Questions tagged [integration]
For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.
1,275
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On the convergence of an Euclidean path integral in 0+0D
Suppose we have an integral $$Z(\lambda) = \frac{1}{\sqrt{2\pi}}\int^{+\infty}_{-\infty} dx e^{-\frac{x^2}{2!}-\frac{\lambda}{4!}x^4}.$$ To my knowledge this is a possible integral which can arise in ...
- 183
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votes
1
answer
63
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Electrostatic potential of finite charged wire
So I was trying to find the electric potential at any point $\boldsymbol{x}$ of a charged wire of length $L$ at the $z$ axis, from $-L/2$ to $L/2$, and I had to write it down in terms of elliptic ...
1
vote
0
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26
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How to derive elliptic integral of the first kind from $\int_{-\pi}^{\pi}\frac{1}{\sqrt{(\eta^{2}/2 - 3/2 -\cos 2\theta)^2+\cos^2\theta}}\ d\theta$ [migrated]
Spent several days already trying to figure out how to convert the given integral to this: $$\frac{8}{\sqrt{(\eta-1)^3(\eta+3)}}K\left(\sqrt{\frac{16\eta}{(\eta-1)^3(\eta+3)}}\right)$$
where
$$
K\left(...
- 11
1
vote
0
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Integration by parts on generic tensors
I try to rephrase here a my question (https://math.stackexchange.com/q/4661784/), explaining more specifically the case.
Given a lagrangian $L=L(\theta_{\mu\nu},\phi)$ , the conserved Noether current ...
- 21
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0
answers
26
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Integral in $\mathbf{k}$-space regarding excitonic physics
I have an integration for the expectation value of the kinetic energy in the following form, in regard to excitons.$$ \int d^2 \mathbf{k}_e d^2 \mathbf{k}_h\left[\varepsilon_e\left(\mathbf{k}_e\right)+...
- 55
0
votes
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51
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Variation of Action and Border Terms
I need to compute the following very general (piece of) variation:
\begin{equation}
\int d^4x \delta (\sqrt{-g} R ) f
\tag{1}
\end{equation}
where $R$ is Ricci scalar and $f$ a generic scalar ...
- 21
3
votes
1
answer
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How to integrate out the Goldstone phase in effective Ginzburg–Landau (GL) action for BCS?
In page 293 of Altland and Simons' "Condensed Matter Field Theory", just above equation (6.38), in the process of deriving the London equations from the BCS path integral, the authors say, &...
2
votes
1
answer
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Riemann-Lebesgue lemma in Faddeev-Kulish approach
I am learning about the established formalism used in the literature of IR divergences and dressed states, and I invariably come across an argument of the following form when evaluating a (photon) ...
- 313
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votes
1
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48
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Dot product of displacement vector with gradient [closed]
I have an integral like Eq. E.21 in Giamarchi's book (Appendix E, Quantum Physics in 1-D) :
$I=\int d^2R \int d^2r (r\cdot\nabla_R \phi(R))^2 e^{-f(r)}$
where, $r=r_1-r_2, R=\dfrac{r_1+r_2}{2}.$
How ...
- 427
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0
answers
29
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Taylor expansion in momentum integral
On Ashok Das' book "Finite temperature field theory", page 21, the book introduces the thermal mass correction to scalar field.
$$
\begin{aligned}
\Delta m^2 & =\Delta m_0^2+\Delta m_T^2 ...
- 1,133
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votes
1
answer
71
views
Doubt on 3d analog on gaussian integral for QFT
It is a well known fact that $$\int_{-\infty}^{\infty}dx e^{-x^2}=\sqrt{\pi}.$$ These types of integrals are commonly encountered in the study of Quantum Mechanics and Quantum Field Theory. I am ...
- 365
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votes
2
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47
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Work-Energy Theorem for a path that is not smooth
In the analysis of Newtonian Mechanics for a single particle, we come across the definition of work and also the Work-Kinetic Energy theorem:
For a single particle, the work done on a particle by a ...
