The equation of Dirac is a equation formulated by Paul Dirac in 1928 within the framework of its Mécanique quantum relativist of the electron. It acts at the beginning of an attempt to incorporate relativity restricted in quantum models, with a linear writing in the mass and the impulse.

Explanation

This equation describes the behavior of elementary particles of Spin S half-entireties, like the electrons. Dirac sought to transform the equation of Schrödinger in order to make it invariant by the action of the group of Lorentz, in other terms to make it compatible with the principles of the restricted Relativité.

This equation takes into account in a natural way the concept of spin introduced little time before and made it possible to predict the existence of the Antiparticule S. Indeed, in addition to the solution corresponding to the electron, he discovers a new solution corresponding to a particle of negative energy and load opposed to that of the electron.

In 1932 Carl Anderson, whereas he studied photons of high energy coming from the space, notes that the interaction of these Photon S with the Bubble chamber produces a particle which is identified with the particle conjectured by Dirac, the Positon.

It is in addition completely astonishing that the operator of Dirac, discovered for reasons absolutely physical (and theoretical) will have in Mathématiques a glorious future by its essential use in one of the deepest results of the century, the theorem of Atiyah and Singer shown in the Years 1960.

Mathematical formulation

Its exact formulation is:

where m is the Masse of the particle, C the Speed of light, $\backslash \; hbar$ the reduced Constante of Planck, X and T cooordonnées in the space and the Temps, and ψ ( X , T ) a Fonction of wave to four components. (The function of wave must be formulated by a Spineur with four components, rather than by simple a Scalaire, because of the requirements of the restricted Relativité.) Finally $\backslash \; alpha\_i,\; \backslash \; i=0,1,2,3$ is matrix S of dimension $4\; \backslash \; times\; 4$ acting on the spinor $\backslash \; psi\; \backslash ,$ and called Matrices of Dirac. In term of the Matrice of Pauli $\backslash \; vec\; \backslash \; sigma$ one can write the matrices of Dirac, in the Représentation of Dirac (of others are possible, like the Représentation of Weyl or the Représentation of Majorana), in the form

It is common in quantum mechanics to consider the operator Quantité of movement $\backslash \; vec\; p\; \backslash \; equiv\; -\; I\; \backslash \; hbar\; \backslash \; vec\; \backslash \; nabla\; \backslash ,$ and in this case the equation of Dirac rewrites in a condensed way

$I\; \backslash \; hbar\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; psi\}\; \{\backslash \; partial\; T\}\; (\backslash \; mathbf\; \{X\},\; T)\; =\; \backslash \; left\; (mc^2\; \backslash \; alpha\_0\; +\; C\; \backslash \; vec\; \backslash \; alpha.\; \backslash \; vec\; p\; \backslash ,\; \backslash \; right)\; \backslash \; psi\; (\backslash \; mathbf\; \{X\},\; T)$

Moreover, it is natural to seek a covariante formulation, which one makes by posing $\backslash \; gamma^0=\; \backslash \; gamma\_0=\; \backslash \; alpha\_0$ and $\backslash \; gamma^i=-\; \backslash \; gamma\_i=\; \backslash \; alpha\_0\; \backslash \; alpha\_i$, in which case there is (by adopting conventions c=1 and $\backslash \; hbar=1$) a notation even more compact:

$\backslash \; left\; (\backslash \; displaystyle\; \{\backslash \; not\}\; p-m\; \backslash \; right)\; \backslash \; psi\; (\backslash \; mathbf\; \{X\},\; T)\; =0$

where one adopted the notation of Feynman $\backslash \; displaystyle\; \{\backslash \; not\}\; a=a^\; \backslash \; driven\; \backslash \; gamma\_\; \backslash \; mu$

See too

Related articles

• Mechanical quantum relativist
• Paradox of Klein
• Zitterbewegung
• Equation of Klein-Gordon
• Quantum theory of the fields

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