Dirac Equation

The Dirac Equation was developed by Paul Dirac in 1928 and was the first theory to combine both the principles of Quantum Mechanics and the theory of Special Relativity.

Essentially it describes the properties of a relativistically-moving electron. In so doing, it also provides an explanation for the origin of Spin and predicted the fine details of the Hydrogen Spectrum.

The equation has the following form:

\left ( \beta mc^2 + \sum_{k = 1}^3 \alpha_k p_k \, c\right) \psi  = i \hbar \frac{\partial\psi}{\partial t}

Understanding the consequences of the Dirac Equation is not easy, but for our purposes, the key predictions are:

  1. Electrons with both positive and negative charges
  2. Positive and negative energy states for each of the above

The first of those predicitons leads to the concept of Anti-Matter as well as Matter and is universally accepted throughout mainstream physics.

The second prediction was interpreted by Dirac via a sea of negative energy which came to be known as the Dirac Sea. This treatment effectively forbids electrons from occupying the negative energy states, by saying that they are already occupied.

The alternative approach adopted by many physicists is to simply ignore the negative energy states and treat them as non-physical, despite the fact that they are mathematically valid predictions of the Dirac equation.

We will take a different approach and accept all 4 solutions of the Dirac Equation at face value (as advocated by Donald Hotson).

Rather than getting tied up in arguments about whether positive energy electrons would emit photons and spontaneously decay into the negative energy states, we will move away from the wave description altogether. Instead, we will use the 4 solutions as a starting point for building a particle-based theory. We prefer to just see where the mathematics will lead, rather than getting hung up on arguments about whether this makes any sense given the traditional understanding of mass and energy.

In most cases, we will not need the relativistic Dirac Equation and can analyse low-speed situations with the traditional laws of Electromagnetism.

In so doing, we accept that we're ignoring the other predictions of the Dirac Equation (such as the hydrogen spectrum) for the time being.

Experimental Confusion

If Dirac was right and there are 4 kinds of electron then why haven't we observed them in experiments?

The answer to this may be that we have already observed them in various experiments, but with current technology we are unable to tell them apart.

Cloud and Bubble Chambers

Cloud Chambers were a standard pieces of apparatus for observing sub-atomic particles during the 1920s.

More, recently Bubble Chambers are typically used for the same purpose.

Charged particles have an ionising effect on the gas or liquid in the chamber, leaving either a vapour trail or a sequence of bubbles respectively. The curvature of the trail is caused by an external electromagnetic field and depends on the ratio of charge to mass.

For the 4 electrons predicted by the unconstrained Dirac Equation, the ratio of mass to charge is the same, differing only in sign.

All 4 particles are charged and will therefore leave a trail.

A positive charge with a negative mass would behave in the same way as a negative charge with a postive mass, at least in response to the electromagnetic field. We would call both of them Electrons

Likewise a positive charge with a positive mass would behave the same as a negative charge with a negative mass. We would call both of them Positrons.

Therefore we would only see 2 types (either electrons or positrons) in a cloud chamber and indeed this is what is observed.

The fact that the individual electrons taking a particular path may have different charges (according to our model) would not be noticed in the cloud chamber experiment. Likewise for the positrons.

Neutral Particles

If the electrons and positrons pair up and have opposite responses to an external eletromagnetic field, we would expect them to take a straight line in a cloud chamber, assuming the field is not strong enough to rip them apart. This is the characteristic of a neutral particle. Note that the composite pair has no overall charge and so would not leave a trail (unless the individual constituents were widely-spaced, in which case they would be seen as 2 separate charged particles).

Double Particles

In contrast, we expect doubly-charged pairs to take the same path through a cloud chamber as either a single electron or positron respectively because the ratio of charge to mass is the same in each case. In this case, even a strong external field will not rip them apart because each constituent has the same response. So again these might be confused with electrons or positrons.

Reactions

Clearly electrons and positrons have different behaviour when they interact with other particles.

The trouble is that we always identify the incoming particles based on the reaction products, without checking what route they would have taken in a cloud chamber.

Uncertainty

In each case, we are leaving the door open to misinterpretation, because we cannot check what an individual electron or positron would do in both tests.

This is known as the Uncertainty Principle and therein could lie an explanation for why we haven't observed (or think we haven't observed) the 4 kinds of electron to date.

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