- 3,078
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1
answer
75
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Gaussian integral identity over Grassmann numbers
I am reading Zee's Quantum Field Theory in a Nutshell. In the section about Grassmann numbers, there is an identity:$$\int dx\int dy\,e^{yAx}=\det A\tag{II.5.13}$$ where $x=(x_1,x_2,\dots,x_N),y=(y_1,...
- 487
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answers
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Mathematical result for one-loop diagram of $\phi^3$ theory in four dimensions
The one-loop diagram in $\phi^3$ theory in four dimensions gives the following integral (I have suppressed factors of $2\pi$)
\begin{equation}
\tag{0}
\Gamma(s) = \Gamma(p^2) = \int {d^4 k} \frac{1}{[(...
- 429
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votes
2
answers
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Why do we integrate over an entire charged object to find the total electric field in physics (what does integrating actually do?)?
I'm taking first-year university electricity and magnetism and this concept is on charged rods/rings/discs. The textbook tells us that integrating over the whole charged object gives us the total ...
- 3
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votes
1
answer
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I dont get how to get the last equality in this calculation
In the calculation of equal-time spatial correlation of the electric field i have an idea of how to get the second equality (even if i miss that factor i/r) but I'm struggling to understand how the ...
- 11
2
votes
0
answers
35
views
Derivation of moment of inertia for $1-x^2$, $-1\le x\le 1$ through $y$ axis [closed]
Say we have a uniform density wire in the shape of the parabola $1-x^2$ from $-1$ to $1$, rotated about $x=0$ (y axis). I'm attempting to calculate the moment of inertia for this object.
I have the ...
- 157
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How to integrate a function multiplied for a sign function?
I am studying QFT and I found this integral on my lecture notes (for the context: we're trying to show that the covariant commutation relations are Lorentz invariant)
$$∫\frac{d^{3}p dp_{0}}{(2\pi)^{3}...
- 75
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votes
1
answer
44
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Why does different nondimensionalizations give different results? Although the results should be the same
I have some problems with the non-dimensionalization of the Hamiltonian of motion in a Coulomb field.
The Hamiltonian has a following form:
$$H=-\frac{\hbar^2}{2\mu^*} \Delta_r-\frac{e^2}{\epsilon_0 r}...
- 189
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1
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Potential due to line charge: Incorrect result using spherical coordinates
Context
This is not a homework problem. Then answer to this problem is well known and can be found in [1]. The potential of a line of charge situated between $x=-a$ to $x=+a$ ``can be found by ...
- 285
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answers
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Why dot in $W = \int \mathbf F\cdot \mathrm d\mathbf s$?
Question essentially in title. Why do we use the dot symbol when writing $W = \int \mathbf F\cdot \mathrm d\mathbf s$? I understand that $\mathbf F$ and $\mathrm d\mathbf s$ are vectors and that ...
0
votes
0
answers
54
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How to interpret $\int\mathrm{d}^2z$? [duplicate]
In chapter 6 of Tong's lecture notes on string theory when calculating the Virasoro-Shapiro/4-point Tachyon amplitude he arrives at the integral
\begin{align*}
C(a, b) = \int\mathrm{d}^2z\ |z|^{2a-2}|...
- 1,758
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votes
2
answers
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How to correctly integrate the following equation (Laplace Equation) twice [closed]
I want to solve the Laplace equation and prove the potential formula for a point charge. I selected spherical coordinates to reduce the number of dimensions in the equations ending up with:
$$\Delta\...
- 1
0
votes
1
answer
43
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Maxwell Boltzmann speed distribution: why isn't speed element integrated when converting from velocity distribution? [duplicate]
Maxwell Boltzmann velocity distribution is given by $$f_{\vec v}(v_x,v_y,v_z)=A^{3/2}\exp{[B(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})]}$$
To convert the velocity distribution into speed distribution, spherical ...
