Hilbert Book Model Project/Dirac Equation

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The Dirac equation

In its original form, the Dirac equation is a complex equation that uses spinors, matrices, and partial derivatives.

Dirac was searching for a split of the Klein-Gordon equation into two first order differential equations.

${\frac {\partial {}^{2}f}{\partial {t}^{2}}}-{\frac {\partial {}^{2}f}{\partial {x}^{2}}}-{\frac {\partial {}^{2}f}{\partial {y}^{2}}}-{\frac {\partial {}^{2}f}{\partial {z}^{2}}}=-m^{2}f$

(1)

$(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )f=\Box f=-m^{2}f$

(2)

Here $\Box$ is the d’Alembert operator.

Dirac used a combination of matrices and spinors to reach this result. He applied the Pauli matrices to simulate the behavior of vector functions under differentiation.

The unity matrix $I$ and the Pauli matrices $\sigma _{1},\sigma _{2},\sigma _{3}$ are given by [15]:

$I={\begin{bmatrix}1&0\\0&1\end{bmatrix}}$; $\sigma _{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}$; $\sigma _{2}={\begin{bmatrix}0&-\mathbb {I} \\\mathbb {I} &0\end{bmatrix}}$; $\sigma _{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}$; $\mathbb {I} ={\sqrt {-1}}$

(3)

For one of the potential orderings of the quaternionic number system, the Pauli matrices together with the unity matrix $I$ relate to the quaternionic base vectors $1,{\vec {i}},{\vec {j}},{\vec {k}}$.

$I\mapsto 1,\quad \sigma _{1}\mapsto \mathbb {I} {\vec {i}},\quad \sigma _{1}\mapsto \mathbb {I} {\vec {j}},\quad \sigma _{1}\mapsto \mathbb {I} {\vec {k}}$

(3)

$\sigma _{1}\sigma _{2}-\sigma _{2}\sigma _{1}=2\mathbb {I} \sigma _{3};\quad \sigma _{2}\sigma _{3}-\sigma _{3}\sigma _{2}=2\mathbb {I} \sigma _{1};\quad \sigma _{3}\sigma _{1}-\sigma _{1}\sigma _{3}=2\mathbb {I} \sigma _{2}$

(4)

$\sigma _{1}\sigma _{1}=\sigma _{2}\sigma _{2}=\sigma _{3}\sigma _{3}=I$

(5)

Instead of the usual$\left\{{\frac {\partial {}}{\partial {\tau }}},{\vec {i}}{\frac {\partial {}}{\partial {x}}},{\vec {j}}{\frac {\partial {}}{\partial {y}}},{\vec {k}}{\frac {\partial {}}{\partial {z}}}\right\}$ we want to use operators $\nabla =\{\nabla _{r},{\vec {\nabla }}\}$. The subscript $_{r}$ indicates the scalar part. The operator $\nabla$ relates to the applied parameter space. This means that the parameter space is also configured of combinations $q=\{q_{r},{\vec {q}}\}$of a scalar $q_{r}$ and a vector ${\vec {q}}$. Also the functions $f=\{f_{r},{\vec {f}}\}$ can be split in scalar functions $f_{r}$ and vector functions ${\vec {f}}$.

The different ordering possibilities of the quaternionic number system correspond to different symmetry flavors. Half of these possibilities offer a right handed external vector product. The other half offer a left-handed external vector product.

The Pauli matrices implement the cross product behavior of three dimensional vectors. A 4X4 dimensional matrix can implement the choice between right handed and left handed vector product.

We will use the momentum operator to represent the nabla operator:

$p_{\mu }=-\mathbb {I} \,{\frac {\partial }{\partial {q_{\mu }}}}$

(6)

$p_{\mu }\,\sigma _{\mu }=-\,e_{\mu }{\frac {\partial }{\partial {q_{\mu }}}}$

(7)

$\langle {\vec {\sigma }},{\vec {p}}\rangle \Leftrightarrow \mathbb {I} {\vec {\nabla }}$

(8)

$(\nabla _{r}+{\vec {\nabla }})(\nabla _{r}-{\vec {\nabla }})=\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle =\boxdot$

(9)

$(\nabla _{r}+\mathbb {I} {\vec {\nabla }})(\nabla _{r}-\mathbb {I} {\vec {\nabla }})=\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle =\Box$

(10)

These two equations hide the fact that the cross product can be right or left handed.

$\phi =(\nabla _{r}+\mathbb {I} {\vec {\nabla }})\psi$

(11)

$\zeta =(\nabla _{r}-\mathbb {I} {\vec {\nabla }})\phi$

(12)

$\zeta =(\nabla _{r}+\mathbb {I} {\vec {\nabla }})(\nabla _{r}-\mathbb {I} {\vec {\nabla }})\psi =(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi =\Box \psi$

(13)

Thus this corresponds with the Klein-Gordon equation if

$\zeta =\Box \psi =-m^{2}\psi$

(14)

We can split this into two first order partial differential equations.

$D_{+}\,f_{A}=(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f_{A}=m\,\mathbb {I} \,f_{B}$

(15)

$D_{-}\,f_{B}=(\nabla _{r}-\mathbb {I} {\vec {\nabla }})f_{B}=m\,\mathbb {I} \,f_{A}$

(16)

$\Box \,f_{A}=D_{-}D_{+}\,f_{A}=(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )f_{A}=(\nabla _{r}-\mathbb {I} {\vec {\nabla }})(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f_{A}=m\,\mathbb {I} \,(\nabla _{r}-\mathbb {I} {\vec {\nabla }})f_{B}=-m^{2}\,f_{A}$

(17)

Similarly

$\Box \,f_{B}=D_{+}D_{-}\,f_{B}=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )f_{B}=(\nabla _{r}+\mathbb {I} {\vec {\nabla }})(\nabla _{r}-\mathbb {I} {\vec {\nabla }})f_{B}=m\,\mathbb {I} \,(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f_{B}=-m^{2}\,f_{B}$

(18)

 Dirac’s approach

The original Dirac equation uses 4x4 matrices ${\vec {\alpha }}$ and $\beta$.

