# 狄拉克場

## 數學公式

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

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

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

${\displaystyle \{\psi _{a}({\textbf {x}}),\psi _{b}^{\dagger }({\textbf {y}})\}=\delta ^{(3)}({\textbf {x}}-{\textbf {y}})\delta _{ab},}$

${\displaystyle \{a_{\textbf {p}}^{r},a_{\textbf {q}}^{s\dagger }\}=\{b_{\textbf {p}}^{r},b_{\textbf {q}}^{s\dagger }\}=(2\pi )^{3}\delta ^{3}({\textbf {p}}-{\textbf {q}})\delta ^{rs},\,}$

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

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

${\displaystyle \psi (x)}$，我們可以建構出狄拉克場的費曼傳播子

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

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

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

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

## 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.