0
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0
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25
views
How to integrate "recursive" pressure/temperature relations?
I hope the term recursive is correct in this context.
The Clausius-Clapeyron relation says that:
$\frac{dP}{dT} = \frac{L}{T\Delta v}$
Where P is the pressure, L is the latent heat of vaporization, T ...
- 323
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votes
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answers
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Integration along real axis with singularities
I'm trying to calculate Green function of wave equation
$\begin{align}
\bigg(\nabla^2 - \frac{\partial ^2}{\partial t^2}\bigg)G(\textbf{x},t;\textbf{x'},t')=\delta^3(\textbf{x-x'})\delta(t-t')
\end{...
- 125
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votes
1
answer
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What does it mean for a field to be defined by a measure?
In Quantum Physics by Glimm and Jaffe they mention on p. 90 that
The Euclidean fields are defined by a probability measure $d\mu(\phi) = d\mu$ on the space of real distributions. Here $d\mu$ plays ...
- 1,830
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1
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How does uncertainty propagate when a quantity is integrated?
Consider a quantity $f(x)$ that has an absolute error $\delta x(x)$. The parentheses indicates that $\delta x$ varies with $x$. What is the uncertainty in $\int_0^x f(x) dx$?
For simplicity let's ...
- 171
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Proof of Landau Conditions
I was reading this Quantum Field Theory book written by Claude Itzykson and Jean-Bernard Zuber. In section 6-3, page 304, the authors introduced analytic properties of Feynman integrals.
Consider the ...
1
vote
1
answer
96
views
Heaviside function in the form of an integral
I am currently reading Optimal storage properties of neural network models by E. Gardner. (DOI 10.1088/0305-4470/21/1/031)
In appendix 1, the Heaviside function is expressed in integral form eq A1.1
$$...
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Solution for forced harmonic motion with non-constant frequency
Is there any integral form of the solution for the equation below?
$$
\ddot{y}+\omega^2(t) y = f(t)
$$
where it's basically the equation for forced harmonic motion with non-constant frequency.
If $\...
- 203
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votes
0
answers
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Inconsistency Regarding Commutators and Integration?
I have this following confusion regarding the ordering of integration and commutator.
Consider an operator $\mathcal{O}$ defined as
$$
\mathcal{O}(t) \equiv \int d^3 x' \ \phi(t, \vec{x}') \nabla'^2 \...
- 21
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1
answer
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Relation between acoustical parameters Definition and Clarity [closed]
In room acoustics, clarity ($C_{50}$) is defined as
$$ C_{50} = 10*\log_{10}{\frac{\int_0^{50ms}p² (t)dt}{\int_{50}^{\infty}p² (t)dt}} $$
and definition ($D_{50}$) is
$$ D_{50} = \frac{\int_0^{50ms}p² ...
- 3
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0
answers
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How to calculate impulse of gravity and tension on a pendulum string on the pendulum mass (released from rest at horizontal position)?
Consider a pendulum of mass $m$ and length $l$, released from rest in a horizontal position. At some time $t_1>t_0=0$, the situation looks like the following
I am aware that we can use ...
- 251
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votes
1
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Field due to objects if inverse square law does not hold [closed]
There is a hypothetical question about what will happen to the field of objects if inverse square law is replaced with something else.
For example, it can be proved that if force is proportional to ...
- 19
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1
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Why did my rearrangement with chain rule end up equating velocity to position?
We all know acceleration is the time-derivative of velocity which in turn is the time-derivative of position. Vice versa: position is the integration of velocity and velocity itself is the integration ...
- 373
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Question on non-holonomic constraints (This is different to the others)
I know there are many posts on non-holonomic constraints and also many on this exact one but I feel that there is still some confusion on it.
"Consider a disk which rolls without slipping across ...
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votes
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Why do we Wick rotate before regularizing Feynman diagrams?
In Folland's Quantum Field Theory he mentions that we can apply Feynman's formula (Feynman parameterization) to either the Wick rotated integrals or the non-Wick rotated integrals corresponding to ...
