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Dirac spinor
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== Chiral basis == In the '''chiral''' (or '''Weyl''') representation of <math>\gamma^\mu</math>, the solution space for the Dirac equation can be parameterized by two‐component complex spinors <math>\xi_s</math> and <math>\eta_s</math>. The general Dirac spinor solutions in this representation are often written as<ref>{{Cite book |last=Peskin |first=Michael Edward |title=An introduction to quantum field theory |last2=Schroeder |first2=Daniel V. |date=2019 |publisher=CRC Press, Taylor & Francis Group |isbn=978-0-367-32056-0 |series=The advanced book program |location=Boca Raton London New York}}</ref><ref>{{Cite book |last=Schwartz |first=Matthew Dean |title=Quantum field theory and the standard model |date=2014 |publisher=Cambridge university press |isbn=978-1-107-03473-0 |location=New York}}</ref> <math>u_s(p) = \begin{pmatrix}\sqrt{p \cdot \sigma}\,\xi_s\\ \sqrt{p \cdot \bar\sigma}\,\xi_s\end{pmatrix}, \quad \quad v_s(p) = \begin{pmatrix}\sqrt{p \cdot \sigma}\,\eta_s\\ -\sqrt{p \cdot \bar\sigma}\,\eta_s\end{pmatrix},</math> where <math>\sigma^\mu = (I_2, \sigma^i),~ \bar\sigma^\mu = (I_2, -\sigma^i)</math> are '''Pauli 4-vectors''' and <math>\sqrt{\cdot}</math> denotes the Hermitian matrix square-root. In many practical calculations, it is convenient to choose <math>\mathbf{p}</math> along the <math>z</math> axis. With this choice, the contractions read as <math>p \cdot \sigma \equiv p_{\mu} \sigma^{\mu} = \begin{pmatrix} E - p_z & 0 \\ 0 & E + p_z \end{pmatrix}, \quad p \cdot \bar \sigma \equiv p_{\mu} \bar\sigma^{\mu} = \begin{pmatrix} E + p_z & 0 \\ 0 & E - p_z \end{pmatrix}.</math> Since the matrices are diagonal, their square roots are <math>\sqrt{p_{\mu} \sigma^{\mu}} = \begin{pmatrix} \sqrt{E - p_z} & 0 \\ 0 & \sqrt{E + p_z } \end{pmatrix}, \quad \sqrt{p_{\mu} \bar\sigma^{\mu}} = \begin{pmatrix} \sqrt{E + p_z} & 0 \\ 0 & \sqrt{E - p_z} \end{pmatrix}.</math> The most convenient choice for the two‐component spinors is: <math> \xi_{+\frac{1}{2}} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \eta_{+\frac{1}{2}} \quad \quad \xi_{-\frac{1}{2}} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \eta_{-\frac{1}{2}}.</math> Then the four independent solutions take the explicit forms <math>u_{+\frac{1}{2}}(p_z) = \begin{pmatrix}\sqrt{E - p_z}\,\\ 0 \\ \sqrt{E+p_z}\, \\ 0 \end{pmatrix}, \quad u_{-\frac{1}{2}}(p_z) = \begin{pmatrix}0\\ \sqrt{E+p_z}\,\\ 0 \\ \sqrt{E - p_z}\, \end{pmatrix},</math> <math> v_{+\frac{1}{2}}(p_z) = \begin{pmatrix}\sqrt{E-p_z}\,\\ 0 \\ -\sqrt{E+p_z}\, \\ 0 \end{pmatrix}, \quad v_{-\frac{1}{2}}(p_z) = \begin{pmatrix}0\\ \sqrt{E+p_z}\, \\ 0 \\ -\sqrt{E-p_z}\, \end{pmatrix}.</math> The conventional normalization conditions for these spinors are <math>\begin{align} \overline u_s (p) u_{s'} (p) &= 2m\delta_{ss'} & \overline u_s (p) v_{s'} (p) &= 0 \\ \overline v_s (p) v_{s'} (p) &= -2m\delta_{ss'} & \overline v_s (p) u_{s'} (p) &= 0 \end{align}</math> while the completeness (spin‐sum) relations are <math> \begin{align} \textstyle \sum_s \displaystyle u_s (p) \overline u_{s} (p) &= {p\!\!\!/} + m \\ \textstyle \sum_s \displaystyle v_s (p) \overline v_{s} (p) &= {p\!\!\!/} - m. \end{align} </math>
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