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2

First, expand the logarithm into its Taylor series: \begin{align} S&\equiv \sum_{n\ge 1}n\log(1-e^{-nx}) \\ &= -\sum_{n\ge 1} n \sum_{k\ge 1} \frac{e^{-knx}}{k} \\ &= -\sum_{k \ge 1} \frac{1}{k}\sum_{n\ge 1}n\,e^{-nkx} \end{align} To sum the inner series, differentiate the following identity with respect to $\beta$, $$\frac{1}{1-e^{-\beta}}-1=\... 1 Well, isn't it the case that any sequence a_n that vanishes as n \to\infty will give rise to the same pressure as the model with no field? Indeed, one can bound$$ e^{-\beta ha_nn^2} Z_{\Lambda_n,\beta,0} \leq Z_{\Lambda_n,\beta,ha_n} \leq e^{\beta ha_nn^2} Z_{\Lambda_n,\beta,0}, $$so that$$ -\beta h a_n + \frac1{n^2}\log Z_{\Lambda_n,\beta,0} \leq \...

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If you're looking for intuition about the relationship between forces and connections, I think the best you can do is to think long and hard about the following elementary example (adapted from Moriyasu's book 'Elementary Primer for Gauge Theory'): Consider a vector at position $x$. Call its length $f(x)$. We want to know how this length changes as we go ...

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See: John David Logan, “First Integrals in the Discrete Variational Calculus,” Æquationes Mathematicæ 9, no. 2 (June 1, 1973): 210–20. DOI: 10.1007/BF01832628.The intent of this paper is to show that first integrals of the discrete Euler equation can be determined explicitly by investigating the invariance properties of the discrete Lagrangian. The result ...

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I think you just need to read more widely to encounter a broad range of number systems being useful in at least theoretical physics. I'll hyperlink to discussions of applications, but just mention the number systems themselves, as some have multiple applications. There are uses for $p$-adic numbers, split-complex numbers, dual numbers, quaternions, split-...

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The short answer is that $\sin^2 (ax) /ax^2$ becomes increasingly localized at zero. The effective domain shrinks like $1/a$ while its value at zero is $a$. Moreover, $\int_{-\infty}^\infty \sin^2 (ax) /ax^2 = \pi$. The rest is math.

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On page 50, it says we note that for a one-dimensional manifold that is special Kahler, the Ricci scalar is related to the invariant coupling by $R+4=2\gamma_\text{inv}^2$ and we present a three-dimensional plot of the Ricci scalar in fig. 10 On page 55, it says for large $\psi$, the Ricci scalar of the moduli space differs from its limiting value by ...

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It may sound old but 0."An introduction to statistical physics- by A.J. Pointon" is a very handy book to absorb the concept of calculation over phase space from the very beginning. The book is suitable for a one semester course, designed for last year undergraduate and beginning graduate students. The exposition of this book is exceptionally clear. It ...

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Consider a real function $\;f(x)\;$ of the real variable $\;x\in\mathbb{R}\;$ for which \begin{align} f(x)\boldsymbol{=}0 \quad & \text{for any} \quad x\boldsymbol{\ne} x_{0} \quad \textbf{and} \tag{01a}\label{01a}\\ \mathcal{I}\boldsymbol{=}\!\!\!\!\int\limits_{\boldsymbol{x_{0}-\varepsilon}}^{\boldsymbol{x_{0}+\varepsilon}}\!\!\!f(x)\mathrm dx\...

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This may not fully answer your question, but there is a general theorem that the lowest order (ell) non-vanishing moment is independent of origin. Let's see how this helps. If the monopole moment does not vanish, then its value is independent of origin. This means that the dipole moment is not independent of origin. If we change the origin through \$x^i\...

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Perhaps the simplest argument is the following. Since the Legendre transformation should be $$L+H ~=~P_1\dot{Q}_1+ P_2\dot{Q}_2 ,\tag{1}$$ then $$\frac{\partial (L+H)}{\partial Q_1}~=~0. \tag{2}$$ Now the higher Lagrange equation reads $$\frac{\partial L}{\partial Q_1} - \frac{d}{dt} \frac{\partial L}{\partial \dot{Q}_1} + \frac{d^2}{dt^2} \frac{\partial ... 1 Hints: Eq. (2.9) expanded into more steps reads:$$ \begin{align} \frac{1}{k\ell}\{ {\rm tr}L^k,{\rm tr}L^{\ell} \} ~=~& \frac{1}{k\ell}{\rm tr}_1{\rm tr}_2\{ L_1^k,L_2^{\ell} \} \cr ~=~& {\rm tr}_1{\rm tr}_2\left(L_1^{k-1} \{ L_1,L_2 \} L_2^{\ell-1}\right) \cr \stackrel{(2.8)}{=}& {\rm tr}_1{\rm tr}_2\left(L_1^{k-1}( [r_{12}, L_1] - [r_{21}, ...

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