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}

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