Dirac equation in the algebra of physical space

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The Dirac equation, as the relativistic equation that describes spin 1/2 particles in quantum mechanics can be written in terms of the Algebra of physical space (APS), which is a case of a Clifford algebra or geometric algebra that is based in the use of paravectors.

The Dirac equation in APS, including the electromagnetic interaction, reads

i \bar{\partial} \Psi\mathbf{e}_3 + e \bar{A} \Psi = m \bar{\Psi}^\dagger

Another form of the Dirac equation in terms of the Space time algebra was given earlier by David Hestenes.

In general, the Dirac equation in the formalism of geometric algebra has the advantage of providing a direct geometric interpretation.

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[edit] Relation with the standard form

The Dirac equation can also be written as

i \partial \bar{\Psi}^\dagger \mathbf{e}_3 + e A \bar{\Psi}^\dagger = m \Psi

The standard form is obtained multiplying the spinor with the projector P = \frac{1}{2}( 1 + \mathbf{e}_3) . Without electromagnetic interaction, the following matrix is obtained from the two previous 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}

This column of projected spinors are related to the spinors in the Weyl representation. This is more evident in identifying the right and left Weyl spinors as

\bar{\Psi}^\dagger P_3 = \psi_L
\Psi P_3^{ } = \psi_R

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}

In the matrix representation each expression is replaced by a 2 by 2 matrix, including ψL and ψR. The nabla operator can be written in terms of the Pauli matrices as

\nabla \rightarrow \sigma \cdot \nabla.

On the other hand, only the first column of each spinor is taken

\psi_{L,R} \rightarrow FirstColumn(\psi_{L,R}),

so the Dirac equation becomes

\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},

from which, the standard relativistic covariant form of the Dirac equation is found

i \gamma^{\mu} \partial_{\mu} \psi = m \psi .

[edit] Spinor expansion in a null basis

The spinor Ψ can be expanded in a null basis as follows

\Psi = \psi_1^* \bar{P}_3 - \psi_2^* P_3 \mathbf{e}_1 + \psi_3 P_3 + \psi_4 \mathbf{e}_1 P_3,

where each coefficient is extracted from the spinor such that

\psi_1 = 2 \langle P_3 \bar{\Psi}^\dagger P_3 \rangle_S
\psi_2 = 2 \langle \mathbf{e}_1 \bar{P}_3 \bar{\Psi}^\dagger P_3 \rangle_S
\psi_3 = 2 \langle P_{3} \Psi_{{}_{ }} P_3 \rangle_S
\psi_4 = 2 \langle \mathbf{e}_1 \bar{P}_3 \Psi P_3 \rangle_S

This expansion is cleanly related with the colum spinor in the Weyl representation so that the Weyl spinor components are

\Psi \rightarrow \begin{pmatrix} \psi_1 \\\psi_2 \\ \psi_3 \\ \psi_4 \end{pmatrix}

Similarly, the spinor components in the Pauli-Dirac representation are calculated as

\Psi \rightarrow \begin{pmatrix} \psi_3 + \psi_1 \\ \psi_4+\psi_2 \\ \psi_3 -\psi_1 \\ \psi_4 -\psi_2 \end{pmatrix}

[edit] Current

The current is defined as

J = \Psi\Psi^\dagger,

which satisfies the continuity equation

\left\langle \bar{\partial} J \right\rangle_{S}=0

[edit] Second order Dirac equation

An application of the Dirac equation on itself leads to the second order Dirac equation

(-\partial \bar{\partial} + A \bar{A}) \Psi - i( 2e\left\langle A \bar{\partial} \right\rangle_S + eF) \Psi \mathbf{e}_3 = m^2 \Psi

[edit] Free particle solutions

[edit] Positive energy solutions

A solution for the free particle with momentum p = p^0 + \mathbf{p} and positive energy p0 > 0 is

\Psi = \sqrt{\frac{p}{m}} R(0) \exp(-i\left\langle p \bar{x}\right\rangle_S \mathbf{e}_3).

This solution is unimodular

\Psi \bar{\Psi} = 1

and the current resembles the classical proper velocity

u = \frac{p}{m}
J = \Psi {\Psi}^\dagger = \frac{p}{m}

[edit] Negative energy solutions

A solution for the free particle with negative energy and momentum p = -|p^0| - \mathbf{p} = - p^\prime is

\Psi = i\sqrt{\frac{p^\prime}{m}} R(0) \exp(i\left\langle p^\prime \bar{x}\right\rangle_S \mathbf{e}_3) ,

This solution is anti-unimodular

\Psi \bar{\Psi} = -1

and the current resembles the classical proper velocity u = \frac{p}{m}

J = \Psi {\Psi}^\dagger = -\frac{p}{m},

but with a remarkable feature: "the time runs backwards"

\frac{d t}{d \tau} = \left\langle \frac{p}{m} \right\rangle_S < 0

[edit] Dirac Lagrangian

The Dirac Lagrangian is

L = \langle i \partial \bar{\Psi}^\dagger \mathbf{e}_3 \bar{\Psi} - e A \bar{\Psi}^\dagger \bar{\Psi} -m \Psi \bar{\Psi} \rangle_0

[edit] See also

[edit] References

[edit] Textbooks

  • Baylis, William (2002). Electrodynamics: A Modern Geometric Approach (2th ed.). Birkhäuser. ISBN 0-8176-4025-8
  • W. E. Baylis, editor, Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering, Birkhäuser, Boston 1996.

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