Questions tagged [mp.mathematical-physics]
Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.
2,123
questions
6
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Time decay for Hartree equation with Coulomb potential
Are there any time-decay results for the solution of the Hartree equation
\begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3\end{equation} in $L^p$...
- 63
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0
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120
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Can an ellipse roll down a tilted sine curve without jumping?
Background
Assume that we have a solid ellipse with uniform density, and that it rolls along a curve.
In the following MO question, I asked along what curve an ellipse rolls down fastest. It was ...
- 4,575
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0
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187
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Does a fusion ring with F and R-symbols uniquely determine a braided tensor category?
Background : In mathematical physics, 'anyons' in (2+1) dimensional systems are described by braided tensor categories. The anyon types correspond to the irreducible objects of the category. From such ...
- 61
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115
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Can two eigenfunctions be almost linearly dependent in a region?
Consider the Schrödinger operator $H=-\Delta+|x|^a$ on $\mathbb{R}$, where $a>0$. Since the potential is growing at $\infty$, we have compact resolvent thus the eigenvalues are discrete and tend to ...
- 869
6
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0
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156
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Nonlinear-PDE arising from flat conformal Chebyshev nets
Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. ...
- 134
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0
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223
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What is a large field problem?
I was reading Constructive Renormalization Group by V. Rivasseau and I got some points which I would like to clarify.
On page 2, Rivasseau talks about the large field problem and, if I understood it ...
- 1,265
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0
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419
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“Cohomological equation” in dynamical systems
Let $$\dot{x}=Ax+v_r(x)+v_{r+1}(x)+ \dots$$ with $x \in \mathbb{C}^n$ and $v_r: \mathbb{C}^n \to \mathbb{C}^n$ a homogenous, polynomial function of order $r.$
Then, being able to find a suitable $h$ ...
- 1,484
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0
answers
480
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Yang–Mills existence and mass gap official statement on Euclidean $\mathbb{R}^4$, why not Minkowski $ \mathbb{R}^{3,1}$?
Yang–Mills existence and mass gap problem is officially stated by Clay Mathematics Institute:
Yang–Mills Existence and Mass Gap.'' Prove that for any compact simple gauge group G, a non-trivial ...
- 10.3k
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278
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Two questions about Fock spaces
Let $\mathscr{H}$ be a complex Hilbert space and denote $\mathscr{H}_{n}$ the tensor product $\overbrace{\mathscr{H}\otimes\cdots\otimes\mathscr{H}}^{\text{n}}$. Denote by $\Pi_{\pm}$ the projection ...
- 3,151
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203
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What is known about "dimension two" vertex algebras?
In the paper Chiral Koszul duality, Gaitsgory and Francis develop a notion of a chiral algebra living on an arbitrary variety $X$. When $X=\mathbf{A}^1$ and the chiral algebra is translation invariant,...
6
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141
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What is Ryu-Takayanagi Entanglement Entropy?
I have a question about how to think about the Ryu-Takayanagi entanglement entropy mathematically.
For simplicity, let's work in the simplified setting of a time-symmetric slice of $AdS_4$ space -- i....
- 2,422
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172
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Descendent Gromov-Witten invariants and Frobenius manifolds
I've heard it said that genus $0$ descendent Gromov-Witten invariants of a smooth projective variety $X$ can be encoded in the structure of a Frobenius manifold on the cohomology $H^*(X,\mathbb{C)}$. ...
- 11
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366
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What are some results that assume the Connes' embedding conjecture or any of its reformulations?
As you all (may) know, the Connes embedding conjecture was disproven last year. Also, as its Wikipedia page shows, there are multiple reformulations (but it is definitely not an exhaustive list):
...
- 265
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223
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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
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118
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Deriving (Gaussian) curvature bounds from bounds on the metric
I am trying to understand a bound in Christodoulou's 2008 paper on black hole formation.
The paper considers a spacelike surface $S$ diffeomorphic to a sphere, with two metrics:
the induced metric $\...
- 409
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0
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154
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Fourier transformation of a distribution
We have no idea how to tackle the following Fourier transformation of a distribution:
$$
\lim_{\epsilon\to0^+}\int_{-\infty}^{\infty}\mathrm{d} t\int_{\mathbb{R}^{d-1}} \mathrm{d}^{d-1}\vec{r} e^{-\...
