Quantum Field Theory
... no spooky action at a distance (Einstein)
Early Results
Relativistic Quantum Mechanics Klein-Gordon Equation
Dirac Equation
The Dawn of QFT Spinors
Feynman Slash Notation
Klein-Gordon Field
Dirac Field
Grassman Variable
Conformal Field Theory
Countdown to the Standard Model
From a framework to a model Yang-Mills Theory
Quantum Electrodynamics
Quantum Chromodynamics
Electroweak Theory
Higgs Mechanism
Standard Model
Semi-Classical Gravity and the Dark Age Hawking Radiation
Chandrashekhar Limit
Problems with the Standard Model
Beyond the Standard Model Beyond the Standard Model
Quantum Gravity
Theory of Everything
Related De Donder-Weyl Theory

The Dirac Equation is an attempt to make Quantum Mechanics Lorentz Invariant, i.e. incorporate Special Relativity. It attempted to solve the problems with the Klein-Gordon Equation. In Quantum Field Theory, it is the field equation for the spin-1/2 fields, also known as Dirac Fields.

StatementEdit this section

Dirac Equation

$ \left( i\hbar \not{\partial }-m{{c}_{0}} \right)\Psi =0 $

Derivation of the Dirac Equation from the Klein-Gordon Equation

The idea is to try to take the square root of $ -\frac{1}{c_{0}^{2}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{z}^{2}}} $ We want this equation to be first-order in both space, and in time.

We therefore propose that:

$ -\frac{1}{c_{0}^{2}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{z}^{2}}}={{\left( {{\gamma }^{0}}\frac{\partial }{\partial x}+{{\gamma }^{1}}\frac{\partial }{\partial y}+{{\gamma }^{2}}\frac{\partial }{\partial z}+\frac{i}{c}{{\gamma }^{3}}\frac{\partial }{\partial t} \right)}^{2}} $

Here, $ \gamma^\mu $ are certain scaling numbers, probably. We obviously do not want terms like $ \frac{{{\partial }^{2}}}{\partial x\partial y} $, so we will need to impose the following restrictions on the "gamma"s:

$ \begin{align} & \left\{ {{\gamma }^{j}},{{\gamma }^{k}} \right\}=0, \\ & {{\left( {{\gamma }^{j}} \right)}^{2}}=1 \\ \end{align} $

Note that $ j $ and $ k $ are distinct, and that $ \left\{ u,v \right\}= uv+vu $ is the anti-commutator bracket, which measures the amount of anti-commutativity. Equivalently, $ \left\{ {{\gamma }^{j}},{{\gamma }^{k}} \right\}=\eta^{jk} $.

This can be achieved if $ {{\gamma }^{j}} $ are matrices. This would have the implication, that the wavefunction would have multiple complex components. This is key in the discovery of Quantum Field Theory.

So, basically, one can imagine that $ {{\left( {{\gamma }^{j}} \right)}^{2}}={{\mathbf{I}}_{4}} $

We can now write down the following equation:

$ \left( {{\gamma }^{1}}\frac{\partial }{\partial x}+{{\gamma }^{2}}\frac{\partial }{\partial y}+{{\gamma }^{3}}\frac{\partial }{\partial z}+\frac{i}{{{c}_{0}}}{{\gamma }^{0}}\frac{\partial }{\partial t} \right)\Psi =r\Psi $

Multiplying by $ {{\gamma }^{1}}\frac{\partial }{\partial x}+{{\gamma }^{2}}\frac{\partial }{\partial y}+{{\gamma }^{3}}\frac{\partial }{\partial z}+\frac{i}{{{c}_{0}}}{{\gamma }^{0}}\frac{\partial }{\partial t} $,

$ \left( -\frac{1}{c_{0}^{2}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{z}^{2}}} \right)\Psi ={{r}^{2}}\Psi $

$ -{{\hbar }^{2}}c_{0}^{2}\left( -\frac{1}{c_{0}^{2}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{z}^{2}}} \right)\Psi =-{{\hbar }^{2}}c_{0}^{2}{{r}^{2}}\Psi $

At this point, we need to realise that even though the Klein-Gordon Equation does not completely restrict the wavefunction to what it should be, it still should be obeyed by every solution to the Dirac Equation. Therefore, we must require that:

$ \begin{align} & -{{\hbar }^{2}}c_{0}^{2}{{r}^{2}}\Psi =-{{m}^{2}}c_{0}^{4}\Psi \\ & {{r}^{2}}=\frac{{{m}^{2}}c_{0}^{2}}{{{\hbar }^{2}}} \\ & r=\pm \frac{mc_{0}^{{}}}{\hbar } \\ \end{align} $

$ \left( {{\gamma }^{1}}\frac{\partial }{\partial x}+{{\gamma }^{2}}\frac{\partial }{\partial y}+{{\gamma }^{3}}\frac{\partial }{\partial z}+\frac{i}{{{c}_{0}}}{{\gamma }^{0}}\frac{\partial }{\partial t} \right)\Psi =\pm \frac{m{{c}_{0}}}{\hbar }\Psi $

Note, that the problems in the Klein-Gordon Equation can be solved if we enforce this $ \pm $ to be a $ + $.

$ \left( {{\gamma }^{1}}\frac{\partial }{\partial x}+{{\gamma }^{2}}\frac{\partial }{\partial y}+{{\gamma }^{3}}\frac{\partial }{\partial z}+\frac{i}{{{c}_{0}}}{{\gamma }^{0}}\frac{\partial }{\partial t} \right)\Psi =\frac{m{{c}_{0}}}{\hbar }\Psi $

To satisfy the earlier conditions that the gamma matrices multiply to yield the Minkowski Metric Tensor, we see that these matrices must take on the following values.

