What is Antimatter?
Dirac predicted it, nobody understands it.
In my previous article
I showed how and why Dirac arrived at his beautiful formula. But soon after its publication on Feb. 1, 1928, some of his colleagues pointed out that two of the four possible solutions contained negative energies.
Usually, this is no problem. When you solve any wave equation, two solutions emerge. The retarded wave describes a wave that starts here and now, propagating outward and into the future. The other solution, the advanced wave, describes a wave coming from the future and from outer space, arriving here and now. This, of course, is nonsense, unless you believe in time travel, which Feynman and Wheeler did, see here:
Physicists disregard the second solution; nature does not have to adapt to mathematics, it’s the other way round. Nature is master, mathematics its slave. At least its servant.
But in the case of Dirac’s exiting equation, this was not possible. The second pair of solutions, the ones with negative energies, couldn’t be ignored. Electrons could jump from positive to negative energies without restrictions. Electrons would fall into increasingly lower energy states, thereby contradicting reality, where matter is mostly stable. Dirac’s equation, despite its obvious beauty, seems to be false. Then what?
Let’s go back six years. In 1922, Russian physicist Alexander Friedmann presented a solution to Einstein’s field equations of gravity and electromagnetism, better known as the fundamental formula of General Relativity. The universe that emerged from Friedmann’s calculations was expanding. This fact contradicted everything that was known at the time. According to scientific consensus, the cosmos was static, unmovable, eternal.
When Einstein heard of Friedmann’s solution, he hastily added a term to his equation, the cosmological constant Λ (“Lambda”), a kind of pressure counteracting the universe’s expansion. Seven years later, Edwin Hubble published the result of his observations: Nearly all galaxies were receding from our position. Surely, the Universe expands, just as Einstein’s formula predicted. Einstein hastily withdrew his additional term, later calling its introduction “the biggest blunder of my life”. He should have trusted his own invention!
And what did Dirac do in such a similar situation? His equation was correct because it was beautiful. Nature had to adapt to mathematics. Although he knew and accepted that
The mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature.
he added:
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. … The two subjects will ultimately unify, every branch of pure mathematics then having its physical application, its importance in physics being proportional to its interest in mathematics.
So he invented the sea of negative energies, named after him. In this sea, all negative levels are already filled with particles. Sometimes, there is a hole in this sea that behaves like a particle with opposite charge. Therefore, this sea is filled with positive particles, which were identified by Dirac as protons.
Soon, Heisenberg, Oppenheimer, and Dirac himself realized that the holes couldn’t be filled with protons. It had to be particles with the same mass as electrons, called “antielectrons” by Dirac. Dirac was really lucky, because these particles, now called positrons, were actually discovered in 1933 by Carl Anderson in cosmic radiation, and later produced in accelerators. Dirac was hailed as a genius, on an equal footing with Newton, Maxwell, and Einstein.
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But what of the negative energies? The only possibility to make the kinetic energy T = mv²/2 or the rest energy E = mc² negative consists in making mass m negative. Inserting this negative mass in Newton’s equations, we get:
- The acceleration of a negative mass, F = -ma, would push an object to the pusher; the force would act in the opposite direction of the acceleration, something absurd and never observed.
- The gravitational effect of the Earth (m1) on a particle with negative mass (m2), according to F ~ m1*m2/r², would result in a repulsion of antimatter in Earth’s gravitational field. Observations on anti-particles and anti-atoms revealed: antimatter behaves the same as matter, it is always attracted. Then what’s the secret of Dirac’s negative energy?
It’s no secret: They still don’t know.
There are a lot of explanations outside of quantum physics (as it was known at Dirac’s time). Here is one from the physics forum discussions:
For quantum fields, it’s no problem to write a creation operator in front of the negative-frequency modes and flip the momentum in the corresponding integral over momenta. This means that you have now reinterpreted the negative-frequency modes as one-particle states with positive energy, and the Hamiltonian gets bounded from below (after normal ordering!), provided you quantize half-integer-spin particles as fermions and integer-spin particles as bosons.
Is it really that simple? What about this one:
When quantizing the field, you can write for an annihilation and for a creation operator. In order to make the solution easier to handle concerning Lorentz covariance you flip the sign in the negative-frequency modes. Doing the analysis with the Poincare generators (Noether’s theorem leading to the expressions for energy, momentum, and angular momentum) and the operators for the U(1) global gauge symmetry (invariance under multiplication with a phase factor), leading to a conserved current leads to the conclusion that the KG field must be quantized as bosons in order to get a positive Hamiltonian (after normal ordering to get rid of all irrelevant divergences). They annihilate two different kinds of particles, which are however closely related, because they both are necessary to create a theory, where the field operator behaves under Poincare transformations as the corresponding classical field, i.e., in a local way.
Indeed. And what had the inventor of negative energy to say? In a lecture in 1941, he declared:
Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative sum of money.
“Mathematically” maybe. But what about physics? There’s still some research to do!
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