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Display information for equation id:math.2662.31 on revision:2662
* Page found: Klein Gordon im (Vektor)Potential, Eichinvarianz (eq math.2662.31)
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Hash: 3697350916e03c9fd438ca58c8a7a958
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\begin{align}
& {{{\underline{D}}}_{\varphi }}=\underline{\nabla }+{{{\underline{f}}}_{\varphi }}\left( \underline{x},t \right)\quad \leftrightarrow \quad \underline{D}=\underline{\nabla }+{{{\underline{f}}}_{\varphi }}\left( \underline{x},t \right)+\mathfrak{i} \nabla \varphi \left( \underline{x},t \right) \\
& {{D}_{0}}={{\partial }_{t}}+{{g}_{\varphi }}\left( \underline{x},t \right)\quad \leftrightarrow \quad {{D}^{0}}={{\partial }_{t}}+{{g}_{\varphi }}\left( \underline{x},t \right)+\mathfrak{i} {{\partial }_{t}}\varphi \left( \underline{x},t \right) \\
\end{align}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><msub><mrow data-mjx-texclass="ORD"><munder><mi>D</mi><mo>_</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mi>φ</mi></mrow></msub><mo>=</mo><mrow data-mjx-texclass="ORD"><munder><mi mathvariant="normal">∇</mi><mo>_</mo></munder></mrow><mo>+</mo><msub><mrow data-mjx-texclass="ORD"><munder><mi>f</mi><mo>_</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mi>φ</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><munder><mi>x</mi><mo>_</mo></munder></mrow><mo>,</mo><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mspace width="1em"></mspace><mo>↔</mo><mspace width="1em"></mspace><mrow data-mjx-texclass="ORD"><munder><mi>D</mi><mo>_</mo></munder></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><munder><mi mathvariant="normal">∇</mi><mo>_</mo></munder></mrow><mo>+</mo><msub><mrow data-mjx-texclass="ORD"><munder><mi>f</mi><mo>_</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mi>φ</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><munder><mi>x</mi><mo>_</mo></munder></mrow><mo>,</mo><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="fraktur">i</mi></mrow></mrow><mi mathvariant="normal">∇</mi><mi>φ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><munder><mi>x</mi><mo>_</mo></munder></mrow><mo>,</mo><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><msub><mi>D</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo>=</mo><msub><mi>∂</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mo>+</mo><msub><mi>g</mi><mrow data-mjx-texclass="ORD"><mi>φ</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><munder><mi>x</mi><mo>_</mo></munder></mrow><mo>,</mo><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mspace width="1em"></mspace><mo>↔</mo><mspace width="1em"></mspace><msup><mi>D</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo>=</mo><msub><mi>∂</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mo>+</mo><msub><mi>g</mi><mrow data-mjx-texclass="ORD"><mi>φ</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><munder><mi>x</mi><mo>_</mo></munder></mrow><mo>,</mo><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="fraktur">i</mi></mrow></mrow><msub><mi>∂</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mi>φ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><munder><mi>x</mi><mo>_</mo></munder></mrow><mo>,</mo><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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