${\vec {\alpha }}$ and $\beta$ are matrices that implement the quaternion arithmetic behavior including the possible symmetry flavors of quaternionic number systems and continuums.

  

$\alpha _{\mu }={\begin{bmatrix}0&\sigma _{\mu }\\\sigma _{\mu }&0\end{bmatrix}}$

(19)

 

$\beta ={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}$

(20)

$\beta \beta =1$

(21)

The interpretation of the Pauli matrices as a representation of a special kind of angular momentum has led to the half-integer eigenvalue of the corresponding spin operator.

Dirac’s selection leads to  

$(p_{r}-\langle {\vec {\alpha }},{\vec {p}}\rangle -\beta mc){\begin{bmatrix}\varphi _{1}\\\varphi _{2}\\\varphi _{3}\\\varphi _{4}\end{bmatrix}}=0$

(22)

$[\varphi ]$ is a four-component spinor.

This equation does not split into two first order partial differential equations. Using gamma matrices cures this.

$\quad \gamma _{0}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}};\quad \gamma _{1}={\begin{bmatrix}0&\sigma _{1}\\-\sigma _{1}&0\end{bmatrix}};\quad \gamma _{2}={\begin{bmatrix}0&\sigma _{2}\\-\sigma _{2}&0\end{bmatrix}};\quad \gamma _{3}={\begin{bmatrix}0&\sigma _{3}\\-\sigma _{3}&0\end{bmatrix}}$

(23)

$\gamma _{\mu }=\beta \,\alpha _{\mu };\quad \gamma _{0}=\beta ;\quad \gamma _{5}=\mathbb {I} \,\gamma _{0}\,\gamma _{1}\,\gamma _{2}\,\gamma _{3}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}$

(24)

${\biggl (}\gamma _{5}{\frac {\partial }{\partial {\tau }}}-\gamma _{1}{\frac {\partial }{\partial {x}}}-\gamma _{2}{\frac {\partial }{\partial {y}}}-\gamma _{3}{\frac {\partial }{\partial {z}}}-{\frac {m}{\mathbb {I} \hbar }}{\biggr )}[\varphi ]=0$

(25)

${\biggl (}{\begin{bmatrix}0&1\\1&0\end{bmatrix}}{\frac {\partial }{\partial {\tau }}}-{\begin{bmatrix}0&\sigma _{1}\\-\sigma _{1}&0\end{bmatrix}}{\frac {\partial }{\partial {x}}}-{\begin{bmatrix}0&\sigma _{2}\\-\sigma _{2}&0\end{bmatrix}}{\frac {\partial }{\partial {y}}}-{\begin{bmatrix}0&\sigma _{3}\\-\sigma _{3}&0\end{bmatrix}}{\frac {\partial }{\partial {z}}}-{\frac {m}{\mathbb {I} \hbar }}{\biggr )}{\begin{bmatrix}\varphi _{A}\\\varphi _{B}\end{bmatrix}}=0$

(26)

This time $[\varphi ]$ is a two component spinor. The equation splits into 

$D_{+}\,\varphi _{A}=(\nabla _{r}+\mathbb {I} {\vec {\nabla }})\varphi _{A}=m\,\mathbb {I} \,\varphi _{B}$

(27)

$D_{-}\,\varphi _{B}=(\nabla _{r}-\mathbb {I} {\vec {\nabla }})\varphi _{B}=m\,\mathbb {I} \,\varphi _{A}$

(28)

The spinors ${\textstyle \varphi _{A}}$ and ${\textstyle \varphi _{B}}$ are not quaternionic fields. Instead they are combinations of fields that show different symmetry. The fields differ in the way that they handle the direction of progression.

The spinors ${\textstyle \varphi _{1}}$, ${\textstyle \varphi _{2}}$, ${\textstyle \varphi _{3}}$, and ${\textstyle \varphi _{4}}$ also differ in the right or left handed treatment of the cross product. In this way they form four different combinations of handling progression and handiness of the cross product.

The symmetry flavors of the parameter spaces combine 16 different symmetry flavors. These differences cover the four differences of the spinors. The spinors do not distinguish anisotropy.

Interpretation

According to conventional physics, the split divides the Klein-Gordon second-order partial differential equation into a first order partial differential equation for the electron and a first order partial differential equation for the positron. However, these equations are no quaternionic first order partial differential equations.

Instead, the equation

$\zeta =\nabla ^{*}\nabla \psi =\nabla \nabla ^{*}\psi =(\nabla _{r}+{\vec {\nabla }})(\nabla _{r}-{\vec {\nabla }})\psi =\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi ={\frac {\partial {}^{2}\psi }{\partial {t}^{2}}}+{\frac {\partial {}^{2}\psi }{\partial {x}^{2}}}+{\frac {\partial {}^{2}\psi }{\partial {y}^{2}}}+{\frac {\partial {}^{2}\psi }{\partial {z}^{2}}}=\boxdot \,\psi$

(29)

splits into two quaternionic first order partial differential equations.

$\phi =\nabla \psi =(\nabla _{r}+{\vec {\nabla }})\psi$

(30)

$\zeta =\nabla ^{*}\psi =(\nabla _{r}-{\vec {\nabla }})\phi =\nabla ^{*}\nabla \psi =\boxdot \psi$

(31)

In conventional physics this equation has not yet found a proper interpretation.