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The Klein-Gordon Equation

Differentiating the wave function twice, and using the mass shell condition,
This is the Klein-Gordon equation». It is a vector identity and constrains time evolution but it is not a first order equation as is required of an equation of motion by Stone’s theorem.

Spin

Differentiating the wave function with respect to time gives the equation of motion,
In terms of the wave function, the equation of motion is
This is not covariant. A covariant equation should reduce to this equation in the rest frame, when time is proper time and momentum is p = (m, 0, 0, 0). This requires a scalar product involving i∂_a and the wave function. It has the form, for some vector operator Γ,
There is no invariant equation in this form for scalar f and the theory breaks down. To rectify the problem, Dirac added an index, μ, to the wave function,


I will use Greek subscripts for spin, and the summation convention for repeated spin indices. I will only use orthogonal spin states, for which it is not necessary to raise and lower indices. The braket becomes

The Dirac Equation

In 1928, Dirac factorised the Klein-Gordon Equation,
and found the the Dirac equation»,
where γ^a are a set of four 4 × 4 matrices known as the Dirac γ matrices obeying the defining condition:


Proof:  By Clairaut’s theorem, ∂_a∂_b = ∂_b∂_a. So,
by interchanging a and b and summing. The anticommutation relation, { γ^a, γ^b } ≡ γ^aγ^b + γ^bγ^a  = 2g^ab is necessary and sufficient that this is the Klein-Gordon equation. Dirac also required that the Dirac equation can be written in the form of the Schrödinger equation, i.e.
where H is Hermitian. Rewriting the Dirac equation, and using bold font for 3-vectors,
The anticommutation relation gives us γ⁰γ⁰ = 1, so that
We require that γ^aγ⁰ and γ⁰ are Hermitian. Then,
Postmultiplying by γ⁰ gives the conjugacy relation,

The smallest matrices which can be found satisfying the defining relations are 4 × 4. In a fundamental theorem due to Pauli (omitted), it can be shown that any two different sets of matrices satisfying the defining relations (anticommutation and conjugacy) are unitarily equivalent - in other words they can be transformed into each other by a change of spin basis. I will use the Dirac representation:
Using the 2 × 2 identity matrix and the Pauli spin matrices», σ¹, σ², σ³,
the Dirac γ matrices can be written more succinctly,

Covariance of the Dirac Equation

The Dirac gamma matrices are not a vector, and do not define a direction in space, but they have properties under Lorentz transform which enable them to be treated as vectors. Specifically, if ψ is a Dirac spinor, then γ^aψ transforms as a vector multiplied by a Dirac spinor. This is because, under a Lorentz transform, , the transformed gamma matrices also obey the defining relation,
It follows from the fundamental theorem that the transformed gamma matrices are simply another representation of the gamma matrices, and hence that the Dirac equation is covariant.

Dirac Particles

The positive energy solutions to the Dirac equation are
where p satisfies the mass shell condition and u is a Dirac spinor» having the form, for r = 1, 2, and for two-spinors ζ(r) normalised so that
(proof). Spinor normalisation has been chosen so that (proof). F(p,r) is the momentum space wave function, calculated at time x⁰ = 0,

This normalisation preserves the formulae for the inner product,
and
The resolution of unity is
It is common to choose a relativistic normalisation by multiplying the spinors by √2p⁰. Then, using the generalised scaling property of the delta function applied to the mass shell condition, the integrals in momentum space are replaced with the invariant integral
This makes no fundamental difference to the theory. Since we ultimately divide by normalisation to calculate probability, the normalisation used here simplifies the formulae.

Clearly, at non-relativistic energies, p ⁄ m ≈ 0, the Dirac spinor effectively reduces to a two spinor. Thus the theoretical requirements of a relativistic quantum theory lead naturally to Pauli’s» empirically based theory of spin». The Dirac equation describes spin ½ particles. According to the spin statistics theorem», proved by Wolfgang Pauli» in 1940, these are necessarily Fermions.

