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P.A.M. Dirac's Theory of Electrons and Positrons

This is a presentation of Paul Adrien Maurice Dirac's theory of the electron and positron as he described it in his acceptance speech upon receiving the Nobel Prize in Physics in 1933.

Background

In classical (nonrelativistic) mechanics the total energy of a particle is equal to the sum of its kinetic energy K and its potential energy V. The kinetic energy is ½m0v² where m0 is the mass of the particle and v is its velocity. The linear momentum p of the particle is mv and therefore kinetic energy can be expressed in terms of momentum as p²/(2m0). For a particle not tied to a potential field, a so-called free particle, the kinetic energy E is then

K = p²/(2m0)
and hence
K² = p4/(4m0²)

Relativistic Classical Mechanics

In relativistic classical mechanics the mass of a particle depends upon its velocity according to the equation

m = m0/(1−(v/c)²)½

where c is the speed of light. This leads to the formula for total energy E of

E = mc²

Total energy includes rest-mass energy m0c² as well as kinetic energy.

When velocity v is small compared with c

E ≅ m0c² + ½m0

but in general it is not true that kinetic energy is equal or closely approximated by ½m0v².

Note that relativistic momentum p is given by

p = mv
and hence
p² = m²v² = m0²v²/(1−(v/c)²)

Note also that (m0c²)² is equal to m0²c4. Thus

(pc)² + (m0c²)² = m0²c4[(v²/c²)/(1−(v/c)²) + 1]
which reduces to
(pc)² + (m0c²)² = (m0c²)² [(v/c)² + 1 − (v/c)²]/(1 − (v/c)²)
and further to
(pc)² + (m0c²)² = (m0c²)²/[1 − (v/c)²]

On the other hand

E² = (m0c²)²/(1−(v/c)²)

Thus

E² = (pc)² + (m0c²)²
which also may be expressed as
E²/c² = p² + (m0c)²

This is known as the Relativistic Momentum-Energy Equation.

Dirac's Analysis

Dirac starts with the equation

K²/c² − p² − m0²c² = 0

which looks like the Relativistic Momentum-Energy Equation but with kinetic energy replacing total energy, which includes rest-mass energy as well as kinetic energy.

Dirac notes that according to standard procedure the above would lead to a quantum mechanical equation for the wave function ψ of the form

(K²/c² − p² − m0²c²)ψ = 0

with K converted to the operator ih(∂/∂t) where i is the imaginary unit, h is Planck's constant and t is time. Momentum p is converted to the operator −ih(∂/∂x).

Dirac notes that this wave equation will not do because the wave equation must be linear in (∂/∂t). Furthermore for relativistic invariance the wave equation must be linear in (∂/∂x). Dirac proposed a wave equation of the form

(K/c − αrpr − α0m0c)ψ = 0

where αr is a vector of some sort to match the vector of momenta pr and the zeroeth component term (m0c). The term αrpr represents the sum of the products of the components of p and α for r=1 to 3.

Dirac worked by proposing equation and then looked for variables that would satisfy the equation. In this case the α vector turned out to be a vector of matrices. These component matrices have elements which are in turn matrices.

The properties which the αr components had to satisfy are

α² = 1 for μ = 0, 1, 2, 3
and
αμαν = −αναμ
for μ≠ν and μ, ν = 0, 1, 2, 3

Further details of the nature of the α variables are given in Dirac's electron.

Experimentally it was found in the 1920's that an electron has a spin of half a quantum and a magnetic moment of one Bohr magnetron in reverse direction to the spin. Dirac asserts that the αr variables imply the spin and magnetic moment found empirically for an electron.

Dirac also asserts that the αr variables give rise to what Erwin Schrödinger called Zitterbewegung, a trembling motion of electrons at high frequency and low amplitude which involves electrons moving at the speed of light.

(To be continued.)

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