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Dirac equation

The Dirac equation is a relativistic quantum mechanical wave equation which describes the behavior of relativistic fermions, invented by Paul Dirac in 1928. It is a generalization of the Schrödinger equation which satisfies special relativity. For several years the meaning of the Dirac equation was unclear because no one understood the meaning of the negative energy levels that the equation predicted. Finally, these negative energy levels were interpreted as antimatter.

In the following article, we will use the index notation developed in the article tensor, with the Einstein summation convention (i.e. repeated indices are implicitly summed over.) Following the typical relativistic notation, Greek indices {μ,ν,ρ,...} run over the values 0, 1, 2, and 3, where 0 is the time component and 1, 2, and 3 are the three spatial components. Roman indices <math>\{i, j, k,...\}</math> run over just the spatial components 1, 2, and 3. Finally, we adopt Planck units, wherein <math>\hbar = c = 1</math>.

With these conventions, the Dirac equation for a particle in free space is:

<math>\left( i \gamma^\mu \partial_\mu - m \right) \psi(x) = 0 </math>

m is the particle's mass, and <math>\psi(x)</math> is a four-component wave function:

<math> \psi(x) = \begin{pmatrix} \psi_1(x)\\ \psi_2(x)\\ \psi_3(x)\\ \psi_4(x) \end{pmatrix} </math>

It is important to note that these four components are internal "degrees of freedom" of the particle, not spacetime components. ψ(x) is not a vector; it belongs to a family of objects known as spinors, and is known as a "Dirac spinor".

The Dirac matrices <math>\gamma_\mu</math> are any choice of <math>4 \times 4</math> representation of the complex Clifford algebra:

<math>\{\gamma^\mu , \gamma^\nu \} \equiv \gamma^\mu \gamma ^\nu + \gamma^\nu \gamma^\mu = 2 g^{\mu\nu} \cdot I </math>

where <math>g^{\mu\nu}</math> is the metric of spacetime.

A useful choice of <math>\gamma</math> matrices, known as the Weyl representation, is:

<math>\gamma^0 =
\begin{pmatrix} 0&&I\\ I&&0 \end{pmatrix} </math>
<math>\gamma^j =
\begin{pmatrix} 0&&\sigma^j\\ -\sigma^j&&0 \end{pmatrix} </math>

where I is the <math>2\times 2</math> identity matrix, and <math>\sigma^j</math> are the set of Pauli matrices.

One of the consequences of the Dirac equation is the prediction of antimatter. Unlike the Schrödinger equation which has a lowest energy level, the Dirac equation has no lowest energy level. This caused difficulties in interpreting the equation. The way that the equation was finally interpreted was to regard all of the negative energy levels as being filled. With enough energy one can move a particle from the filled energy level to an unfilled one. This produces a particle and a hole. The hole can then be regarded as an antiparticle.



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