$ \begin{align} & {{\gamma }^{0}}=\left( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{matrix} \right) \\ & {{\gamma }^{1}}=\left( \begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ \end{matrix} \right) \\ & {{\gamma }^{2}}=\left( \begin{matrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \\ \end{matrix} \right) \\ & {{\gamma }^{3}}=\left( \begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{matrix} \right) \\ \end{align} $

We call these matrices Dirac Gamma Matrices.

Our equation can then be written as:

$ i\hbar {{\gamma }^{\mu }}{{\partial }_{\mu }}\Psi -m{{c}_{0}}\Psi =0 $

If we use Feynman Slash Notation, that $ \not P=\gamma^\mu P_\mu $,

$ \left( i\hbar \not{\partial }-m{{c}_{0}} \right)\Psi =0 $

Relationship with Klein-Gordon EquationEdit this section

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The sense in which the Dirac Equation can be regarded as the square root of the Klein-Gordon Equation is made clear by the use of the Feynman Slash Notation.[1]. The Klein-Gordon Equation can be factored:

$ 0 = (\hbar^2\partial^\mu \partial_\mu + (mc)^2)\psi = ((\hbar\partial\!\!\!/)^2 + (mc)^2)\psi = (i\hbar\partial\!\!\!/ + mc)(-i\hbar\partial\!\!\!/ + mc)\psi \,. $

The last factor, $ (-i\hbar\partial\!\!\!/ + mc)\psi \, $, is simply the Dirac equation. Hence any solution to the Dirac Equation is automatically a solution to the Klein-Gordon Equation:

$ (-i\hbar\partial\!\!\!/ + mc)\psi = 0 \rightarrow (\hbar^2\partial^\mu \partial_\mu + (mc)^2)\psi = 0 \,. $

But the converse is not true; not all solutions to the Klein–Gordon equation solve the Dirac Equation.

In a PotentialEdit this section

In a potential, the Dirac Equationw takes the form:

$ \left(i\hbar\not\nabla-mc_0\right)\psi=0 $

Where $ \nabla^\mu=\partial^\mu+ig A_\mu $ where $ g $ is the Coupling Constant.

Free Particle SolutionEdit this section

The Free Particle Solution takes the form: $ \psi = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2E_{p}}}\sum_{s} \left( a^{s}_{\textbf{p}}u^{s}(p)e^{-ip \cdot x}+b^{s \dagger}_{\textbf{p}}v^{s}(p)e^{ip \cdot x}\right).\, $

Relationship with spin and the Pauli TheoryEdit this section

Main Article: Spin

Due to the implication of these complex scalar-valued Wavefunctions being replaced by spinor-valued fields, one finds that there are operators called spin operators. These spin operators' eigenvalues become the ordinary spin quantum numbers, and we see that they are quantised as $ \frac{n}{2} $ where $ n\in\mathbb Z $, and does not exceed 4. The spins are therefore

$ 0, \frac12, 1, \frac32, 2 $. \

Note, that a $ \frac32 $ spin is possible only by Supersymmetry. This is in fact an argument in favour of Supersymmetry.'

We furthermore see that bosons have integer spins, whereas fermions have half-integer spins. The actual spins can be shown to be representations of $ SO(n+1) $ or $ SO(n+\frac12) $ where $ n $ is the spin quantum number.

ImplicationsEdit this section

AntimatterEdit this section

It is seemingly paradoxical that the Dirac Equation allows for the existence of negative energies. This seems very strange at first glance. However, with resolute faith in Mathematics, one must accept the existence of these weird creatures. This became known as the model of Antimatter.

Antimatter is matter with negative energy. The positron, for example, is a particle of antimatter, whereas the electron is one of matter. They both have the same mass, but the opposite charge.

SpinorsEdit this section

One sees that the solutions to the Dirac Equation take the form $ \psi^\mu $, and become spinorial. Therefore, the Dirac Equation introduces Spinors into Physics.

Quantum Field TheoryEdit this section

As we have seen previously, the Dirac Equation acts indeed paradoxically if we continue to interpret the $ \psi $ as an ordinary wavefunction.

Therefore, we must interpret it as a Field. This leads to the birth of Quantum Field Theory.

Maximum Atomic Number for atomsEdit this section

In the Dirac Equation, one may calculate the eigenstates and find the eigenvalues of the Hamiltonian, one sees that the Energies of an electron 'bound to the nucleus are quantised as:

$ E=mc_0^2\sqrt{1-\alpha^2Z^2} $

Here, $ Z $ is the number of electrons, $ \alpha $ is the Fine-Structure Constant, the coupling constant for Electromagnetism. If $ Z>137 $, this becomes imaginary.

Classically/(Purely Special Relativistically), this is interpreted as an Electron going faster than light, as can be seen from the apparent mass, energy, etc., of an object moving at superluminal speeds.

Therefore, this means that the atom cannot have more than 137 electrons. For neutral atoms, this is equivalent to stating that the atomic number is never more than 137. For cations, however, it is possible to have a larger atomic number, since it has more protons than electrons.

ReferencesEdit this section

  1. Particle Physics (3rd Edition), B. R. Martin, G.Shaw, Manchester Physics Series, John Wiley & Sons, ISBN 978-0-470-03294-7