Stamp of Dirac

The matrices of Dirac are matrix S which were introduced by Paul Dirac, at the time of the search for a relativistic equation of wave of the electron.

Interest

The natural generalization of the equation of Schrödinger is the equation of Klein-Gordon. Unfortunately, this one described of the particles of Spin 0 and is not appropriate for the electrons which are of spin 1/2. Dirac then tried to find an equation linear like that of Schrödinger in the form:

$i \ frac {\ partial \ psi} {\ partial T} = \ left (\ frac {1} {I} \ mathbf {\ alpha} \ cdot \ nabla+ \ beta m \ right) \ psi \ equiv H \ psi$

where $\ psi$ is a Fonction of vectorial wave , $m$ the Masse of the particle, $H$ the Hamiltonien and $\ mathbf {\ alpha}, \ beta$ is respectively a vector of square matrices and a square Matrice. The equation of Dirac must respect the three following points:

the components of $\ psi$ must satisfy the equation of Klein-Gordon, a Onde planes whose solution is:
: $E^2= \ mathbf {p} ^2+m^2$ being a solution.
There exists a quadrivector Densité of current which is preserved and the temporal component is a positive density (identified with the electric charge).
the components of $\ psi$ should not satisfy any auxiliary condition, i.e. at a given moment they are functions independent of $x$ .

Matrices of Dirac

Dirac proposed that the square matrices are anticommutantes and of square equal to one. I.e. they obey the following Algèbre:

$\ left \ {\ alpha_i, \ alpha_k \ right \} =0 \ I \ k$

\ left \ {\ alpha_i, \ beta \ right \} =0

\ alpha_i^2: \ beta^2=I

where the hooks are the anticommutator $\ left \ {has, B \ right \} =AB+BA$ .

By raising the equation of Dirac squared, one checks immediately that the first condition is satisfied. One introduces then the matrices of the Dirac $\ gamma^ \ mu$ themselves:

$\ gamma^0= \ beta$

\ gamma^i= \ beta \ alpha^i \ i=1,2,3

\ left \ {\ gamma^ \ driven, \ gamma^ \ naked \ right \} =2g^ {\ driven \ naked}

The slash of Feynman

One introduces also the “ slash ” of Feynman:

$\ not has \ equiv a_ \ driven \ gamma^ \ mu$

The equation of Dirac takes the form then:

$\ left (I \ gamma^ \ driven \ partial_ \ mm \ right) \ psi \ equiv \ left (I \ not \ partial-m \ right) =0$

A representation explicit, known as “standard representation”, is given by:

$\ gamma^0= \ begin {pmatrix} I & \ mathbf {0} \ \ \ mathbf {0} & - I \ end {pmatrix}$

\ gamma^i= \ begin {pmatrix} \ mathbf {0} & \ sigma^i \ \ - \ sigma^i & \ mathbf {0} \ end {pmatrix}

\ beta= \ begin {pmatrix} I & \ mathbf {0} \ \ \ mathbf {0} & - I \ end {pmatrix}

\ alpha^i= \ begin {pmatrix} \ mathbf {0} & \ sigma^i \ \ \sigma^i & \ mathbf {0} \ end {pmatrix}

where $I$ is the matrix unit 2×2 and $\ sigma^i$ is the Matrices of Pauli.

This representation is particularly practical because it highlights the spinor character (due to the Spin half-entirety) of the Fonction of wave of the electron and it separates the components from energy positive and negative. Thus, by writing the function of wave like a Bispineur:

$\ psi= \ begin {pmatrix} \ phi \ \ \ chi \ end {pmatrix}$

where $\ phi$ and $\ chi$ is two Spineur S, the equation of Dirac becomes:

$i \ frac {\ partial \ phi} {\ partial T} =m \ phi+ \ frac {1} {I} \ mathbf {\ partial sigma} \ cdot \ nabla \ chi$

i \ frac {\ \ chi} {\ partial T} =-m \ chi+ \ frac {1} {I} \ mathbf {\ sigma} \ cdot \ nabla \ phi

By introducing the function of wave combined like:

$\ bar \ psi= \ psi^ \ dagger \ gamma^0$

One finds:

$\ bar \ psi \ left (I \ overleftarrow {\ not \ partial} \ right) =0$

And with the equation of Dirac, that gives:

$\ bar \ psi \ left (\ overleftarrow {\ not \ partial} + \ overrightarrow {\ not \ partial} \ right) \ equiv \ partial_ \ driven \ left (\ bar \ psi \ gamma^ \ driven \ psi \ right) =0$

What gives a preserved current:

$j^ \ mu= \ bar \ psi \ gamma^ \ driven \ psi$

Whose temporal component $j^0= \ rho= \ bar \ psi \ gamma^0 \ psi= \ psi^ \ dagger \ psi$ is positive. One defines also the matrix:

$\gamma^5=\gamma^0\gamma^1\gamma^2\gamma^3$

The use of $\ gamma^5$ thus makes it possible to build various types of combinations such as, for example, of the vectors:

$\ bar \ psi \ gamma^ \ driven \ psi$

Pseudoscalaire S:

$\ bar \ psi \ gamma^5 \ psi$

One easily checks the relativistic Covariance of all this formalism.

Representations

The matrices of Dirac are completely determined by the relation:

$\ gamma^ \ driven \ gamma^ \ nu+ \ gamma^ \ naked \ gamma^ \ mu=2 \ eta^ {\ driven \ naked}$

where $\ eta^ {\ driven \ naked}$ is the Tenseur of Minkowski. There is also $\ gamma_ \ driven \ gamma^ \ mu=4$ . There exists an infinity of possible representations of the matrices of Dirac. I.e. an infinity of possible solutions to the preceding relation.

Let us quote for example the Représentation of Majorana obtained starting from the preceding representation by exchanging $\ alpha_2$ and $\ beta$ and by changing the sign of $\ alpha_1$ and $\ alpha_3$ . It with the property interesting to make the equation of Dirac real whose solutions are combinations of real solutions. Let us quote also the chiral representation:

$\ gamma^0= \ beta= \ begin {pmatrix} \ mathbf {0} & - I \ \ - I & \ mathbf {0} \ end {pmatrix}$

\ mathbf {\ alpha} = \ begin {pmatrix} \ mathbf {\ sigma} & \ mathbf {0} \ \ \ mathbf {0} & - \ mathbf {\ sigma} \ end {pmatrix}

\ mathbf {\ gamma} = \ begin {pmatrix} \ mathbf {0} & \ mathbf {\ sigma} \ \ - \ mathbf {\ sigma} & \ mathbf {0} \ end {pmatrix}

Its advantage is that the two spinors change independently under the Rotation S and the Translation S. It is particularly useful for particles without mass, the equations being simplified considerably. It was used for the Neutrino although it is known now that this one has an extremely small but nonnull mass.

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