Questions tagged [integration]

For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.

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8
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478 views

Cutoff-Scheme Renormalization and Order of Integration in QFT

The following is the result of Fubini's Theorem, describing when you can replace a double integral with an iterated integral safely: For a set $X \times Y \subset \mathbb{R}^2$, if $\iint |f(x,y)| d(...
8
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0answers
1k views

Gaussian Integrals : Functional determinant expressed as a trace

Be $A_{ij}$ a symmetric matrix. Then I can easily write $$ \int \exp\left(-\frac{1}{2}\sum_{i,j}x_i A_{ij} x_j+\sum_{i} B_i x_i\right)\; d^nx= \sqrt{(2\pi)^n}\exp\left\{-\frac{1}{2}\mathrm{Tr}\log A\...
7
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1answer
189 views

Integration of Differential Forms

I want to understand what it actually means to integrate a differential form on a manifold. Being a mathematician, the explanation I always get is that they simply follow the right transformation rule....
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269 views

Feynman rules for this perturbative expansion in Grassmann variables

I'm given the integral $$ Z[w] = \frac{1}{ (2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \overline{\theta}_i d \theta_i \exp \left( - \overline{\theta}_i \partial_j w_i (x) \theta_j - \frac{1}{2} w_i(x) ...
4
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1answer
66 views

What are the scalar equations for velocity and displacement if acceleration obeys the inverse-square law?

In basic high school physics/calculus you learn that you can formulate equations for velocity and displacement under constant acceleration as: $a(t) = a_0$ $v(t) = a_0t + v_0$ $x(t) = \frac{1}{2}...
4
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0answers
65 views

Inserting a trace property into a divergent loop integral - what exactly is being done here?

I'm reading through "H. Kleinert and V. Schulte-Frohlinde" notes for "Critical Properties of $\phi^{4}$-Theories", and I've reached this point in the lecture notes: $\ $ $\ $ The trace property ...
4
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3answers
802 views

Finding the illuminance from a triangular light source

Since most light sources in games are point-like, it's pretty difficult to approximate area light sources with point sources. As triangles are a universal form to represent 3D models (thus area light ...
3
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0answers
82 views

Why can I not asymptotically expand a Feynman integral this way?

I would like to asymptotically expand a series of Feynman diagrams in Euclidean space, and as a toy I started with the following integral, for which I know the full solution in $4d$ ($\omega \to 2$): ...
3
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1answer
100 views

Analytical continuation in QFT

My question is quite basic and generic. It is known that scaleless integrals that appear in QFT such as $\int \frac{d^dk}{(k^2)^2} = \frac{1}{\epsilon_{\mathrm{UV}}} - \frac{1}{\epsilon_{\mathrm{IR}}} ...
3
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1answer
76 views

Regulating a divergent integral in QED

When we try to regulate a divergent integral, we introduce another parameter, say $\lambda$ and then compute the integral. We finally take a limit (either $\lambda \rightarrow 0, \infty $) to restore ...
3
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0answers
70 views

How to write integral to calculate the area where a field is between two limits?

I would like to express the following quantity in a mathematical form, but cannot think how I would write the integral. "The area on the surface of a cylinder where the magnetic field is between two ...
3
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1answer
206 views

Density of states (DOS) integral when surface is not closed

According to the density of states (DOS) formula $$\rho(\varepsilon)\propto \int_{\varepsilon=\text{const}}\frac{dS}{|\nabla_k \varepsilon_k|}.$$ Since there is an integral on the constant energy ...
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652 views

Propagator in massless scalar field theory

Suppose we have the following Lagrangian: $\mathcal{L} = \frac{1}{2} \phi \Box \phi + V(\phi)$, where $\Box = \partial _ {\mu} \partial ^ {\mu}$ and $V$ is the interaction term. We use the $(-+++)$ ...
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113 views

Coordinate Systems in Loop Integrals

Let us consider a two-point two-loop integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f(k _ {1}, k _ {2}, p)$, where $k _ {1}$, $k _ {2}$ and $p$ are four-dimensional vectors in ...
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109 views

The question about MTW 4-momentum integral expression and lorentz nature

In section 5.8 of Misner, Thorne, and Wheeler's "Gravitation" there is a proof that 4-momentum determined as $$ \tag 1 p^{\mu} = \int T^{\mu 0}\,\mathrm{d}^{3}\mathbf r , \quad \partial^{\mu}T_{\mu \...
2
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0answers
90 views

Integral in direct calculation of anomalies

I am trying to follow Weinberg's triangle diagram calculations in section 22.3 of volume II of The Quantum Theory of Fields. He reduces the calculation to evaluating the integral \begin{equation} \...
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56 views

What does it mean to integrate with respect to matrices?

