Dirac spinor

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Solutions to the Dirac equation for free-particles have the form of a plane-wave:

\psi = \omega e^{-i p \cdot x} \,

where ω is a four-component spinor (Dirac spinor) which is not a function of x.

This spinor can be written

\omega = \begin{bmatrix} \phi \\ \frac{\mathbf{\sigma \cdot p}}{E + m} \phi \end{bmatrix} \,
where
\phi \, is a two-spinor,
\sigma \, are the Pauli matrices,
E, m, p are the Energy, mass, and four-momentum of the particle respectively.

Contents

[edit] Derivation from Dirac equation

The Dirac equation has the form

\left(-i \alpha \cdot \nabla + \beta m \right) \psi = i \frac{\partial \psi}{\partial t} \,

In order to derive the form of the four-spinor ω we have to first note the value of the matrices α and β:

\alpha = \begin{bmatrix} \mathbf{0} & \mathbf{\sigma} \\ \mathbf{\sigma} & \mathbf{0} \end{bmatrix} \quad \quad \beta = \begin{bmatrix} \mathbf{I} & \mathbf{0} \\ \mathbf{0} & -\mathbf{I} \end{bmatrix} \,

These two 4x4 matrices are related to the Dirac gamma matrices. Note that 0 and I are 2x2 matrices here.

The next step is to look for solutions of the form,

\psi = \omega e^{-i p \cdot x},

While at the same time splitting ω into two two-spinors:

\omega = \begin{bmatrix} \phi \\ \chi \end{bmatrix} \,.

[edit] Results

Using all of the above information to plug into the Dirac equation results in

E \begin{bmatrix} \phi \\ \chi \end{bmatrix} = \begin{bmatrix} m \mathbf{I} & \mathbf{\sigma \cdot p} \\ \mathbf{\sigma \cdot p} & -m \mathbf{I} \end{bmatrix} \begin{bmatrix} \phi \\ \chi \end{bmatrix} \,.

That matrix equation is really two coupled equations:

  • \left(E - m \right) \phi = \left(\mathbf{\sigma \cdot p} \right) \chi \,
  • \left(E + m \right) \chi = \left(\mathbf{\sigma \cdot p} \right) \phi \,

Solve the 2nd equation for \chi \, and then one can then write

\omega = \begin{bmatrix} \phi \\ \chi \end{bmatrix} = \begin{bmatrix} \phi \\ \frac{\mathbf{\sigma \cdot p}}{E + m} \phi \end{bmatrix} \,

[edit] Details

[edit] Two-spinors

The most convenient definitions for the two-spinors are:

\phi^1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad \quad \phi^2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \,

and

\chi^1 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad \quad \chi^2 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \,

[edit] Pauli matrices

The Pauli matrices are

\sigma_1 = \begin{bmatrix} 0&1\\ 1&0 \end{bmatrix} \quad \quad \sigma_2 = \begin{bmatrix} 0&-i\\ i&0 \end{bmatrix} \quad \quad \sigma_3 = \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}

Using, these one can calculate:

\mathbf{\sigma \cdot p} = \sigma_1 p_1 + \sigma_2 p_2 + \sigma_3 p_3 = \begin{bmatrix} p_3 & p_1 - i p_2 \\ p_1 + i p_2 & - p_3 \end{bmatrix}

[edit] Four-spinor for particles

Particles are defined as having positive energy. The normalizatin for the four-spinor ω is chosen so that \omega^\dagger \omega = 2 E \,. These spinors are denoted as u:

u(\mathbf{p}, s) = \sqrt{E+m} \begin{bmatrix} \phi^{(s)}\\ \frac{\mathbf{\sigma} \cdot \mathbf{p} }{E+m} \phi^{(s)} \end{bmatrix} \,
where s = 1 or 2 (spin "up" or "down")

Explicitly,

u(p, 1) = \sqrt{E+m} \begin{bmatrix} 1\\ 0\\ \frac{p_3}{E+m} \\ \frac{p_1 + i p_2}{E+m} \end{bmatrix} \quad \mathrm{and} \quad u(p, 2) = \sqrt{E+m} \begin{bmatrix} 0\\ 1\\ \frac{p_1 - i p_2}{E+m} \\ \frac{-p_3}{E+m} \end{bmatrix}

[edit] Four-spinor for anti-particles

Anti-particles are defined as having negative energy anti particles

v(p,s) (\mathbf{p}) = \sqrt{E+m} \begin{bmatrix} \frac{-\mathbf{\sigma} \cdot \mathbf{p} }{E+m} \chi^{(s)}\\ \chi^{(s)} \end{bmatrix} \,

Explicitly,

v(p, 1) = \sqrt{E+m} \begin{bmatrix} \frac{-p_1 + i p_2}{E+m} \\ \frac{p_3}{E+m} \\ 0\\ 1 \end{bmatrix} \quad \mathrm{and} \quad v(p, 2) = \sqrt{E+m} \begin{bmatrix} \frac{-p_3}{E+m} \\ \frac{-p_1 - i p_2}{E+m} \\ 1\\ 0\\ \end{bmatrix}

[edit] Completeness relations

The completeness relations for the four-spinors u and v are

\sum_{s=1,2}{u^{(s)}_p \bar{u}^{(s)}_p} = p\!\!\!/ + m \,
\sum_{s=1,2}{v^{(s)}_p \bar{v}^{(s)}_p} = p\!\!\!/ - m \,
where
p\!\!\!/ = \gamma^\mu p_\mu \,      (see Feynman slash notation)
\bar{u} = u^{\dagger} \gamma^0 \,


[edit] See also

[edit] References

  • Aitchison, I.J.R.; A.J.G. Hey (Sept 2002). Gauge Theories in Particle Physics (3rd ed.). Institute of Physics Publishing. ISBN 0750308648. 

αβ ωω

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