Dirac Equation in The Algebra of Physical Space - Relation With The Standard Form

Relation With The Standard Form

The spinor can be written in a null basis as


\Psi = \psi_{11} P_3 - \psi_{12} P_3 \mathbf{e}_1 + \psi_{21} \mathbf{e}_1 P_3 + \psi_{22} \bar{P}_3,

such that the representation of the spinor in terms of the Pauli matrices is


\Psi \rightarrow \begin{pmatrix} \psi_{11} & \psi_{12} \\ \psi_{21} & \psi_{22}
\end{pmatrix}
 \bar{\Psi}^\dagger \rightarrow \begin{pmatrix} \psi_{22}^* & -\psi_{21}^* \\ -\psi_{12}^* & \psi_{11}^*
\end{pmatrix}

The standard form of the Dirac equation can be recovered by decomposing the spinor in its right and left-handed spinor components, which are extracted with the help of the projector

such that

 \Psi_L = \bar{\Psi}^\dagger P_3
 \Psi_R = \Psi P_3^{ }

with the following matrix representation

 \Psi_L \rightarrow \begin{pmatrix} \psi_{22}^* & 0 \\ -\psi_{12}^* & 0
\end{pmatrix}
 \Psi_R \rightarrow \begin{pmatrix} \psi_{11} & 0 \\ \psi_{21} & 0
\end{pmatrix}

The Dirac equation can be also written as

Without electromagnetic interaction, the following equation is obtained from the two equivalent forms of the Dirac equation


\begin{pmatrix}
0 & i \bar{\partial}\\
i \partial & 0
\end{pmatrix}
\begin{pmatrix} \bar{\Psi}^\dagger P_3 \\ \Psi P_3
\end{pmatrix}
= m
\begin{pmatrix} \bar{\Psi}^\dagger P_3 \\ \Psi P_3
\end{pmatrix}

so that


\begin{pmatrix}
0 & i \partial_0 + i\nabla \\
i \partial_0 - i \nabla & 0
\end{pmatrix}
\begin{pmatrix} \Psi_L \\ \Psi_R
\end{pmatrix}
= m
\begin{pmatrix} \Psi_L \\ \Psi_R
\end{pmatrix}

or in matrix representation


i \left(
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix} \partial_0 +
\begin{pmatrix}
0 & \sigma \\
-\sigma & 0
\end{pmatrix} \cdot \nabla
\right)
\begin{pmatrix} \psi_L \\ \psi_R
\end{pmatrix}
= m
\begin{pmatrix} \psi_L \\ \psi_R
\end{pmatrix},

where the second column of the right and left spinors can be dropped by defining the single column chiral spinors as

 \psi_L \rightarrow \begin{pmatrix} \psi_{22}^* \\ -\psi_{12}^*
\end{pmatrix}
 \psi_R \rightarrow \begin{pmatrix} \psi_{11} \\ \psi_{21}
\end{pmatrix}

The standard relativistic covariant form of the Dirac equation in the Weyl representation can be easily identified 
i \gamma^{\mu} \partial_{\mu} \psi = m \psi,
such that

 \psi_= \begin{pmatrix} \psi_{22}^* \\ -\psi_{12}^* \\ \psi_{11} \\ \psi_{21}
\end{pmatrix}

Given two spinors and in APS and their respective spinors in the standard form as and, one can verify the following identity


\phi^\dagger \gamma^0 \psi = \langle \bar{\Phi}\Psi + (\bar{\Psi}\Phi)^\dagger \rangle_S
,

such that


\psi^\dagger \gamma^0 \psi = 2 \langle \bar{\Psi}\Psi \rangle_{S R}

Read more about this topic:  Dirac Equation In The Algebra Of Physical Space

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