In Random matrix theory, the following definition of a partition function for an ensemble is common. $$Z=\int dM \exp [-N Tr(M^2)]$$ where $M$ is a Random matrix of dimension $N \times N$. In general,...
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98 views

Regarding Rayleigh-Sommerfeld Diffraction Integral

While studying the Rayleigh-Sommerfeld diffraction formula I get the standard result for the following integration given at serial no. 11 under section 8.421 of "Table of Integrals" by Gradshteyn and ...
2
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0answers
34 views

Total amount of diffusely reflected light off of a sphere?

I have a numerical simulation that uses ray tracing to calculate the total amount of light picked up by a sensor, after diffusely reflecting off of an object. To validate this simulation, I'd like to ...
2
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0answers
96 views

Can we do one-loop integrals in the unitary gauge?

$\hspace{5cm}$ Imagine we want ot compute one of the diagrams for the self-energy of the quark $u$, with external momentum $p$. Inside the loop, we would have a $W^+$ and a $d$-quark propagator, with ...
2
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1answer
175 views

Regularising the Green's function in 2D

The Green's function for the 2D Helmholtz equation satisfies the following equation: $$(\nabla^2+k_0^2+\mathrm{i}\eta)\,{\mathsf{G}}_{2\mathrm{D}}(\mathbf{r}-\mathbf{r}',k_o)=\delta^{(2)}(\mathbf{r}-\...
2
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0answers
110 views

Integrating to find net pressure and force on a sphere by a fluid

How would I go about solving for the net force on a sphere exerted by air pressure as a result of pressure being dependent only on the vertical position of the ball? When the sphere it is placed in a ...
2
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0answers
58 views

Polylogarithmic integrals

NB: I was sent here from Math.SE, stating that polylog integrals are more common in physics and someone here might have an answer. I have asked in Phys.SE chat whether it was okay to post here but no ...
2
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0answers
94 views

An involved Feynman integral

Working with QCD, I have found the following integral from Feynman diagrams to solve $$ I(p)=\int\frac{d^4p_1}{(2\pi)^4}\int\frac{d^4p_2}{(2\pi)^4}\frac{1}{p_2^2-m_0^2} \left(\frac{p\cdot p_1-p_1\cdot ...
2
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0answers
66 views

Vector potential of a partially-known magnetic field

let's consider a three-dimensional space permeated by a known magnetic field $\vec{B}$. Let's consider in this space a topologically spherical surface $\mathcal{S}$ centred in the origin. I put a ...
2
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1answer
92 views

Unusual Feynman Integrals at Two-Loops

I'm studying a very particular conformal field theory where unusual Feynman integrals appear when I'm trying to evaluate a two-loop correlator (in position space). These integrals are on the form $\...
2
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0answers
125 views

Integrals of the form $\int\frac{d^Dk}{(2\pi)^D} \frac{1}{k^{2n}}$ in $D=4-2\varepsilon$ dimensions?

In a massless theory we often get integrals of the form $$\int\frac{d^Dk}{(2\pi)^D} \frac{1}{k^{2n}} \tag{*}$$ where $D=4-2\varepsilon$. I have tried to calculate this in two ways in Minkowski space ...
2
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0answers
226 views

Jacobian in path-integral

I am reading Inomata's "Alternative Exact-Path-Integral Treatment of the Hydrogen Atom" and I think I've worked too long because I got stuck near the end and cant for the life of me figure it out. ...
2
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0answers
397 views

Torque of a distributed load: Integrating with dF vs using resultant force

We have a rod that rotates around an fixed point A, which coincides with the end of the rod. The mass distribution along the rod is uniform. We know that the torque generated by the force field at ...
2
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0answers
939 views

How do I calculate the overlap integral of H2?

I'm given a problem where I need to "evaluate the overlap integral for two 1s orbitals as a function of interatomic spacing, R". This is what I think I need to do: $$S = \int \psi^*_a (\vec{r})\...
2
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0answers
143 views

Calculate expectation value of plane wave

I'm trying to calculate: $$<\Phi_{n'l'm'}| e^{-ib\cdot r}|\Phi_{nlm}>,$$ where the hydrogen wave functions are $\Phi_{nlm}$ and $\Phi_{n'l'm'}$. If I use the Rayleigh plane-wave expansion: $$...
2
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0answers
79 views

Passing from a discrete summation to a continous integal

I'm having trouble understanding the math behind a step in an explanation of BCS theory. At one point the superconductor gap $\Delta$ is defined as \begin{equation} 1 = V \sum_q \frac{1}{\xi_q^2+\...
2
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0answers
186 views

How to write this functional Jacobian?

