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Yes, the strategy is right. If one has a one-dimensional current $I$ in the thin wire, the volume density of the current is $$\vec J(\vec r) = I\cdot \delta^{(2)}(\vec r - \vec R_{\rm nearest}) \cdot \vec n$$ where $\vec R_{\rm nearest}$ is the point on the wire that is closest to the point $\vec r$. There are other ways to write the current but this is ...

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The delta function is not really a function, it is a distribution, In the strict sense both $\delta (x)$ and $e^{ikx}$ are not normalizable when $n=m$ One way to prove your equations is to use fourier transforms Using Placherels theorem the fourier transform $F([f(x)]k)$ for the function $f(x)$ is given by ...

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I) The Dirac delta distribution (and derivative thereof) in the dipole field $$\Phi ~=~\frac{1}{4\pi\varepsilon}\frac{\vec{p}\cdot \vec{r}}{r^3} \tag{1}$$ $$\Downarrow$$ $$\vec{E}~=~-\vec{\nabla}\Phi ~=~ \frac{1}{4\pi\varepsilon}\frac{3(\vec{p}\cdot \vec{r})\vec{r}-r^2\vec{p} }{r^5} -\frac{\vec{p}}{3\varepsilon}\delta^3(\vec{r}) \tag{2}$$ $$\Downarrow$$ ...

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"The divergence of any electric field created by any surface charge distribution would be zero." No it's not. Consider a charged conducting sphere with uniform surface charge density and a Gaussian sphere of radius greater than the original one. The electric field is diverging through the surface of the Gaussian sphere. So the divergence cannot be zero. ...

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When you write $n(x)=\psi^\dagger(x)\psi(x)$, you basically claim that $n(x)$ is a composite operator. Such naively defined composite operators in QFT suffer from UV-divergences, and this is essentially what you observe. In order to have a well-defined $n(x)$, you need to renormalize it, and the standard approach for free theories is to take a normal-ordered ...

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You are not missing anything: the commutator $[\psi(0),\psi^\dagger(0)]$ is ill defined. This is related to the fact that operators are actually distributions, not functions of $x$, so taking $x=0$ is meaningless.

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'Mathematical Physics' by Kusse and Westwig is just the thing you need. The fifth chapter is devoted to the Dirac-delta function. The book is fairly easy to understand and provides the proofs of the theorems that are stated in Arfken-Weber. After having read this, you can read the appendices I and II in Cohen-Tannoudji (Quantum Mechanics) on Fourier ...

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