- 1,830
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vote
0
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66
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Gaussian integral over Grassmann numbers
I'm trying to evaluate a Gaussian integral over Grassmann numbers but not sure if I've made a mistake.
What I want to evaluate is
\begin{equation}
\left(\prod^N_i\int d\theta^*_i d\theta_i\right)\...
- 11
1
vote
1
answer
81
views
Peskin and Schroeder eqn 9.14 [closed]
I am not familiar with functional integral, and in the text like $$\int D\phi D\pi \exp [i\int^T_od^4x(\pi\dot{\phi}-\frac{1}{2}\pi^2-\frac{1}{2}(\nabla \phi)^2-V(\phi))].\tag{9.14}$$ I try to compile ...
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Understanding Feynman parameterization
On Wikipedia, it gives the Feynman parametrization formula as
$$\frac{1}{A_1\cdots A_n} = (n-1)!\int_0^1 du_1\cdots \int_0^1 du_n \frac{\delta(1-\sum_{k=1}^n u_k)}{(\sum_{k=1}^n u_kA_k)^n}
\\ = (n-1)!...
- 1,830
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How can a definite acceleration integral be useful in mechanics and why is an indefinite integral not used?
We have an acceleration function and in order to find the displacement function, it would be logical to take an indefinite integral 2 times. Then we would get a function. Why is it proposed here to ...
- 21
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vote
1
answer
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Integrating over a charge distribution without using Radon-Nikodym
Goodmorning,
I'm studying for a basic Electrostatic course and I have a doubt about how to justify in terms of measure theory the physicists' writing
$dq=\rho \ d\tau$ or $\rho = \frac{dq}{d\tau}$,
...
2
votes
1
answer
63
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On the finiteness of worldsheet area
It is commom to define the wordlsheet of a classical open string, for example, as the $2$-dimensional smooth manifold with boundary as $\mathbb{R} \times [0,\pi]$. With the appropriate embedding $X: \...
- 962
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votes
0
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Modified Boltzmann statistics
I want to compute the following integral:
$$
p(E_i, \lambda) =\frac{L}{Z} \int_{-\infty}^{\infty}dt e^{it}\frac{1}{\lambda E_i +it}e^{-L\mathrm{tr}\log(\lambda H +it)}
$$
in the limit of large $L$, ...
- 53
1
vote
2
answers
94
views
Fermionic measure in path integral
When writing the fermionic path integral one arrives at an expression containing $\mathcal{D}\bar{\psi}$ and $\mathcal{D}\psi$:
$$
\int \mathcal{D}\bar{\psi} \mathcal{D}\psi e^{iS}
$$
Usual ...
- 125
3
votes
0
answers
93
views
An Integral from Feynman & Hibbs [closed]
In Appendix "Some Useful Definite Integrals" of Feynman & Hibbs "Quantum Mechanics and Path Integrals" they use some integral formulas(1) that I'm struggling to derive, and ...
- 51
1
vote
1
answer
81
views
How do I assign momenta for internal loops of a Feynman diagram?
I've been working on the one-loop corrections, and encountered the following diagrams:
[a] and [b] come from the Yukawa Lagrangian $\phi\bar\psi\psi$. We can assign the momentum $k$ to one of the ...
- 1,305
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vote
1
answer
80
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Why does Ronald Ruth integrate momentum as velocity in his 1983 paper "A canonical integration technique"?
I am trying to apply a time integration technique for a system of discrete particles using Hamiltonian symplectic mechanics.
In his relatively well-known paper (CERN,PDF) on symplectic integrators, ...
- 57
-1
votes
2
answers
89
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Integral of absolute values [closed]
Work done is given by the integral
$$\int \vec F\cdot d\vec r$$
Where $\vec F$ is force and $d\vec r$ is displacement. Writing displacement in terms of velocity, we get
$$\int\vec F\cdot\vec v\,dt=\...
