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.
2,124
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Open-closed string correspondence
Recently, after many years of searching for the right source, I came across the excellent lecture by Aspinwall, "Some Applications of Commutative Algebra
to String Theory", in Eisenbud's Festschrift. ...
38
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4
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A family of polynomials whose zeros all lie on the unit circle
I had posted the following problem on stack exchange before.
Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of the ...
5
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Paving property
In their famed paper (https://arxiv.org/abs/math-ph/0011053), Bourgain and Goldstein conjecture what they call the paving property:
Let $H_{jk}=\delta_{j,k+1}+\delta_{j,k-1}+v(\theta+j\omega)\delta_{...
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Topological field theories and their path integrals
Examples of topological field theories are the cohomological field theories as they were initially defined by Witten [1]. Such examples include the Donaldson-Witten theory in 4d or the Gromo-Witten ...
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Barnes's double $\Gamma$-function, $\gamma_{h}(0) = -\frac{1}{12}$?
I apologize if I may have gotten the name of the function incorrectly. The function $\gamma_h(x)$ is defined in Appendix A of the paper SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, https://arxiv.org/...
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Finite sum of spherical Bessel functions
In L.G. Afanasyev, A.V. Tarasov, Breakup of relativistic pi+pi- atoms in matter, Phys. At. Nucl. 59 (1996) 2130 the following identity is given for the spherical Bessel functions $j_n(z)=\sqrt{\frac{\...
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Physical significance of some killing vector fields on a $G_2$ manifold
I have recently calculated three linearly independent Killing vector fields on a G^2 manifold with a specific metric, which can be found in this paper:
https://arxiv.org/pdf/hep-th/0011256.pdf
The ...
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1
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Question on Witten’s paper “Supersymmetry and Morse theory”
EDIT. I am trying to read the article “Supersymmetry and Morse theory” by E. Witten (JDG 17 (1982)).
This well known article applies some tools developed by physicists (e.g. path integrals) to ...
3
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Nekrasov Partition function and the leading term of Prepotential
I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY
AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.
In (4.25) the author expressed the partition function ...
3
votes
1
answer
230
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Casimir force without cheating
If memory serves me right, some years ago I heard rumors about developing methods
of computing the Casimir force which are rigorous (by the standards of constructive quantum field theory). I ...
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Which operators constructed from 10d gamma matrices commute with $SO(1,2)\times SO(3)\times SO(3)$?
In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of boundary conditions in N=4 Super Yang-Mills in four dimensions in ...
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3
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Graph of a Lie super algebra
Let $A$ be a generalized Cartan matrix and let $\mathfrak{g}$ be the Kac-Moody Lie algebra associated to $A$. There is an associated graph of $\mathfrak{g}$ which is known as the Dynkin diagram of $\...
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Does Helmholtz's decomposition give an over-determined rotational flow?
From Helmholtz's decomposition,
$v=v_{\scriptscriptstyle IR} +v_{\scriptscriptstyle R} $
where $\nabla\times v_{IR} =0$ and $\nabla\cdot v_R=0$
when apply this to the linearized Navier-Stokes ...
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Why the unreasonable applicability of complex numbers in physics/engineering? [duplicate]
After years of using complex numbers in every kind of analysis of physical and electrical engineering problems I am starting to wonder: why is this particular algebra so effective in modelling the ...
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Frobenius structure for A_n singularities
I need to compute monodromy matrices $M(v)$, associated to a Frobenius structure for $A_n$ singularity with flat coordinates $v_1,\dots,v_n$, that is, $f(x)=x^{n+1}$. (due to Saito, Dubrovin etc.) ...
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A problem about closed 2-forms on Minkowski space
The problem is:
For any closed 2-form in the Minkowski space $\mathbb{R}^{3,1}$ satisfiying $dF=0$ and $\delta F \ne 0$ (with $\delta$ denoting the codifferential), does there exist a Lorentz ...
3
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2
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References on thin film equation: derivation and properties
Where can I find
a derivation of the thin film equation
$$u_t = - \mathrm{div} (u^m\nabla\Delta u)$$ from a physical model?
a good introduction to its properties (e.g. conserved quantities and ...
9
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1
answer
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Fedosov vs. Kontsevich deformation quantization : a beginner survey
I'm a condensed matter physicist who tries to understand the details of deformation quantization.
In my self-made training, I've found two huge pieces of work, namely
Fedosov, B. V. (1994). "A ...
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0
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80
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How to calculate the exact probability $p$ at which maxima occurs in the curves in an infinite system?
I'm writing with respect to this paper: Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice
Here's a ...
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Is this correct: Inflection points of Euler number graph in Island-Mainland transition correspond to spanning cluster site percolation threshold?
I'm writing with respect to the paper Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice.
Here's a link to a PDF ...
13
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A modified Heisenberg uncertainty relation
Let $\psi\colon\mathbb R\to\mathbb C$, and set
$$
F[\psi]:=\pi\ \frac{\displaystyle\int_{\mathbb R} |x|\;|\psi(x)|^2\,\mathrm dx}{\displaystyle\int_{\mathbb R}|\psi(x)|^2\,\mathrm dx}\frac{\...
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Integral involving square of associated Laguerre polynomial and sperical bessel function
In a quantum mechanical problem I encountered the integral
$$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$ ...
