Käännösesimerkit
Nämä kenttäyhtälöt ovat ∂ μ ( ∂ L ∂ ( ∂ μ ψ ) ) = ∂ μ ( i ψ ¯ γ μ ) {\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)=\partial _{\mu }\left(i{\bar {\psi }}\gamma ^{\mu }\right)\,} ∂ L ∂ ψ = − e ψ ¯ γ μ A μ − m ψ ¯ {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \psi }}=-e{\bar {\psi }}\gamma _{\mu }A^{\mu }-m{\bar {\psi }}\,} Laittamalla nämä kaksi takaisin Eulerin-Lagrangen yhtälöön (2), jolloin saadaan i ∂ μ ψ ¯ γ μ + e ψ ¯ γ μ A μ + m ψ ¯ = 0 {\displaystyle i\partial _{\mu }{\bar {\psi }}\gamma ^{\mu }+e{\bar {\psi }}\gamma _{\mu }A^{\mu }+m{\bar {\psi }}=0\,} ja kompleksikonjugaatti i γ μ ∂ μ ψ − e γ μ A μ ψ − m ψ = 0. {\displaystyle i\gamma ^{\mu }\partial _{\mu }\psi -e\gamma _{\mu }A^{\mu }\psi -m\psi =0.\,} Jos keskimmäinen termi laitetaan oikealle puolelle, saadaan: Vasemmanpuoleinen on kuten alkuperäinen Diracin yhtälö ja oikeanpuoleinen on vuorovaikutus sähkömagneettisen kentän kanssa.
Using the field-theoretic Euler–Lagrange equation for ψ, The derivatives of the Lagrangian concerning ψ are ∂ μ ( ∂ L ∂ ( ∂ μ ψ ) ) = ∂ μ ( i ψ ¯ γ μ ) , {\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)=\partial _{\mu }\left(i{\bar {\psi }}\gamma ^{\mu }\right),} ∂ L ∂ ψ = − e ψ ¯ γ μ ( A μ + B μ ) − m ψ ¯ . {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \psi }}=-e{\bar {\psi }}\gamma _{\mu }(A^{\mu }+B^{\mu })-m{\bar {\psi }}.} Inserting these into (2) results in i ∂ μ ψ ¯ γ μ + e ψ ¯ γ μ ( A μ + B μ ) + m ψ ¯ = 0 , {\displaystyle i\partial _{\mu }{\bar {\psi }}\gamma ^{\mu }+e{\bar {\psi }}\gamma _{\mu }(A^{\mu }+B^{\mu })+m{\bar {\psi }}=0,} with Hermitian conjugate i γ μ ∂ μ ψ − e γ μ ( A μ + B μ ) ψ − m ψ = 0. {\displaystyle i\gamma ^{\mu }\partial _{\mu }\psi -e\gamma _{\mu }(A^{\mu }+B^{\mu })\psi -m\psi =0.} Bringing the middle term to the right-hand side yields The left-hand side is like the original Dirac equation, and the right-hand side is the interaction with the electromagnetic field.
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