Antiparticles

The Dirac equation has four solutions, two with positive energy and two with negative energy. The negative energy solutions cannot simply be discarded because, when interactions are included, the emission of energy leads to transition to lower energy states, and there would be nothing to stop positive energy particles falling to an unobserved negative energy state. Dirac’s hole theory interpretation proposed the existence of an infinite sea of electrons in negative energy states. For Fermions, each energy state can only be occupied by one particle, so if all the negative energy states are already full, this would prevent further transitions. In hole theory, the absence of a negative energy electron would be seen as an antiparticle». Originally Dirac thought that there might be some unknown lack of symmetry and the proton might be the antiparticle, but the possibility was refuted by Herman Weyl». Dirac modified his position and predicted the existence of the positron» in 1931. Positrons were discovered experimentally by Carl Anderson» in 1932. Once clearly identified, it was recognised that many previous instances of positrons in cloud chambers had been mistaken for electrons travelling in the opposite direction.

The description of an infinite sea of particles is problematic, practically as well as philosophically. It would have an infinite electric charge and it is not mathematically possible to stop particles interacting with the sea, creating unobserved effects. There are further problems because the concept of a sea depends on the property that each energy state can only be occupied by one particle. This cannot work for antiparticles of vector bosons which are now known to exist. I will describe an alternative interpretaion, proposed independently by Stückelberg» and by Feynman».

Classically the path followed by a particle is traced out by parallel transport of the energy-momentum vector in the direction in which it points (geodesic motion). A negative time-like component defines vector pointing backwards in time, so the path traced by parallel transport of a particle with negative energy represents a particle travelling backwards in time.

This interpretation will become more meaningful in the mathematical formalism of interactions, in which particles are created and annihilated. When a particle is created with negative energy, it will go backwards in time and the mathematical formalism describes the annihilation of an antiparticle with positive energy. Likewise, when a negative energy particle is annihilated, it will have come from the future and the formalism describes the creation of a positive energy antiparticle.

In relativity, we found that the most physically meaningful form of time is proper time, time measured along the path of particular matter — coordinate time is based on proper time for a reference clock. If every particle has its own proper time, and if there is no other fundamental time, then it is not unnatural to think that one particle's proper time can become reversed compared to another; antimatter is matter whose proper time is inverted compared to proper time for surrounding matter.

A sign is lost in the mass shell condition, due to the squared terms, but it is natural to describe a time-like vector with a negative time-like component as having negative mass, m < 0. In this case, permissible solutions of the Dirac equation have positive energy E = p⁰ > 0 when m is positive and negative energy when m is negative. Complex conjugation reverses time, and the direction of momentum, while maintaining the probability interpretation and restoring positive energy. We also change the sign of mass, m → −m. Thus the negative energy solution of the Dirac equation is transformed to a positive energy solution of
where is the complex conjugate, . This is simply another representation of the Dirac equation. The positive energy solution is the wave function for the antiparticle,
where p satisfies the mass shell condition, is the complex conjugate of the Dirac spinor, for r = 1, 2, and for normalised two-spinors ζ(r),
normalised so that . The proof is as for the particle spinor and is omitted. F(p,r) is the momentum space wave function given by

Conserved Current

The Dirac adjoint» of a Dirac spinor is the Hermitian adjoint multiplied by the time-like γ matrix. Its definition is motivated by the requirement to form well-behaved, measurable quantities out of Dirac spinors.


Since quantum logic is based on probability theory, we need a relativistic statement that probability is conserved. This will take the same form as other continuity equations. That is, we must define a vector current density j^a such that
and the probability density for finding a particle at x is

Since the Dirac matrices multiplied by a spinor transform as a vector multiplied by a spinor, this leads us to the definition of a vector current density.


We have,
as is required of a conserved current density.

The Dirac Equation ↑ States of Many Particles →