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Questions tagged [integration]

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

147 questions with no upvoted or accepted answers
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8
votes
1answer
383 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\...
6
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1answer
150 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....
4
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1answer
50 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
votes
1answer
374 views

Massive versus Massless $\phi^4$ Sunset Diagram - does $\frac{1}{\epsilon^2}$ term vanish for $m=0$?

In a real scalar massive $\phi^4$-interacting theory consider the amputated sunset diagram. This is the integral out of Kleinert and Schulte-Frohlinde Critical Properties of $\phi^4$-Theories: The ...
4
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0answers
194 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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0answers
60 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
votes
3answers
775 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
votes
1answer
61 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
69 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
votes
1answer
149 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 ...
3
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0answers
627 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 $(-+++)$ ...
3
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0answers
109 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 ...
3
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0answers
103 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
31 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
votes
0answers
74 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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0answers
74 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
56 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
89 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
64 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
votes
1answer
81 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
186 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
386 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
795 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
133 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
77 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
172 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
862 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
735 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
618 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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0answers
362 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
votes
0answers
253 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 ...
1
vote
1answer
27 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 $B$. ...
1
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0answers
64 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
vote
0answers
37 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)...
1
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0answers
54 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=...
1
vote
1answer
99 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}-\...
1
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1answer
55 views

Integral of the divergence of a vector field multiplied by the component of another vector field

In Forces in Molecules by Richard Feynman (Phys. Rev. 56, 340 (1939)), eq. (5) implies that $$\int(\nabla\cdot \textbf{F})E_\mu^\alpha dv=-\int F_\mu(\nabla\cdot E_\mu^\alpha)dv,$$ being $\textbf{F}...
1
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1answer
51 views

Solving Lagrangian given initial and final coordinate

Consider a Lagrangian $$L=L\left(q, \dot{q}\right)$$ I can use the Euler-Lagrange equation to find an expression $$\ddot{q}=A\left(q,\dot{q}\right).$$ Let's assume that the equation can be ...
1
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0answers
35 views

What is the magic behind Sector Decomposition?

I have a question regarding Sector Decomposition, which is briefly introduced in this paper arXiv: 0803.4177. So far I played around with a toy example and applied the Sector Decomposition method to ...
1
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0answers
62 views

Integration by parts in dimensional regularisation

I have a question concerning integration by parts identities in dimensional regularisation. Appearently, almost every textbook about dimensional regularisation claims that $$\int d^Dl_1...d^Dl_L \...
1
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0answers
94 views

Explicit integration of the time-dependent Schrodinger equation when eigenvalues are unknown

Let's consider the hydrogen atom Hamiltonian $$H = - \frac{1}{2}\Delta - \frac{1}{r}$$ The solution for the corresponding time-dependent Schrodinger equation is the following: $$\psi = \psi (t = 0){...
1
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0answers
49 views

Closed form of Iterated integrals arising in Fredholm's integral equation solution in the context of Nonequilibrium Quantum field Theory?

While solving a Non-equilibrium quantum field theory problem I came across this class of $2n_{}^{\text{th}}$ order iterated integral : $$F(T_{}^{},T_{0}^{},\epsilon)=\int_{T_{0}^{}}^{T_{}^{}}dt_{1}^{}...
1
vote
0answers
37 views

Zero Temperature non relativistic fermionic propagator

As an assigmente i am trying to prove the following result: $$\hbar \int \frac{d\omega}{2\pi}\frac{e^{i\omega0^+}}{\hbar\omega-\zeta(\vec{k})+isgn(\omega)\eta}=i\theta(-\zeta(\vec{k}))$$ Where $\...
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0answers
99 views

is it true that $\lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x)$?

Is the following statement true? $$ \lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x) $$ where $\mathscr{P}$ is the Cauchy principal value. The above ...
1
vote
0answers
108 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 ...
1
vote
1answer
249 views

Renormalization of a Feynman diagram with zero bare mass

Consider the Feynman diagram below: in the case of $\phi^4$ theory where there is no bare mass: $$\mathcal{L}=\frac{1}{2} \partial_\mu \phi\partial^\mu\phi-\frac{\lambda}{4!} \phi^4$$ the ...
1
vote
0answers
196 views

Computing Feynman propagator with momentum-dependent form factor

Can anyone help me to solve the following problem? I want to compute the integral \begin{equation}\label{eq:q0} \int_{-\infty}^{+\infty} dq_0 \, \frac{f(q^2)\,\exp[-i q_0(t_1-t_2)]}{[q_0-(|\vec{q}|-...
1
vote
1answer
271 views

Hamiltonian for single-mode field in cavity

Given that the Hamiltonian for a single-mode field is $$ H = \frac{1}{2}\int dV \left[ \epsilon_0E_x^2(z,t) + \mu_0^{-1}B_y^2(z,t) \right],$$ with \begin{align} E_x(z,t) =& \sqrt{\frac{2\omega^2}{...