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 with the, relation with, relation, standard and/or form:

“There is a constant in the average American imagination and taste, for which the past must be preserved and celebrated in full-scale authentic copy; a philosophy of immortality as duplication. It dominates the relation with the self, with the past, not infrequently with the present, always with History and, even, with the European tradition.”
—Umberto Eco (b. 1932)

“[Man’s] life consists in a relation with all things: stone, earth, trees, flowers, water, insects, fishes, birds, creatures, sun, rainbow, children, women, other men. But his greatest and final relation is with the sun.”
—D.H. (David Herbert)

“In relation to God, we are like a thief who has burgled the house of a kindly householder and been allowed to keep some of the gold. From the point of view of the lawful owner this gold is a gift; From the point of view of the burglar it is a theft. He must go and give it back. It is the same with our existence. We have stolen a little of God’s being to make it ours. God has made us a gift of it. But we have stolen it. We must return it.”
—Simone Weil (1909–1943)

“... the meanest life, the poorest existence, is attributed to God’s will, but as human beings become more affluent, as their living standard and style begin to ascend the material scale, God descends the scale of responsibility at a commensurate speed.”
—Maya Angelou (b. 1928)

“Modern women are squeezed between the devil and the deep blue sea, and there are no lifeboats out there in the form of public policies designed to help these women combine their roles as mothers and as workers.”
—Sylvia Ann Hewitt (20th century)