- 373
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201
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Deformation quantization of infinite dimensional Poisson manifolds
In 1999, Karaali wrote a review of formal deformation quantization for a class she took with Weinstein.
She ends the paper with the following remark:
Another question that remains involves the ...
- 517
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209
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References for superhomology
This question concerns topological string theory.
It was known sice its outset, that the BRST-cohomology ("observables") of the weakly coupled topological string B-model on a Calabi-Yau ...
- 502
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355
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What is the status of the Born Rule in axiomatic QM?
While physicists have tried multiple times and failed to derive the Born Rule (for example: https://arxiv.org/pdf/quant-ph/0409144.pdf). I was wondering what axiomatic Quantum Mechanics had to say ...
- 237
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258
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Branching rules for E6 into SU(3)^3
I am very confused about what are the branching rules for representations of $E6$ into a $SU(3)\times SU(3)\times SU(3)$ subgroup. At least in the physics literature, there seems to be a serious ...
- 469
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198
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Conjecture for a certain Cauchy-type determinant
Given the Cauchy-like matrix
$$
\mathbf X_M(q) = \left[ \frac{2}{\pi} \frac{
\Gamma\!\left(m - \frac{1}{2} \right)\Gamma\!\left(n + \frac{1}{2} \right)
}{
\Gamma(m)\,\Gamma(n)
}
\frac{m-\frac{3}{4}}
{\...
- 2,705
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answers
298
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Hints on an expository article about Kardar-Parisi-Zhang (KPZ)
It seems the KPZ is the next big thing in mathematical physics and probability. The skeletal idea is probably that while classical averages are in the Gaussian universality class, lots of other ...
- 267
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373
views
What’s the limit of a vector bundle?
In geometric measure theory, there’s an answer to the question “what’s the limit of a family of submanifolds”, namely there’s some kind of object called an integral current.
In the geometric ...
- 8,643
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196
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A 'Fock'-type construction on a $C^*$-algebra
In very rough terms, let $A$ be a complex unital $C^*$-algebra. Assume it nuclear for convenience, but it doesn't matter much. Consider the 'Fock'-type $C^*$-algebra (don't know a better name for it)
$...
- 1,939
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433
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Integrality of the Chern-Simons form and normalization of the action
I'm somewhat confused about normalization of the Chern-Simons action (for arbitrary compact gauge group). If we have a trivial principal bundle we write
$$S(A)=\frac{k}{8\pi^2}\int_M\text{Tr}\left(A\...
- 241
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315
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Freed-Hopkins-Lurie-Teleman topological boundary conditions v.s. Lagrangian subspace
This question concerns the comparison of topological boundary conditions of TQFTs on a manifold with some boundary.
For example, we can consider defining the TQFT on a $D^3$ ball with a topological ...
- 10.3k
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113
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Does the Hodge *-operator act on the tangent space at 0 to the space of integral (n-1)-cycles in a conformal manifold of dimension d=2n?
Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$.
Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral
$k$-currents in $M$
and write ${\cal D}^{\mathit{int}}...
- 351
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477
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intuitive connection between The KdV equations and the Virasoro bott group
I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group....
- 949
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516
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Time-separation function on "globally hyperbolic" spacetimes with everywhere timelike boundary
It is well-known that, in globally hyperbolic spacetimes, the time separation function $\tau$ (aka Lorentzian distance function) enjoys the following property: fix a point $p$ and a point $q \in I^-(p)...
- 473
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199
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Spectral theory for Dirac Laplacian on a funnel
I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...
- 313
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0
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392
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semiclassical proof of Wigner semicircle
In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix.
\[ \int ...
- 22.6k
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255
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Given that a conditional measure is Gaussian, how bad can the original measure be?
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
- 8,392
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559
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Jones Polynomial and Quantum Field Theory
I am trying Witten's paper but unable to re-produce the computations presented in the paper.
I tried few things on internet but all these tutorials explicitly don't show the calculations and refer to ...
- 61
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406
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Birth-Death Process associated with Orthogonal Polynomials
I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...
