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

Famous quotes containing the words relation, standard and/or form:

    “You know there are no secrets in America. It’s quite different in England, where people think of a secret as a shared relation between two people.”
    —W.H. (Wystan Hugh)

    “Gentlemen, those confederate flags and our national standard are what has made this union great. In what other country could a man who fought against you be permitted to serve as judge over you, be permitted to run for reelection and bespeak your suffrage on Tuesday next at the poles.”
    —Laurence Stallings (1894–1968)

    “Ambivalence reaches the level of schizophrenia in our treatment of violence among the young. Parents do not encourage violence, but neither do they take up arms against the industries which encourage it. Parents hide their eyes from the books and comics, slasher films, videos and lyrics which form the texture of an adolescent culture. While all successful societies have inhibited instinct, ours encourages it. Or at least we profess ourselves powerless to interfere with it.”
    —C. John Sommerville (20th century)