility of negative energy appeared, which Dirac initially called holes. He saw a particle the same size as the electron, but with a positive charge instead of a negative one. It was not easy for him to see this, but eventually he did. The antiparticle of the electron, called the positron, was discovered in 1932 in a cosmic ray experiment. The antiproton was found in 1955.21

There are two important sets of statistics that describe the behavior of particles, one is the Bose-Einstein statistics and the other is the Fermi-Dirac statistics. Bose-Einstein statistics relate to particles called bosons, which have integral spin and are the carriers of force. Fermi-Dirac statistics are used for particles that have half-integer spin called fermions. These particles make up matter, that is to say, make up the particles (electrons, protons and neutrons) that are the constituents of atoms. Dirac's work in this area was important enough to have his name attached to these fundamental statistics.

Dirac's life was dedicated to physics with no interests outside of his work, but, besides quantum mechanics, he did work on isotope separation, magnetic monopoles, large-number hypothesis and other physics areas. The large-number hypothesis was based on Dirac's belief that very large constants should not exist in nature. Somehow these large constants that did exist were a consequence of the age of the universe.22 One of the interesting implications of his work that predicted the positron was the prediction of a magnetic monopole. It is common knowledge that a magnet has a north and a south pole, where opposites attract and sameness repels. The idea that a pole could exist in isolation is quite foreign. Although theory predicts its existence, none has ever been found. His work in isotope separation was a step from his theoretical world into the world of experimental physics. He had done some work in the 1930s, but stopped when his colleague, Peter Kapitza, found himself unable to leave the Soviet Union, because Stalin had revoked the necessary exit permit.23

In the 1940s the war effort dragged Dirac back into isotope separation. A group at Oxford was looking for an efficient means to do it. Dirac's method worked, but it was not considered the most cost effective. However, he did continue to contribute to the effort, and even wrote a report on the statistical method of isotope separation that contained concepts still used today.24

Dirac views on religion were very restricted. He seemed to have believed that nothing was as important as his physics. Heisenberg related a story of an exchange between Dirac and Wolfgang Pauli where Dirac expressed his agnostic views. Pauli responded with "Dirac has a new religion. There is no God and Dirac is his prophet."25 Dirac was a member of the Pontifical Academy of Sciences at the Vatican, having written many papers for them. He was not anti-religious. His wife maintained that he was deeply religious, but he has shown no evidence for it.26

Footnotes

  1. R.H. Dalitz, "Paul Adrien Maurice Dirac:8 August 1902-20 October 1984," Biographical Memoirs of Fellows of the Royal Society 32 (1986): 139-185.
  2. Helge Kragh, Dirac: A Scientific Biography (Cambridge: Cambridge University Press, 1990), 115.
  3. Dalitz, 176.
  4. Dalitz, 176.
  5. Dalitz, 167.
  6. John Mauldin, Particles in Nature (Blue Ridge Summit, PA: Tab Books Inc., 1986), 148.
  7. Dictionary of National Biography.
  8. Physics Today 1990.
  9. Dalitz, 156.
  10. Kragh, 30.
  11. Victor Weisskopf, The Joy of Insight (New York: Basic Books, 1991), 25.
  12. Weisskopf, Joy of Insight, 42.
  13. Robert Crease, The Second Creation (New York: Macmillan, 1986), 82.
  14. Kursunoglu 1987 151.
  15. Crease, 79.
  16. Cassidy 1992 206.
  17. Dalitz 1986 164.
  18. Mauldin, 146.
  19. Crease, 84.
  20. Crease, 84.
  21. Tony Hey, The Quantum Universe (Cambridge: Cambridge University Press, 1987), 124.
  22. P. A. M. Dirac,"The cosmological constants," Nature 139 (1937): 323.
  23. Weisskopf, Joy of Insight, 100.
  24. Dalitz, 153.
  25. Crease 1986 81.
  26. Dalitz 1986 163.