Let's look more closely at the Dirac equation's problem of negative energy solutions. We'll
use normal units where the speed of light appears explicitly.
The positive energy solutions have a minimum for momentum p equals zero.
So the lowest positive energy level is E-1 equals m c squared.
Higher energy levels correspond to higher momentum, the electron moving faster and faster.
And, this continues without limit towards E equals infinity.
Suppose an electron has energy E-2. It can drop to energy E-1 with the emission
of a photon. If this is the lowest possible energy, the ground state, then the electron
would remain their indefinitely. The negative-energy solutions mirror the positive
ones. The highest, negative energy is minus E-1,
corresponding to zero momentum, an electron at rest.
Lower energy states correspond to higher momentum, the electron moving faster and faster.
And, this continues without limit towards E equals minus infinity.
The existence of negative energy states means that an electron with energy E-1
can transition to energy minus E-1 with the emission of a photon.
from there it can continue falling to ever lower energy
states and emitting photons without end. So, it seems, every electron in existence would
fall down this negative energy ladder, emitting an infinite amount of radiation in the process.
Thankfully this doesn't happen. But it makes the Dirac equation as it stands unviable.
In 1929 Dirac submitted a work around to this problem in a paper titled A Theory of Electrons
and Protons. In the section subtitled Solution of the Negative
Energy Difficulty after noting the problem
He offered a solution: The Pauli exclusion principle, however, will come into play and
prevent more than one electron going into any one state.
Let us assume there are so many electrons in the world that all the states of negative
energy are occupied except perhaps a few of small velocity.
We shall have an infinite number of electrons in negative-energy states, and indeed an infinite
number per unit volume all over the world, but if their distribution is exactly uniform
we should expect them to be completely unobservable. Only the small departures from exact uniformity,
brought about by some of the negative-energy states being unoccupied, can we hope to observe.
Dirac went on to suggest that these unoccupied negative-energy states might actually be protons.
So, Dirac says, yes, there are an infinite number of negative energy electron states.
But they're all full. And the infinite number of electrons that fill them are invisible.
They have no observable effect. It's hard to overstate how woo woo sounding
this idea is. As far as the negative-energy states go it's
invisible electrons all the way down. So, within any milliliter sized region of space
there are an infinite number of electrons with velocities ranging from zero to arbitrarily
close to the speed of light. Yet we can't see them or otherwise observe any effect due
to their presence. This explanation became known as the Dirac
Sea. Needless to say, it was controversial and was the target of serious criticism.
In the opinion of Werner Heisenberg, quote, The saddest chapter of modern physics is and
remains the Dirac theory. I regard it as learned trash which no one can take seriously.
Here's the thing about science though. Ultimately people's opinions don't matter. It doesn't
matter if an idea seems ridiculous and laughable. There is only one scientific authority and
that's Nature. What matters is if an idea leads to testable predictions and if those
predictions can be verified. Dirac supposed that just as negative energy
electrons could emit photons they could also absorb them.
Therefore a photon of energy 2 m c squared could cause an electron to transition from
the highest, negative energy state to the lowest, positive energy state.
Half of that energy, m c squared would correspond to the mass m of the electron.
The other m c squared would also correspond to a mass m.
Moreover, taking a negative charge away from nothing would leave a positive charge.
Dirac called this empty negative-energy state a hole and predicted it would behave like
a particle with the electron's mass and opposite charge.
Here's a cartoon representation of this prediction. The electrons in the box represent the invisible
Dirac Sea. On the left a photon excites one of the Dirac
Sea electrons into a positive energy state wherein it becomes visible. This leaves a
hole in the Dirac Sea. The energy of the system has increased by
2 m c squared. The electric charge has not changed because we've merely changed the state
of a pre-existing electron. Dirac's claim, shown at right, is that this
will appear the same as if we created two particles with positive energy, an electron
and a positively charged particle with the same mass as the electron, which, for now,
we'll call a positive electron. The energy of this system has also increased by 2 m c
squared and the charge is unchanged since we created both negative and positive charges
which cancel out. At left, suppose the electron above the hole
moves downward. In effect the hole as moved upward.
At right, this would be equivalent to the positive electron moving upward.
As this continues what we would observe is a positive electron
moving through space. In 1931 Dirac submitted his prediction for
publication. He claimed, A hole, if there were one, would
be a new kind of particle, unknown to experimental physics, having the same mass and opposite
charge to an electron. We may call such a particle an anti-electron. And there you have
the prediction of anti-matter. There was no experimental evidence for the existence of
such a bizarre thing. This was motivated solely, one might say, by Dirac wanting to fix a seeming
shortcoming of his equation. He continued. We should not expect to find
any of them in nature, on account of their rapid rate of recombination with electrons,
but if they could be produced experimentally in high vacuum they would be quite stable
and amenable to observation. So, according to Dirac, it should be possible
for the energy of a photon to be transformed into an electron, anti-electron
pair
The electron and anti-electron can be distinguished by applying a magnetic field which causes
moving charges of different signs to curve in opposite directions.
In 1932 Carl Anderson was using a so-called cloud chamber to observe cosmic rays. A charged
particle traveling through a cloud chamber ionizes molecules along its path. Those ions
then cause a supersaturated vapor in the chamber to condense into a visible cloud of droplets.
The result is to make the particle's track through the chamber visible.
Anderson observed an event of the type predicted by Dirac, confirming the existence of the
anti-electron, what we now call a positron. Here's a later result that shows the process
more clearly. At the point marked by the arrow we see two particles diverge.
They curve in different directions in the magnetic field allowing us to identify the
upper particle as a positron and the lower particle as an electron.
At the time Anderson published his result people were using different names for this
particle including the anti-electron, the positive electron and simply the positive.
Anderson suggested the name positron, which has stuck. He also suggested that by symmetry
the electron be called the negatron, a name that didn't stick.
So, in spite of the bizarre, woo woo nature of the Dirac Sea concept, Dirac and his equation
were vindicated. The main weakness of his theory became one of it's greatest strengths.
Still, there's something strange about this Dirac Sea idea.
In the cartoon at left we need the Dirac Sea as a place to first pull an electron out of,
and second as something for the hole to exist and move in. This is what the Dirac equation
predicts. In the cartoon at right, there's no need for
a Dirac Sea. We simply create an electron-positron pair. This is what is experimentally observed.
Isn't this a more elegant explanation? It doesn't require space to be filled with an
infinite number of invisible, negative-energy electrons. So why can't the theory predict
that directly? The problem is with quantum mechanics itself.
First, we started with the assumption that psi is the wavefunction of a single electron.
There's nowhere for other particles to fit in the theory.
Second, in both the Schr�dinger and Dirac equations the probability at any time that
the electron is somewhere in space is always 100 percent. Therefore, projecting this back
in time it can never have been created and projecting it into the future it can never
be destroyed. It follows that if an electron appears, it
must have actually already existed and merely transitioned from an invisible state to a
visible state. And, if it disappears it must have merely
transitioned to an invisible state in which it still exists. The Dirac Sea provides the
required invisible state. And it provides a way to account for the properties of a positron,
as a hole in the Dirac Sea, in a theory that only describes electrons.
What in time became clear is that, to quote Steven Weinberg, relativistic wave mechanics,
in the sense of a relativistic quantum theory of a fixed number of particles, is an impossibility.
Another theoretical framework was needed in which to express these ideas.
Eventually quantum electrodynamics, the first quantum field theory, was able to recast the
Dirac equation into a formalism that gives a rigorous and coherent description of electrons,
positrons, photons and their interactions, without negative energies and without the
Dirac Sea.