All Questions
Tagged with integration mp.mathematical-physics
29 questions
16
votes
2
answers
2k
views
Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory
(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$:
$$
\int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i}
$$
We can generalize this ...
- 2,147
13
votes
1
answer
812
views
Summation of series involving $\sinh$ of a square root
Consider the following series:
$$
S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})}
$$
From the physical ...
user76025
6
votes
2
answers
761
views
Gauge integral versus path integral
According to this paper, "The gauge integral [a.k.a. Henstock-Kurzweil integral] provides the only formal framework that is close to the original development of the Feynman path integral", and also "...
- 42.8k
6
votes
2
answers
3k
views
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Because I still have no idea how it is possible for me to write down seemingly important equations ... that don't make any sense (at least for me) and because I haven't got any helpful comment so far, ...
- 1,026
6
votes
0
answers
150
views
Can this Casimir-effect integral be reduced to a special function?
This integral plays a central role in a physics problem (Casimir effect)${}^\ast$
$$\Omega(\phi,L)=-\frac{1}{\pi}\operatorname{Re}\int_0^\infty \ln\bigl[1+\beta(\omega)^2 e^{i\phi-2\omega L}\bigr]\,d\...
- 188k
6
votes
0
answers
256
views
What is the expected value of the volume of a tetrahedron inscribed in the unit sphere?
Four (non-coincident) points on the unit sphere determine a tetrahedron. What is the expected value of the volume of such a tetrahedron--the volume of the sphere itself being $\frac{4 \pi}{3} \approx ...
- 723
5
votes
4
answers
952
views
Limit of an integral vs limit of the integrand
I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral
$$
I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
5
votes
1
answer
403
views
Matrix integrals in combinatorics, for dummies
This is actually about one particular question that I posted a while ago, "Special" meanders. Among several approaches tried is a huge subclass of approaches which can be generated from ...
- 18.4k
5
votes
1
answer
345
views
rigorous derivation of isoperimetric inequality from ideal gas equation
I'm an undergraduate math student that learned about classical ideal gases and the associated maxwell-boltzmann distribution for particle velocities in a statistical physics course. Now, starting from ...
- 3,871
5
votes
3
answers
1k
views
Perform an integration involving the product of two hypergeometric functions
I've encountered the following product,
\begin{equation}
\, _2F_1\left(3 d+2,3 d+2;6 d+4;1-\frac{1}{t^2}\right) \,
_3F_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;t^2\right)
\...
- 723
4
votes
3
answers
490
views
Positivity of the Coulomb energy in two dimensions
In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{\|\cdot\|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...
- 255
4
votes
3
answers
598
views
Meaning of divergent integrals
In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. ...
4
votes
1
answer
211
views
Perform an integration over the unit interval of a two-parameter expression involving a Gauss hypergeometric function
In a quantum-information-theoretic context, I've encountered the problem
of integrating over $r \in [0,1]$, the function
\begin{equation}
r^{2 d-1} \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2}...
- 723
3
votes
1
answer
1k
views
A Gaussian integral over complex variables by a defined Green's function for a Gaussian ensemble of random matrix
We construct an $N\times N$ matrix $J$ whose elements are drawn from Gaussian distribution with zero mean and variance $\frac{1}{N}$. Since we want to have different variances for different columns, ...
- 325
2
votes
1
answer
119
views
Power series expansion of the order parameter in the Kuramoto model
In this review of the Kuramoto model, Eq. 14 is obtained by expanding the following integral in powers of $K r$,
$$
r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta
$$
...
- 640
2
votes
1
answer
125
views
Is $C_c(\mathbb{R}^2,\mathbb{R}^2)$ dense in the irrotational square integrable functions?
Let $L_D(\mathbb{R}^n)^n$ be the set of square integrable functions which are the weak derivative of a locally square integrable function. That is
$$L_D(\mathbb{R}^n)^n=\{Du\colon u\in H^1_{loc}(\...
- 53
2
votes
0
answers
118
views
What is the justification for using Wiener integrals to integrate over a space of differentiable functions?
In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
- 33
2
votes
0
answers
425
views
Why is the integral of the tautological 1-form equal to the action?
I am having a hard time to understand why the integral of the tautological 1-form is the action of the system.
The tautological one form is defined by :
\begin{align}
\theta_{(q,p)} : T_{(q,p)}T^*Q &...
- 71
2
votes
0
answers
51
views
What restrictions on the form of an integral equation have a unique solution f=0?
We're stuck on the following question in a problem relevant to a physics paper on AdS/CFT that we are working on. Given a Fredholm equation of second kind with the form $f$+$\int_D K f\,dx = 0$, where ...
- 31
2
votes
0
answers
491
views
Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question
This is a prequel to my question:
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Clearly my ...
- 1,026
2
votes
0
answers
463
views
How to perform this matrix integral?
Edit: some backgrouds added.
In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral
$$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} e^{-\...
- 395
1
vote
1
answer
166
views
How can one integrate over the unit cube, subject to certain (quantum-information-theoretic) constraints?
To begin, we have two constraints
\begin{equation}
C1=x>0\land z>0\land y>0\land x+2 y+3 z<1
\end{equation}
and
\begin{equation}
C2=x>0\land y>0\land x+2 y+3 z<1\land x^2+x (3 z-2 ...
- 723
1
vote
0
answers
80
views
Biot-Savart-like integral for a toroidal helix
The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon.
Let us consider a toroidal helix parametrized as follows:
$$
x=(R+r\cos(n\phi))\cos(\phi)...
- 255
1
vote
0
answers
222
views
L2 norm of the diagonal entries of a random rotation of a fixed matrix?
Let $X\in\mathbb{R}^{d\times d}$ be the diagonal matrix with $d/2$ entries equal to $1$ and $d/2$ entries equal to $-1$. Let $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...
- 91
1
vote
0
answers
212
views
Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices
Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm (http://mathworld.wolfram.com/OperatorNorm.html) of the $2\times 2$ ...
- 723
1
vote
0
answers
88
views
Fresnel Integral and his formulas
Below is how Fresnel approximate the eponymously "Fresnel Integral". In his own words:
Let $i$ and $i+t$ be the narrow limits between which it is proposed to integrate $\cos(qv^2) \, dv$. ...
1
vote
0
answers
284
views
Integral involving square of associated Laguerre polynomial and sperical bessel function
In a quantum mechanical problem I encountered the integral
$$I_k=\int_0^\infty x^{2(l+1)-k}j_k(\sigma x)e^{-x}[L_{n-l-1}^{2l+1}(x)]^2 dx,$$
where $j_k(x)$ is a spherical Bessel function, and $\sigma$ ...
- 16.5k
0
votes
1
answer
143
views
Formally confirm a formula for a certain three-dimensional constrained integral over the unit cube
The result of the three-dimensional constrained integration (for the Hilbert-Schmidt two-qubit absolute separability probability) over the unit cube $[0,1]^3$
\begin{equation} \label{one}
\int_0^1 \...
- 723
0
votes
1
answer
123
views
Could variable be still function in x and y after performing Reynolds averaging over area
All,
Let $S(x,y,t)$ be a variable function in $x$, $y$, and $t$. After
performing Reynold averaging over area $\frac{1}{A}\int S(x,y,t) dA$, could $S$ still be a function in $x$, and $y$?
Equations (1-...
- 101