All Questions
24 questions
2votes
1answer
65views
Computing position two-point function and Fourier transform of $p^4 \ln p$
I am computing a two-point correlator in 4D Euclidean space and I am struggling with one particular term. I have found that in momentum space my correlator goes as $$\langle \mathcal{O}(p)\mathcal{O}(...
1vote
0answers
97views
A 1d Feynman integral: How to compute?
I am trying to evaluate the following integral: $$ I_{n_1,n_2,\alpha} \,=\, \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \, \bigg(\,{ \frac{e^{...
- 604
6votes
0answers
127views
Fourier transform of Feynman Integral
In Nastase, Introduction to AdS/CFT, the first chapter talks a little about the star-triangle duality. In fact, it was claimed that the Fourier Transform of a Feynman-like diagram in position space in ...
- 998
0votes
0answers
103views
Fourier transform of Green function using residue theroem
I want to compute the Fourier transform of a Green function in $k$-space : $$ G^R_{n,m}(\omega)=\int_0^{2\pi}\frac{dk}{2\pi}\frac{e^{ik(n-m)}}{\omega+i\eta-\epsilon_k} $$ By substituting $\omega$ and ...
- 101
0votes
1answer
73views
How to integrate a function multiplied for a sign function?
I am studying QFT and I found this integral on my lecture notes (for the context: we're trying to show that the covariant commutation relations are Lorentz invariant) $$∫\frac{d^{3}p dp_{0}}{(2\pi)^{3}...
- 571
7votes
1answer
1kviews
Integration of Laplacian by parts
I'm trying to solve assignment (1.5) in Bellan's "Fundamentals of Plasma Physics" using Fourier transforms, but I'm stuck integrating the Laplacian. Here's the problem: Equation (1.5) is ...
0votes
1answer
62views
How to deduce the energy of a pair of vortices the classical XY model?
Consider a pair of oppositely charged vortices with unit strength, we estimate the energy of a pair of vortices as: $$ E_{\text {pair }}-E_{0} \cong \frac{J}{2} \int d^{2} r(\nabla \theta)^{2}=\frac{J}...
- 103
3votes
2answers
619views
Fourier transform in Minkowski space
Recently, I encountered a difficulty in proving the equation, $$\int \mathrm d^4x\, \frac{e^{-ipx}}{x^4} =\pi^2 \ln(p^2+i\epsilon)\quad .$$ Here, $x$ is the coordinate, $p$ is the momentum in ...
- 127
1vote
1answer
191views
Fourier transform of linear response function
I was studying Linear Response Theory from 'A modern course in statistical physics' by Reichl, and some doubts came up. The response function is defined as $$<\alpha(t)>_{F} = \int_{-\infty}^{+\...
- 653
1vote
0answers
82views
Matching Two Point Function in momentum space using spherical coordinate
Background of the problem: The problem I am currently struggling is related to the momentum representation of Fourier transform. Briefly speaking, the integral in Minkowski under Cartesian coordinate ...
1vote
1answer
168views
A Oscillatory integral in light-cone coordinates
I am trying to evaluate an integral in light-cone coordinates Where light-cone coordinates in 1+1D are defined by $x^+=\frac{x^0+x^1}{\sqrt 2}$ and $x^-=\frac{x^0-x^1}{\sqrt 2}$. The integral that I ...
- 186
1vote
1answer
521views
Fourier Transform of $1/k^4$
I am dealing with a higher derivative theory problem and I have to perform the following integral, \begin{equation} \int \dfrac{d^3k}{(2\pi)^3}\dfrac{e^{i{\bf k}\cdot {\bf r}}}{k^4} \end{equation} ...
1vote
1answer
141views
4-dimensional Fourier transform of $(k\cdot v)^{-1}$
I have been trying to compute, without much success, the following Fourier transform in 4-dimensional Minkowski space $$ I=\frac{1}{(2\pi)^4}\int d^4 k \,\frac{e^{ik\cdot x}}{k\cdot v}, $$ where $v^\...
0votes
2answers
112views
What happens if I change the integration limits of the Fourier transform of $1$?
The Fourier transform of $1$ is the (one-dimensional) Dirac delta function: $$\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dp\ e^{-i p x}. \tag{1}$$ Now I would like to replace the RHS with: $$\...
- 1,753
3votes
3answers
892views
Peskin & Schroeder: Free particle propagation
In Peskin & Schroeder Ch. 2, p. 14, in proving that the NRQM propagation amplitude for a free particle is nonzero everywhere, they move from \begin{equation} U(t)~=~ \frac{1}{(2\pi)^3} \int d^3p \...
- 33
