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So one detail I omitted from the question was that: $$ \psi_{sc}(k)=\frac{g+g I}{2\pi(k^2-p^2)} $$ Where: $$ I=\int^{\infty}_{-\infty}\psi(q)dq \space\space\space\space (1) $$ (I had used in arbitrary prescription in the original description of the problem, this is what I obtain before solving for $I$)$$\\$$ Using equation (1) we can solve for I, obtaining: ...


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The Klein-Gordon equation (($\partial_{\mu}\partial^{\mu} + m^2)\varphi=0$) that you have mentioned is only for free field $\phi$. Now the solution $$\varphi(\vec x,t)=e^{-ip\cdot x}$$ obeys well the free field condition $E^2=\vec{p}^2+m^2$. To verify this put the above solution in (($\partial_{\mu}\partial^{\mu} + m^2)\varphi=0$). ADDENDUM: say you have ...


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Fourier expansion is used as a change of basis method that makes our calculations simpler and more in context. Usually we are working with particles with precise momentum. Momentum is a far more useful quantity than position when doing experiments. Also the Feynman rules can be obtained in terms of momentum as well. Furthermore as ACuriousMind mentioned in ...


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So my question is, is Fourier Analysis essentially what String Theory is? Briefly, no. String theory "is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings." Fourier analysis "is the study of the way general functions may be represented or approximated by sums of ...


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Take the wave equation $$\nabla^2\vec{E} = \frac{1}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2},$$ and let $\vec{E}(\vec{r},t)$ be a solution. Indeed taking the real part $\Re(\vec{E}(\vec{r},t))$ yields the physical significant values. The initial values at $t = 0$ are $\Re(\vec{E}(\vec{r},0))$ and the problem arises here: this does not give you enough ...


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To obtain a limit $$f_k=\int^\infty_{-\infty}f(x)\exp\left(-2\pi i k x\right) dx$$ from a discrete Fourier transform, you must actually look what you mean by the formula $$f_k=\sum_{i=-N/2}^{N/2}f_i \exp\left(-2\pi i k x_i\right).\;\; (*)$$ This formula is taking samples of undetermined step in the $x$ direction and it will not converge to a Fourier ...


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The formula for $N$-DFT should be: $$\tilde{F}[k] = \sum_{n=0}^{N-1} \tilde{f}[n]\exp(-2\pi ikn/N),$$ where $\tilde{f}[n]$ is the discrete input and $\tilde{F}[k]$ is the discrete frequency output. One can optionally scale by $N^{-1/2}$. The indexes $n$ and $k$ are dimensionless. Usually $\tilde{f}[n]$ is obtained from $f(x)$ by sampling, which involves ...


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You're mistakingly assuming that a sequence must have exactly the same unit to be considered the approximation of a signal. In general, a sequence $f_k$ can be considered as an approximation of $f(x)$ if $f_k=\int^{x_{k+1}}_{x_k} f(x) dx $. Clearly the units differ here. Note that by this rule, $f_k=0$ is an approximation of $f(x)=0$, $c*f_k$ is an ...


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I) One mathematical problem is that the function $$\tag{1} f(q)~:=~ \frac{q\sin(rq)}{q^2+u^2}, \qquad q,r,u~>~0, $$ is not integrable $f\notin {\cal L}^{1}(\mathbb{R}_{+})$, because the integral over the absolute value of the integrand is infinite: $$\tag{2} \int_{\mathbb{R}_{+}} \! dq~|f(q)| ~=~\infty.$$ However it is still possible to define the ...


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Spectral Power – Image B (top right). The nature of images C and D is explained by the spectral power image, B. You can see from spectral power image that most energy is concentrated at low frequencies (the brighter middle part of the image). The higher frequencies (moving away from the centre) have less energy in them because there is less brightness in ...


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Rather than post a bunch of images, I'm going to violate protocol and suggest a look at the ImageMagick_Fourier help page on fourier transforms of images. They've got a lot of very nice plots of the Magnitude and Phase results for 2D FFTs and reconstruction from same. That should give you a pretty decent idea of what information each part of the FFT ...



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