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Notation in this answer: In this answer, let $z,z^{\ast}$ denote two independent complex Grassmann-odd numbers. Let $\overline{z}$ denote the complex conjugate of $z$. With this notation OP's Grassmann-odd/fermionic coherent state path integral reads $$\langle\lambda^{\ast}|e^{-i\hat{H}}|\eta\rangle ~=~ \int_{\xi(0)=\eta}^{\bar{\xi}(1)=\lambda^{\ast}} \!{\...


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Consider a rectangular Amperian loop of sides $\Delta w$, $\Delta h$ (as per your right hand diagram), symmetrically placed with respect to the interface. Faraday's law yields $$ (E_{1t} - E_{2t})\Delta w + \frac{1}{2}(E_{1n} + E_{2n})\Delta h - \frac{1}{2}(E_{1n} + E_{2n})\Delta h = -\frac{\partial B}{\partial t}\Delta w\Delta h,$$ where $E_{1n}$ and $E_{...


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Now velocity of sound in solid is significantly greater than air, so solid is rarer medium compared to air, [...] Am i missing something? Yes, you are missing something. You are correct, that speed of sound in a solid is significantly greater than in air. But you are wrong with the densities. A solid is certainly denser, not rarer than air. The ...


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The integral form of Maxwell's equation, i.e., Faraday's law, where the flux and emf are calculated over a surface ${\mathcal S}$ and its boundary contour ${\mathcal{L}=\partial \mathcal S}$, resp., is $$\oint_{\mathcal{L}} \mathbf{E}\cdot d{\mathbf{l}} = -\frac{d}{dt}\iint_{\mathcal S} \mathbf{B}\cdot d\mathbf{s} \tag {1}$$ This $(1)$ is true for any ...


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About symmetry breaking Some remarks that may clarify some of your misconceptions: The finite-volume magnetization density $m_\Lambda^\#(\beta,h)$ is not odd in $h$ in general. Only the limiting quantity (as $\Lambda\uparrow\mathbb{Z}^d$) is odd. Exceptions for which the result does hold for finite systems are the cases of free and periodic boundary ...


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