معادله دیراک

${\displaystyle H_{D}\Psi =E_{D}\Psi }$

${\displaystyle H_{D}=c\;{\vec {\alpha }}\cdot {\vec {p}}+\beta \;mc^{2}}$

${\displaystyle {\vec {\alpha }}^{\dagger }={\vec {\alpha }}\quad ,\quad \beta ^{\dagger }=\beta }$

${\displaystyle (H_{D})^{2}=(c\;p)^{2}+(m\;c^{2})^{2}}$

${\displaystyle (\alpha ^{i})^{2}=1\quad ,\quad \beta ^{2}=1}$

${\displaystyle \lbrace \alpha ^{i},\beta ^{\;j}\rbrace =2\;\delta ^{ij}1}$

${\displaystyle (\gamma ^{0})=\beta \quad ,\quad \gamma ^{j}=\beta \;\alpha ^{j}}$

${\displaystyle {\mathcal {L}}(x)={\overline {\psi (x)}}(\mathrm {i} {\overleftrightarrow {\partial \!\!\!/}}-m)\psi (x)}$

${\displaystyle {\overline {\psi (x)}}=\psi ^{\dagger }(x)\gamma ^{0}\quad ,\quad {\overleftrightarrow {\partial }}\equiv {\frac {{\overrightarrow {\partial }}_{\mu }-{\overleftarrow {\partial }}_{\mu }}{2}}}$

${\displaystyle \partial _{\mu }{\dfrac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\overline {\psi (x)}})}}-{\dfrac {\partial {\mathcal {L}}}{\partial {\overline {\psi (x)}}}}=0}$

${\displaystyle {\mathcal {L}}={\overline {\psi (x)}}{\dfrac {\mathrm {i} }{2}}\gamma ^{\mu }\partial _{\mu }\psi -m{\overline {\psi (x)}}\psi (x)-{\dfrac {\mathrm {i} }{2}}(\gamma ^{\mu }\partial _{\mu }{\overline {\psi (x)}})\psi (x)}$

${\displaystyle \partial _{\mu }[-{\dfrac {\mathrm {i} }{2}}\gamma ^{\mu }\psi ]-[{\dfrac {\mathrm {i} }{2}}\gamma ^{\mu }\partial _{\mu }\psi -m\psi ]=0}$

${\displaystyle -\mathrm {i} \gamma ^{\mu }\partial _{\mu }\psi +m\psi =0\Rightarrow }$

${\displaystyle (\mathrm {i} \partial \!\!\!/-m)\psi (x)=0}$

• Dirac, P.A.M. , Principles of Quantum Mechanics, 4th edition (Clarendon, 1982)
• Shankar, R. , Principles of Quantum Mechanics, 2nd edition (Plenum, 1994)
• Bjorken, J D & Drell, S, Relativistic Quantum mechanics
• Thaller, B. , The Dirac Equation, Texts and Monographs in Physics (Springer, 1992)
• Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley. ISBN 0-201-06710-2.