Paul Dirac and the Positron

A Triumph of Mathematics in the First Anti-Particle

Mar 3, 2008 Isaac M. McPhee

In the late 1920's Paul Dirac it was discovered that "normal" electrons might not be the only ones to exist. There might be an "opposite" particle as well.

In 1928, Paul Dirac (another of the founding fathers of quantum mechanics in the early decades of the twentieth century) made an important use of the “totalitarian principle” which says that in quantum mechanics, anything which is not forbidden is compulsory. This breakthrough occurred when he began attempting to combine the idea of special relativity with quantum mechanics, eventually coming up with the “Dirac Equation,” a crucially important component in any explanation of quantum theory.

Dirac’s Discovery of Negative Energy

Dirac, when examining the equation for mass/energy equivalence in Special Relativity, noticed something about the version of this equation which was used to determine mass-energy equivalence in the case where momentum must be included in the equation (the equation to determine the Relativistic Energy/Momentum relationship): E = m²c^4 + p²c²

Dirac noticed that a fundamental feature of this equation (as in all binomial equations) is that it could be solved in two identical ways (just as the quadratic equation most people learned in high school leaves one with two solutions) – with the value of E being either a positive or a negative number, in which case both results should be valid.

In terms of Einstein’s equation, these negative answers were generally disregarded, set aside and assumed to have no bearing whatsoever on reality. Where most physicists simply accepted the positive value as being the true answer (for what could possibly be meant by an equation which resulted in an answer which seemed to represent negative amounts of energy?), Dirac took a chance and decided to trust the mathematics over the experimental data thus far, toying with the equation using negative answers and the possible meanings therein.

What he found was strange. According to the equation Dirac came up with, within an atom there must be both positive energy levels (occupied by normal electrons) and negative energy levels. In essence, what he found was that all of the empty space within and in between atoms was not really empty space at all. It was merely a sea of an infinite number of electrons existing at ‘negative’ energy levels.

Negative Energy Applied to Atomic Models

According to Bohr’s model of the atom, electrons naturally fall into the lowest possible energy state, which meant that, according to the totalitarian principle, electrons should technically be able to fall into these negative energy states which lay in between the lowest energy level and the nucleus itself (for such a thing would be forbidden by no known law) – unless these energy levels were already occupied. So the only explanation for the observed behavior of atoms is this “sea” of negative energy, which would thus prevent positive energy electrons from falling any further.

But Dirac didn’t stop there.

He then proposed that if one of this negative energy was potentially given enough energy (it would have to be given at least 2mc of energy, according to Dirac’s equation for relativistic quanta), it would jump from being purely negative energy to a positive energy electron, leaving behind an electron-sized ‘hole’ in the negative energy field. Dirac proposed that this “hole” would to any outside observer look just like a normal electron, only exactly carrying exactly the opposite charge. It would have the same mass as an electron, and the same properties, except that it would be positively charged. The “hole” would then move just as a normal electron, zipping about as other negative electrons moved to fill in the gaps, shifting the hole slightly each time.

The Positron

Thus, this hypothetical particle became known as the “positron” – a positive electron

To many this must have seen like mere mathematical “conjecture” with no bearing on actual fact. After all, there was no experimental data to back this up at the time. All Dirac had to work with was his mathematical expertise and the totalitarian principle, which implied that this thing which worked so well on paper simply had to exist in real life, unless there was an as-of-yet undiscovered rule preventing it.

Discovery of the Positron

Dirac and his equations were finally justified four years after his original proposal, when the first traces of real life positrons were found by experimenter Carl Anderson in 1932 while studying experiments performed using cosmic rays. Both Dirac and Anderson eventually won Nobel prizes for their discoveries. (Actually, positrons had been spotted prior to this point, but experimenters, lacking Dirac’s theory, had not known what to look for, and had missed them).

Consider it justification for both the mathematics of quantum mechanics and the totalitarian principle. Dirac has been quoted as saying that, “it is more important to have beauty in one’s equations than to have them fit experiment.”

Controversial, maybe, but well said.

For further information on anti-matter, click here.

References:

Glashow, S. L. (1988). Interactions - A Journey Through the Mind of a Particle Physicist and the Matter of this World. New york: Warner Books.

Gribbin, J. (1994). In Search of Schrodinger's Cat: Quantum Physics and Reality. New York, NY: Bantam Books.

Isaacs, A. (2003). Dictionary of Physics. London: Grange Books.

Kl-Khalili, J. (2003). Quantum: A Guide for the Perplexed. New York, NY: Weidenfield & Nicolson.

The copyright of the article Paul Dirac and the Positron in Physics is owned by Isaac M. McPhee. Permission to republish Paul Dirac and the Positron in print or online must be granted by the author in writing.

Cloud Chamber Photograph of a Positron