Quick Calculation 3.2

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A First Course in String Theory

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Verify that the gauge transformation (3.10) are correctly summarized by (3.21).

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Eq. (3.21):

\displaystyle{ \begin{aligned} A_\nu' &= A_\nu + \partial_\nu \epsilon \\ \end{aligned} }

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\displaystyle{ \begin{aligned} \left( A_0', A_1', ... \right) &= \left( - \Phi + \frac{\partial \epsilon}{\partial x^0}, A^1 + \frac{\partial \epsilon}{\partial x^1}, ... \right) \\ \left( -\Phi', {A^1}', ... \right) &= \left( - \Phi + \frac{1}{c} \frac{\partial \epsilon}{\partial t}, A^1 + \frac{\partial \epsilon}{\partial x^1}, ... \right) \\ \end{aligned} }

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\displaystyle{ \begin{aligned} \Phi' &= \Phi - \frac{1}{c} \frac{\partial \epsilon}{\partial t} \\ \left( {A^1}', {A^2}', {A^3}' \right) &= \left( {A^1}, {A^2}, {A^3} \right) + \left( \frac{\partial}{\partial x^1}, \frac{\partial}{\partial x^2}, \frac{\partial}{\partial x^3} \right) \epsilon \\ \end{aligned} }

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Eq. (3.10):

\displaystyle{ \begin{aligned} \Phi' &= \Phi - \frac{1}{c} \frac{\partial \epsilon}{\partial t} \\ \vec A' &= \vec A + \nabla \epsilon \\ \end{aligned} }

— Me@2022-04-07 07:05:29 PM

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2022.04.07 Thursday (c) All rights reserved by ACHK