- 22.6k
6
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234
views
Generalization of the non-existence of a monostatic planar body
Domokos, Papadopulos, and Ruina showed that there does not exist a convex planar rigid body of uniform density which has only
one orientation of stable equilibrium and one orientation of unstable ...
- 5,926
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186
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what books to read to quickly understand adiabatic approximation
Hi group, I'm a theoretical ecologist with fairly adequate training in applied math (ODE, linear algebra, applied probability, some PDEs). In my current work, I've encountered the use of adiabatic ...
- 61
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1k
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Harmonic maps into compact Lie groups
Consider locally minimizing harmonic maps from D-dimensional Euclidean space into a compact Lie group G. When $D=3$ the general regularity theory due to Schoen-Uhlenbeck,
Schoen, Richard; Uhlenbeck, ...
5
votes
4
answers
982
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Where to start with research regarding maslov index/class
Hi,
I am a physicist and currently doing my bachelor thesis about geometric quantization.
In the book by Bates and Weinstein I encountered the Maslov index, which seems to be very important :-).
But ...
- 5,472
5
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4
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948
views
literature on geometrical viewpoint on calculus of variations for physics
What is a good reference for a geometrical viewpoint on the calculus of variations for physics, using differential forms etc. to derive Yang-Mills equations and other topics of the standard model?
...
- 911
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votes
3
answers
4k
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Physicist trying to understand modern mathematics
I'm a physicist trying to gain a deep understanding of mathematics that is required for my work.I intend to specialize in string theory which is a very math intensive branch of theoretical physics ...
- 117
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2
answers
699
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What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
In the same spirit of this question:
How much of mathematical General Relativity depends on the Axiom of Choice?
I want to go radically further ahead and ask for what remains of mathematical general ...
- 308
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votes
5
answers
991
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What are the local maxima and minima of $\frac{\sin(nx)}{\sin(x)}$
FYI: I asked this question here couple of days ago but got no answer yet.
$n$ is an integer
We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are ...
- 105
5
votes
3
answers
922
views
Number-theoretic congruences with geometry and topology?
There are many examples of $q$-series identities being proven by interpreting them as generating series of geometric invariants like the Donaldson invariants. I would like to know if there are ways ...
- 22.6k
5
votes
3
answers
712
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Meaning of a phrase from "The algebra of grand unified theories".
Motivated by an answer to this mathoverflow question I've been making an effort to understand Baez and Huerta's article "The algebra of grand unified theories".
As far as I can tell, mathematically, ...
- 4,194
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2
answers
2k
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Advice on doing physics under the umbrella of mathematics and the converse
Note: This is a question directly copied from Theoretical Physics SE primarily to get the advice of people indulged in mathematics.
In the current scenario of research in QFT and string theory (and ...
5
votes
2
answers
666
views
Polygamma function in mathematical physics
Are there situations in which the polygamma pops up naturally in a mathematical physics context? In particular: are there examples of potentials having some interest for which the dependence on the ...
- 4,263
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votes
3
answers
2k
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Group cohomology of compact Lie group with integer coeffient
It is known that group cohomology class $H^d[U(1),Z]$ is Z for even d and 0 for odd d.
Do we know $H^d[G,Z]$ for $G=SO(3)$, $SU(2)$ and other compact Lie group?
Also is the Borel-group-cohomology ...
- 4,716
5
votes
2
answers
1k
views
Space with 720° / not 2$\pi$ rotational symmetry?
Is there a space with a 720°, but no 360° rotational symmetry? Possibly one that can be mapped onto something more conventional like R(3) or R(3,1)?
The reason I am asking is because in quantum ...
- 161
5
votes
3
answers
562
views
Morphisms of supermanifolds
I am confused regarding supermanifolds. Suppose I consider R^(1,2) (1 "bosonic", 2 "fermionic"), This map (x,a,b) -> (x+ab, a,b) (a,b are fermionic) is supposed to be a morphism of this supermanifold. ...
- 3,323
5
votes
1
answer
922
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Has a discrete/quantum theory of probability based on the Cournot-Borel principle or something been developed?
In 1930, Émile Borel, the father of measure theory together with his student Lebesgue and a world-class expert in probability theory, published a short note Sur les probabilités universellement ...
- 1,016