狄拉克場

數學公式

$(i\gamma^{\mu} \partial_{\mu} - m) \psi(x) = 0.\,$

$\psi(x) = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2E_{p}}}\sum_{s} \left( a^{s}_{\textbf{p}}u^{s}(p)e^{-ip \cdot x}+b^{s \dagger}_{\textbf{p}}v^{s}(p)e^{ip \cdot x}\right).\,$

$a\,$$b\,$標示了旋量的指標，$s\,$表示自旋，s = +1/2或s=−1/2。前面係數中的能量是為了勞倫茲積分的協變性。由於$\psi(x)\,$可以視作一個算符，每個傅立葉基底的係數也必須是算符。因此，$a^{s}_{\textbf{p}}$以及$b^{s \dagger}_{\textbf{p}}$為作用子。這些算符的性質可以從這些場的性質中得知。 $\psi(x)\,$$\psi(y)^{\dagger}$遵守反對易關係：

$\{\psi_a(\textbf{x}),\psi_b^{\dagger}(\textbf{y})\} = \delta^{(3)}(\textbf{x}-\textbf{y})\delta_{ab},$

$\{a^{r}_{\textbf{p}},a^{s \dagger}_{\textbf{q}}\} = \{b^{r}_{\textbf{p}},b^{s \dagger}_{\textbf{q}}\}=(2 \pi)^{3} \delta^{3} (\textbf{p}-\textbf{q}) \delta^{rs},\,$

$\mathcal{L}_{D} = \bar{\psi}(i\gamma^{\mu} \partial_{\mu} - m)\psi\,$

$\mathcal{L}_{D} = \bar{\psi}_{a}(i\gamma^{\mu}_{ab} \partial_{\mu} - m\mathbb{I}_{ab})\psi_{b}\,$

$\psi(x)$，我們可以建構出狄拉克場的費曼傳播子

$D_{F}(x-y) = \langle 0| T(\psi(x) \bar{\psi}(y))| 0 \rangle$

$T(\psi(x) \bar{\psi}(y)) \ \stackrel{\mathrm{def}}{=}\ \theta(x^{0}-y^{0}) \psi(x) \bar{\psi}(y) - \theta(y^{0}-x^{0})\bar\psi(y) \psi(x)$

$D_{F}(x-y) = \int \frac{d^{4}p}{(2\pi)^{4}} \frac{i(p\!\!\!/ + m)}{p^{2}-m^{2}+i \epsilon}e^{-ip \cdot (x-y)}$

$\frac{i(p\!\!\!/ + m)}{p^{2}-m^{2}}$

參考資料

• Edwards, D. (1981). The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories, International J. of Theor. Phys., Vol. 20, No. 7.
• Peskin, M and Schroeder, D. (1995). An Introduction to Quantum Field Theory, Westview Press. (See pages 35-63.)
• Srednicki, Mark (2007). Quantum Field Theory, Cambridge University Press, ISBN 978-0521864497.
• Weinberg, Steven (1995). The Quantum Theory of Fields, (3 volumes) Cambridge University Press.