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Consider a metal sheet placed in a uniform electrical field normal to the sheet surface. The electrons will be "dragged" by the electrical field and form an excess of negative charges on one side of the metal sheet and an excess of positive charges on the other side of the metal sheet. The excess charges produce an electrical field that cancels the applied ...

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This seems to be a simple matter of confusion regarding which variables are held constant. His notation $$\frac{\partial X}{\partial\tau}(\tau,\sigma_*)=0$$ is misleading. What he really means is $$\frac{\partial}{\partial\tau}\left(X(\sigma_*)\right)(\tau)=0$$ In other words, we fix $\sigma$ to be one of the end points and look at how it changes with ...

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I think physically it is very clear. If the string endpoint ($\sigma=\sigma_1$) is fixed, the variations must vanish there i.e. $\delta X(\tau,\sigma_1)=0$.

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I) Let us here phrase the problem in the context of some position operator $\hat{q}$ of QM for simplicity. The generalization to QFT can formally be achieved by replacing the position operator $\hat{q}$ with a quantum field $\hat{\psi}({\bf x})$. We know that the overlap with Minkowski (M) signature is given as a path integral ...

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