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The Electron Theory of P.A.M. Dirac


Paul Adrien Maurice Dirac

Paul Adrien Maurice Dirac was a physicist of genius but with certain personality eccentricities. Neils Bohr called him the strangest man and that became the title of a biography for him. Bohr also said that Dirac was an Englishman and the Danes had come to expect almost anything from Englishmen. Physicists circulated stories of his strangeness. For example, when someone asked Dirac if he played a musical instrument, he replied, "I do not know; I have never tried." Dirac turned down a British knighthood because he did not want to be called Sir Paul. He signed his name as P.A.M. Dirac and this was the name used for the authorship of his books.

Although the physicists of his acquaintance joked about his peculiarities they also recognized his extraordinary talents. Werner Heisenberg referred to an Englishman that was so smart that it was hopeless to compete with him in theoretical research. Heisenberg was undoubtedly referring to Dirac. Albert Einstein said that in reading Dirac's articles it was hard to tread the narrow path between genius and madness. Wolfgang Pauli described Dirac's form of theorizing as acrobatic.

I myself had an adverse reaction to Dirac's work when I first saw it . In seeing the equations he conjured up my first thought was, "How in the world does he expect there to be any meaningful solutions to such equations?" It was like someone who built strange cages for a zoo and then tried to find creatures to fill them rather than having a collection of creatures and building the cages of a zoo to hold them.

Against all intuition Dirac's approach was the right one. There was no assurance that the explanation of some physical phenomenon would be in terms of the variables that a theorist has in mind. Instead there is some assurances about the form of the equations which would be required. Dirac conjectured about the equations. If a proposed equation implied something contradicted by experiment he discarded it and went on to another equation. An essential part of the analysis was finding out what types of variables would satifiy the equation and this was not easy, but ultimately Dirac found the equations and the types of variables required to satisfy them. This was the exciting element of Dirac's analysis.

The Requirement that the
Equation Conserve Probability

The conservation of probability required that the equation be linear in the time derivative of the wave function. Schrödinger's equation satisfies this requirement.

The Requirement that the Equation be
Compatible with Special Relativity

In the Special Theory of Relativity time is a fourth dimension essentially the same as the three spatial coordinates. But the time dependent Schrödinger equation for quantum mechanics involved the partial derivatives with respect to the spatial coordinates to the second degree but the partial derivative with respect to time only to the first degree.

i(∂ψ/∂t) = H(p²)ψ = H(−∇²)ψ

where i is the square root of negative one, is Planck's constant divided by 2π, p is momentum and ∇² is the Laplacian operator.

In many places in the literature on quantum mechanics, including Dirac analysis which follows below, the units are chosen such that c, the speed of light, is equal to 1. This can be easily achieved by taking the unit of length equal to one light-second. But there is a disadvantage to this ploy because a reader does not know when the mass of a particle appears in an equation whether it represents mass per se or energy mc². Likewise units can be chosen such that is equal to 1, but that also has a disadvantage. The reader would not know whether a frequency ν stands for frequency per se or energy ν. Thus there is a trade-off between the simplification entailed in setting c equal to unity and the obscurring the nature of relationships.

The symbols displayed in red in the following are matrices of larger dimensions than 1×1. (Remember that a vector is a particular kind of matrix.) Time could be taken to be the fourth dimension but there is a certain aesthetic appeal to making it the zeroeth dimension. Thus a point in space-time is identified by the vector Q=(t, x, y, z). There is another vector P whose components are the canonical conjugates of the space-time coordinates. The canonical conjugates to the spatial variables are the corresponding momenta. Two variables are canonical conjugates if their product has the dimensions of action [ML²/T]. The variable which is canonically conjugate to time is energy. Under Special Relativity energy is expressed as

mc²
where
m = m0/(1−(v/c)²)½

where m0 is the rest mass of the particle. Dirac in his development made the zeroeth component energy divided by the speed of light, E/c or, equivalently mc. He did this apparently to keep the units of the components the same. This is unnecessary and introduces an unnecessary complication into the amalysis.

The Energy-momentum Relation in Quatum Mechanics

The energy-momentum equation is

E² = p² + m²

In terms of operators this equation is

−(∂²ψ/∂t²) = (−∇² + m²)
or, equivalently
(∂²ψ/∂t²) = (∇² − m²)

The RHS of this last equation can be respresented as (∇ + m)(∇ − m).

