by Bryan Deaton
Different fields of science must rely on different instruments of investigation. Particle physics is extremely limited in its methods of observation and experimentation due to the diminutive size of the objects being studied. As a result particle physicists have been forced to rely heavily on theory and mathematical models to represent the subatomic world. It is often difficult to translate mathematical results into real world phenomena, but a great number of breakthroughs in particle physics were accomplished in just this manner. Perhaps the best example of an understanding of subatomic particle being extrapolated out of a mathematical equation is the discovery of the positron from the Dirac equation. Paul Dirac himself suggested that "one should allow oneself to be led in the direction which the mathematics suggests." It takes a great deal of faith to be led by mathematics which produces some seemingly meaningless results, but that faith led Dirac to a correct relativistic electron equation and the prediction of the positron.
Before Dirac invented his equation in 1928 the energy of a particle was modeled by the Shrodinger equation. This equation, however, was only valid for particles at nonrelativistic conditions. In order to correctly model a particle such as an electron at velocities close the speed of light a new equation was needed. Dirac also had another problem with Shrodinger's equation. It did not insure that [rho] would be positive. Dirac insisted that there be a definite positive [rho] in order not to violate the transformation theory. The transformation theory, proposed in 1902 by Rutherford and Soddy, introduced the idea that certain elements can spontaneously transmutate. Today we commonly refer to this phenomenon as radioactivity, but at the time it was introduced it was a radical idea. Dirac could not allow himself to give up the transformation theory so he therefore had to have a definite positive [rho].
Dirac found a simple, yet clever, way to accomplish this. He introduced four coefficients a variation of the Schrodinger equation. These coefficients are particularly interesting because they do not commute, or in other words are anti-commuting. This led Dirac to correctly believe that they must be matrices. Each coefficient is in fact a four by four matrix. The final product of his work was a success and at the same time a mystery. It correctly predicted the spin and magnetic moment of an electron. The unusual results were that apparently an electron could have a positive or negative kinetic energy.
Does a negative kinetic energy have any significance in the real world? If negative electrons states do exist the Pauli exclusion principle requires that all the negative states be filled. It is possible to imagine an infinite "sea" of negative energy electrons. This sea is referred to as the Dirac sea. If a member of this negative energy sea is given enough energy it is possible for it to rise into a positive energy state. A resulting hole would be created in the negative energy sea. This hole, which is truly an absence of negative charge, would appear to be positively charged. The hole, in fact, would appear to be a positively charged particle. At time Dirac invented his equation the electron and proton were assumed to be the only two subatomic particles. It was therefore assumed that this positive particle, produced by an electron leaving a negative energy state, was a proton. There was a problem with this theory however. Several scientists showed how the hole in the Dirac sea would have to have the same mass as an electron. A proton obviously is much more massive that an electron. Is it possible then that there could be another positively charged particle?
At the time Dirac came out with his equation no one wanted to, or had the guts to, suggest that there might be a new particle with a positive charge and the mass of an electron. In 1932 experimental results were found to support a new particle. Carl Anderson was experimenting with cosmic rays. He was using a cloud chamber in a strong magnetic field to be able to determine the charge of particles. While studying his results he observed more particles bending upward, indicating positive charge, than would be expected. Anderson then placed thin pieces of lead into the path of the particles to reduce their kinetic energy, and thus allowing for greater curvature and an accurate measurement of mass. He found that these particles had the same mass as an electron. Anderson called these new particles "positrons."
Dirac was rewarded for his confidence in his theoretical equation of a relativistic electron. He accurately interpreted the meaning of an electron with negative kinetic energy even if he did not predict a new particle. The positron has come to be referred to as an anti-electron because when it comes into contact with an electron they annihilate each other. The positron was the first discovered antimatter. Dirac's electron sea clearly explains the nature of the annihilation of a positron and electron. When an electron comes into contact with a hole in the negative energy sea it spontaneously fills the hole and consequently must release the excess energy in the form of radiation. The discovery of the positron marked a high point in particle physics and won Dirac, along with Schrodinger, a Nobel prize in 1933. The success of Dirac's interpretation of a mathematical equation into an explanation of the real world validates the mathematical methods used as well as the interpretation of its results.
Resources:
Beiser, Arthur. Concepts of Modern Physics 4th ed. McGraw-Hill Book Co., New York, 1987.
Brown, Laurie and Lillian Hoddeson. The Birth of Particle Physics. Cambridge University Press, New York, 1986.
Cropper, William. The Quantum Physicists. Oxford University Press, New York, 1970.
Ohanian, Hans. Modern Physics. Prentice-Hall Inc., New Jersey, 1987.
Pais, Abraham. Inward Bound. Oxford University Press, New York, 1986.
Ryder, Lewis. Elementary Particles and Symmetries. Gordon and Breach Science Publishers, New York, 1975.