I'm trying to find the correct Jacobian for an action that is invariant under the transform $X(u)\rightarrow Y(X(u))$. It involves the functional jacobian: $$\Omega[X] = \det\left( \partial_n X^\mu(...
2
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0answers
918 views

The Feynman propagator and the $i\epsilon$ prescription

The Feynman propagator is usually represented in the i-epsilon form and texts solve the integral in this form (as opposed to doing the Feynman (time-ordered) contour on the real axis). Restricting ...
2
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0answers
848 views

Integration by parts with Dirac Delta function

I am having some hard time trying to understand the following "heuristic" integral, involving integration by parts with the Dirac's Delta. We start with the following relation $$ f(x) = \int_{-\infty}^...
2
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0answers
663 views

An integral involving the Bose-Einstein distribution

I'm trying to reproduce the following calculation from the book by Fetter and Walecka (eq. 55.37 and following ones), which represents the temperature dependance of the non-condensate part of a weakly-...
2
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3answers
561 views

Limits of Integration Trig, Mag Field Infinite Length Wire

I don't understand how the limits of integration should be defined when doing basic integrals of trig functions. It seems like it's an arbitrary decision, I don't understand it. Here's the set up: ...
2
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0answers
384 views

Shifting the integration variable in loop integrals

We know that, in four dimensions, shifting the integration variables is valid only for convergent and logarithmically divergent integrals. If we employ a hard cutoff $\Lambda$, is it permissible to ...
2
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0answers
256 views

Finding the moments of the Boltzmann/Gibbs Distribution

I am trying to compute the moments of the Boltzmann distribution using a moment generating function, by taking the Fourier transform of the distribution and then taking derivatives to find the ...
2
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0answers
323 views

Why does this integral come out imaginary?

Im working through Zee and I'm having a little trouble with some integrals. I'm trying to reproduce the analogue of the inverse square law for a 2+1 D universe and I figured I could start with the ...
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0answers
30 views

Product over integral representations and steepest descents

Suppose we have a product of Dirac or Heaviside functions in the context of a spin model and we use an integral representation to express these in order to do some manipulations, more specifically ...
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0answers
30 views

How to calculate the total gravitational potential energy of a vertical object (do we use integration?)

Hello I was reading another question asked by zach466920, and when he was trying to calculate the total GPE of a water 'tower', he used this explanation: He basically used integration to calculate ...
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0answers
30 views

What is the Fourier transform of this expression in $2\omega$ dimensions?

I would like to perform the following Fourier transform in $2\omega$ (Euclidean) dimensions: $$A(x_1,q) = \int d^{2\omega} p_1\ e^{i p_1 \cdot x_1} \frac{\delta^{(1)}(v \cdot (p_1 + q))}{p_1^2 (p_1^2 ...
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0answers
56 views

Infrared divergency in the correction to the electron vertex Function in QED/ Question to Schwartz "Quantum Field Theory and the Standard Model

I'm currently studying Mathew Schwartz's "Quantum Field Theory and the Standard Model". In the chapter on Infrared divergences (Page 359 (7th edition, hard cover)) he calculates the infrared divergent ...
1
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1answer
58 views

Evaluating integral for Friedel oscillation using branch cuts

I am finding some difficulties understanding the following problem. I have the following logarithm for which I have to identify branch cuts: $$\lim_{\epsilon\rightarrow0}\ln{\frac{(p+2p_F)^2+\epsilon^...
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0answers
39 views

Why does integrability imply compatibility?

In mechanics, we have the so called compatibility conditions, which quarantee that when a body deforms, the strains are "compatible" in such a way to no discontinuities or gaps for inside the body as ...
1
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1answer
34 views

Torque experienced by a coplanar loop of current in a uniform magnetic field

There are a lot of posts on this already, but apparent all of them just consider some special case. I am now struggling with this more general case. Let there be a magnetic field with strength $B$. ...
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0answers
105 views

Can I use dimensional regularization with this integral?

I would like to extract the divergence of this integral in 4d Euclidean space: $$\int d^4z \frac{1}{(x-z)^4}\tag{1}$$ This divergence is expected to cancel with other divergences, which I got using ...
1
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0answers
46 views

Neutron mass fraction evolution: approximation

In Mukhanov's "Physical foundations of cosmology" on page 102 the author considers an equation for the evolution of the neutron mass fraction $X_{n}\equiv n_{n}/(n_{p}+n_{n})$: $$ \tag 1 \dot{X}_{n}(t)...
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0answers
59 views

How to evaluate the period of a particle in a system with potential energy $U=-U_0/\cosh^2(\alpha x)$?

I am working through the textbook "Mechanics", from the series "Course of Theoretical Physics " by Landau and Lifshitz. In Chapter 3, where the authors talk about integrating the equation of motion $E=...