Talk:PlanetPhysics/Dirac Equation
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[edit source]%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Dirac equation %%% Primary Category Code: 03.65.Pm %%% Filename: DiracEquation.tex %%% Version: 2 %%% Owner: invisiblerhino %%% Author(s): invisiblerhino %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}
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The Dirac equation is an equation derived by Paul Dirac in 1927 that describes relativistic \htmladdnormallink{spin}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} $1/2$ \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html} (\htmladdnormallink{fermions}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}). It is given by:
\[ (\gamma^\mu \partial_\mu - im)\psi = 0 \] The \htmladdnormallink{Einstein summation convention}{http://planetphysics.us/encyclopedia/EinsteinSummationNotation.html} is used. \subsection{Derivation} Mathematically, it is interesting as one of the first uses of the \htmladdnormallink{spinor}{http://planetphysics.us/encyclopedia/ECartan.html} calculus in \htmladdnormallink{mathematical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html}. Dirac began with the relativistic equation of total \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html}: \[ E = \sqrt{p^2c^2 + m^2c^4} \] As Schr\"odinger had done before him, Dirac then replaced $p$ with its quantum mechanical \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html}, $\hat{p} \Rightarrow i\hbar \nabla$. Since he was looking for a Lorentz-invariant equation, he replaced $\nabla$ with the D'Alembertian or \htmladdnormallink{wave operator}{http://planetphysics.us/encyclopedia/DAlembertOperator.html} \[ \Box = \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \] Note that some authors use $\Box^2$ for the D'alembertian. Dirac was now faced with the problem of how to take the \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} root of an expression containing a differential operator. He proceeded to factorise the d'Alembertian as follows: \[ \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} = (A \frac{\partial}{\partial x} + B \frac{\partial}{\partial y} + c \frac{\partial}{\partial z} + D\frac{i}{c} \frac{\partial}{\partial t})^2 \] Multiplying this out, we find that: \[ A^2 = B^2 = C^2 = D^2 = 1 \] And \[ AB + BA = BC + CB = CD + DC = 0 \] Clearly these \htmladdnormallink{relations}{http://planetphysics.us/encyclopedia/Bijective.html} cannot be satisfied by \htmladdnormallink{scalars}{http://planetphysics.us/encyclopedia/Vectors.html}, so Dirac sought a set of four \htmladdnormallink{matrices}{http://planetphysics.us/encyclopedia/Matrix.html} which satisfy these relations. These are now known as the Dirac matrices, and are given as follows: \[ A = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}, B = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix} \] \[ C = \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{pmatrix}, D = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} \] These matrices are usually given the symbols $\gamma^0$, $\gamma^1$, etc. They are also known as the \htmladdnormallink{generators}{http://planetphysics.us/encyclopedia/Generator.html} of the special unitary \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of order 4, i.e. the group of $n \times n$ matrices with unit \htmladdnormallink{determinant}{http://planetphysics.us/encyclopedia/Determinant.html}. Using these matrices, and switching to natural units ($\hbar = c = 1$) we can now obtain the Dirac equation: \[ (\gamma^\mu \partial_\mu - im)\psi = 0 \] \subsection{Feynman slash notation} Richard Feynman developed the following convenient notation for terms involving Dirac matrices: \[ \gamma^\mu q_\mu = \cancel{q} \] Using this notation, the Dirac equation is simply \[ (\cancel{\partial} - im)\psi = 0 \]
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