Dirac proposed that the equation to replace the Schrödinger equation is

i(∂ψ/∂t) = (−iΣαj(∂/∂xj)) + βm)ψ

where ψ is a vector of wave functions and the symbol α at this point represents a three dimensional row vector whose components are matrices; i.e.,

α = (α1, α2, α3)

The matrices αj are not further specified at this point.

It is typographically convenient to expand α to include a zeroeth component α0 which replaces β, which is of the same nature as the other components of α. Now α is given by

α = (α0, α1, α2, α3)

If the row vector of operators (m, ∂/∂x1, ∂/∂x2, ∂/∂x3) is denoted as P then the equation that Dirac proposed that to replace the Schrödinger equation takes the simplified form

i(∂ψ/∂t) = Hψ = (αP)ψ

where ψ is a vector of wave functions and the symbol αP represents the dot product of α and P.

In the immediate following the expression of the equations is greatly simplified if the zeroeth component of P, m, is denoted as p0.

The energy squared, E², in relativistic terms is then equal to (p0²+p1²+p2²+p3²)=Σjpj². Hitherto the indices j and k ran from 1 to 3 whereas now they run from 0 to 3.

E² is converted to an operator by replacing E with i(∂/∂t) and then according to Dirac's equation i(∂/∂t) is replaced with αP. Thus

E²ψ = ( i(∂/∂t)[ i(∂ψ/∂t)] = i(∂/∂t) αPψ;
= α·Pi(∂ψ/∂t) = (α·P)²ψ

The expression (α·P)² evaluates to the sum of all the cross products; i.e.,

ΣjΣkpjpkαjαk
for j and k
equal to 0, 1, 2, 3.

If this is to reduce to Σjpj²I, where I is an identity matrix then it must be that

αjαk
is an identity
matrix if j=k.

If j≠k then

αjαk + αkαj

must be equal to
a matrix of zeroes
and hence
αjαk = −αkαj

General Properties of αj
Matrices Like Those Above

Hereafter the convention of displaying matrices in red is generally suspended.

An expression AB+BA for two operators is known as their anticommutator and is denoted as {A, B}. If {A, B} is equal to a matrix of zeroes and B² is equal to an identity matrix then

BAB = −A
because
AB = −BA
and hence
BAB = −B²A = −IA = −A

The trace (sum of principal diagonal elements) for such a matrix A is equal to zero because

tr(BAB) = tr(ABB) = tr(AI) = tr(A)
but tr(BAB) = tr(−A) = −tr(A)
so tr(A) = −tr(A)
and hence
tr(A) = 0

Furthermore if B²=I then the eigenvalues of B are ±1. If tr(B)=0 then the dimension of B must be even; i.e., it must have equal numbers of +1 and −1 eigenvalues.

The number of traceless n×n matrices is n²−1. For n=2 there are only three traceless matrices but four are needed. Therefore n has to be at least 4 since it has to be an even number.

Specific Properties of the α Matrices

Dirac found that the αj can be expressed in terms the Pauli matrices. The Pauli matrices are 2×2 complex element matrices so the α are 2×2 matrices with elements which are 2×2 matrices.

The Pauli matrices are:

σ0 = | 10 |
    | 01 |

σ1 = | 01 |
    | 10 |

σ2 = | 0-i |
    | i 0 |

σ3 = | 1 0 |
    | 0-1 |

Dirac's α matrices are then defined as

α0 = | σ0  0  |
    | 0−σ0 |

and for j=1, 2, 3

αj = | 0σj |
    | σj  0 |

Dirac also defined a set of γ matrices where

γ0 = | σ0  0  |
    | 0−σ0|

and for j=1, 2, 3

γj = |  0  σj |
    | −σj  0 |

Dirac's equation for the electron was engraved upon the memorial where he was buried in Tallahasee, Florida in the following simplified form

Having found the form of the equation and having found the nature of the α matrices the next step is find the solutions for ψ which is a two component vector of complex values. The next step after finding solutions for ψ is finding physical implications of such solutions. This is where the question of electron spin arises.

(To be continued.)

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