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Dirac equation

(Definition)

The Dirac equation is an equation derived by Paul Dirac in 1927 that describes relativistic spin particles (fermions). It is given by:

$\displaystyle (\gamma^\mu \partial_\mu - im)\psi = 0$

The Einstein summation convention is used.

Derivation

Mathematically, it is interesting as one of the first uses of the spinor calculus in mathematical physics. Dirac began with the relativistic equation of total energy:

$\displaystyle E = \sqrt{p^2c^2 + m^2c^4}$

As Schrödinger had done before him, Dirac then replaced

with its quantum mechanical operator, $\hat{p} \Rightarrow i\hbar \nabla$ . Since he was looking for a Lorentz-invariant equation, he replaced $\nabla$ with the D'Alembertian or wave operator

$\displaystyle \Box = \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}$

Note that some authors use $\Box^2$ for the D'Alembertian. Dirac was now faced with the problem of how to take the square root of an expression containing a differential operator. He proceeded to factorise the d'Alembertian as follows:

$\displaystyle \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} = (a^0 \... ...^2 \frac{\partial}{\partial z} + a^3\frac{i}{c} \frac{\partial}{\partial t})^2$

Multiplying this out, we find that:

$\displaystyle (a^0)^2 = (a^1)^2 = (a^2)^2 = (a^3)^2 = 1$

And

$\displaystyle a^0a^1 + a^1a^0 = a^0a^2 + a^2a^0 = a^0a^3 + a^3a^0 = a^1a^2 + a^2a^1 = a^1a^3 + a^3a^1 = a^2a^3 + a^3a^2 = 0$

Clearly these relations cannot be satisfied by scalars, so Dirac sought a set of four matrices which satisfy these relations. These are now known as the Dirac matrices, and are given as follows:

$\displaystyle \gamma^0 = -ia^0 = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & ... ...0 & 0 & 1 \ 0 & 0 & 1 & 0 \ 0 & -1 & 0 & 0 \ -1 & 0 & 0 & 0 \end{pmatrix}$

$\displaystyle \gamma^2 = -ia^2 = \begin{pmatrix} 0 & 0 & 0 & -i \ 0 & 0 & i &... ...0 & 1 & 0 \ 0 & 0 & 0 & -1 \ -1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \end{pmatrix}$

These matrices are also known as the generators of the special unitary group of order 4, i.e. the group of $4 \times 4$ matrices with unit determinant. Using these matrices, and switching to natural units ( $\hbar = c = 1$ ) we can now obtain the Dirac equation:

$\displaystyle (\gamma^\mu \partial_\mu - im)\psi = 0$

Feynman slash notation

Richard Feynman developed the following convenient notation for terms involving Dirac matrices:

$\displaystyle \gamma^\mu q_\mu = \cancel{q}$

Using this notation, the Dirac equation is simply

$\displaystyle (\cancel{\partial} - im)\psi = 0$

Relationship with Pauli matrices

The Dirac matrices can be written more concisely as matrices of Pauli matrices, as follows:

$\displaystyle \gamma_0$	$\displaystyle = \begin{pmatrix}\sigma_0 & 0\\ 0 & -\sigma_0 \end{pmatrix}$
$\displaystyle \gamma_1$	$\displaystyle = \begin{pmatrix}0 & \sigma_1\\ -\sigma_1 & 0 \end{pmatrix}$
$\displaystyle \gamma_2$	$\displaystyle = \begin{pmatrix}0 & \sigma_2\\ -\sigma_2 & 0 \end{pmatrix}$
$\displaystyle \gamma_3$	$\displaystyle = \begin{pmatrix}0 & \sigma_3\\ -\sigma_3 & 0 \end{pmatrix}$

"Dirac equation" is owned by Raphanus. [ full author list (2) | owner history (1) ]

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Also defines:

Feynman slash notation, Dirac matrices

Keywords:

relativistic, D'Alembertian

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Cross-references: Pauli matrices, terms, Richard Feynman, determinant, unit, group, order, unitary group, generators, matrices, scalars, relations, differential operator, expression, square root, wave operator, operator, Calculus, spinor, Einstein summation convention, equation
There are 5 references to this entry.

This is version 17 of Dirac equation, born on 2008-03-15, modified 2008-05-19.
Object id is 10407, canonical name is DiracEquation.
Accessed 914 times total.

Classification:

AMS MSC:	35Q40 (Partial differential equations :: Equations of mathematical physics and other areas of application :: Equations from quantum mechanics)
	81Q05 (Quantum theory :: General mathematical topics and methods in quantum theory :: Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations)

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