3
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1
answer
160
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How to show that a map which relates to Donaldson–Thomas invariants is an automorphism?
I am reading the lecture notes INTRODUCTION TO DONALDSON–THOMAS INVARIANTS. I have a question in the end of page 1 about the proof of a map is an automorphism.
Let $m>0$ be an integer. Let $\...
5
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0
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245
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Localization principle in integration over supermanifolds
This post is closely related to the post Localization principle in supersymmetry
and can be considered as a continuation of it, although independent.
In § 9.3 of the book "Mirror symmetry" (K. Hori ...
3
votes
1
answer
228
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wave speed and travelling wave
I have seen a lot of work has been done in the context of travelling wave. For example the work of McKenna and Chen in Journal of Differential Equations Volume 136, Issue 2, 20 May 1997, Pages 325-355....
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117
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Green's third identity potential massive object
Consider a massive object occupying a volume $U$ with boundary $\partial U$. Let the gravitational potential inside be $V_{in}$ and outside $V_{out}$
Normally the gravitational field of a massive ...
6
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372
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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 ...
2
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418
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$SO(6) \to SU(2) \times SU(2) \times U(1)$ branching rules
What do these branching rules mean?
\begin{eqnarray*} SO(6)_E &\to& SU(2)_\ell \times SU(2)_r \times U(1)_\Sigma
\end{eqnarray*}
I am taking these examples from a paper of Gukov (on p.51) ...
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1
answer
694
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Invariance of a vector under parallel transport along an infinitesimal orthogonal loop
I'm not very familiar with differential geometry and am coming from a general relativity background, so would appreciate help with a question from that context. If this question could be posed in a ...
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0
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141
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Why is $\textbf{J}$ called angular momentum?(Quantum) [closed]
Why is $\textbf{J}$ called angular momentum operator? Can anyone explain why the expectation value of J is angular momentum?
Here is how $\textbf{J}$ is defined: The rotation operator
$$
U(\alpha)=\...
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0
answers
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Numerical computation of spectrum for operators on real line with "confining potential"
I am looking to understand the conditions under which one can expect "reasonably" accurate solution to leading eigenvalues/eigenvectors of a second order differential operator posed on the real line.
...
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Reference request: Partition function as a topological invariant of a QFT
I have read (mainly in the articles of Atiyah) that the partition function is the simplest topological invariant of a quantum field theory.
I have an arithmetic geometry background and know ...
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Euclidean or Minkowski Metric for Clustering Spatio-Temporal Data?
Question:
when does using Minkowski metric $\quad\sqrt{x^2+y^2+z^2-t^2}\quad $for clustering $(x,y,z,t)$ data yield better results than using Euclidean metric $\quad\sqrt{x^2+y^2+z^2+t^2}\ $?
I ...
3
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1
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Does the Leclerc-Thibon involution exchange vertex operators of the first and second type?
This question is about $U_q ( \hat{\mathfrak{sl}}_2 )$ representation theory. There is a notion of vertex operators $\Phi_{\pm }(z)$ of first and $\Psi_{\pm}(z)$ of the second type. They are defined ...
12
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3
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Geometrically quantizing real Grassmannians
It seems that the Grassmannian of oriented 2-dimensional planes in $\mathbb{R}^n$
$$ \mathrm{Gr}(n,2) = \frac{\mathrm{SO}(n)}{\mathrm{SO}(n-2) \times \mathrm{SO}(2)} $$
has a symplectic structure ...
3
votes
2
answers
540
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Can one obtain this ODE as an Euler-Lagrange equation?
Some of the second order ODE can be considered as Euler-Lagrange equations for an appropriate Lagrangian. However this is true not for arbitrary second order equation. But some of important equations ...
2
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0
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How exactly are holomorphic maps with hyperkahler targets identified as triholomorphic maps?
In Appendix A of Hyperinstantons, the Beltrami Equation, and Triholomorphic Maps by Fre et al., it is described how holomorphic maps from a Riemann surface to a hyperkahler manifold $\mathcal{N}$, ...
6
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1
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Possibly new solution to equal-mass three-body problem; refinement required
(This is a repost of this question from 18 months ago on the main Mathematics SE site, as the response there has been underwhelming, and I thought here would be a better authority. As you can probably ...
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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)
$...
6
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1
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342
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Toda Hierarchy and Quantum Cohomology of $\mathbb{P}^1$ Frobenius manifolds
People usually say that the quantum cohomology of $\mathbb{P}^1$ Frobenius manifold $QH^*(\mathbb{P}^1)$, corresponds to dispersionless extended Toda hierarchy (e.g. page 6 of https://arxiv.org/pdf/...
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Generalisation of Lyapunov time to stochastic dynamical systems
Might there be useful generalisations of the Lyapunov time to stochastic dynamical systems? In particular, I'm interested in methods for calculating confidence intervals around stochastic analogues of ...
17
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What is an "Instanton" in classical gauge theory? (to a mathematician)
There's already a question about the same topic but I think its aim is different.
Classical (non-quantum) gauge theory is a completely rigorous mathematical theory. It can be phrased in completely ...
8
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Lax pairs in an abstract formalism
I am reading Integrals of Nonlinear Equations of
Evolution and Solitary Waves by Peter Lax and I'm having a hard time. The methods are pioneering, of course, but Lax does not bother much to provide ...
60
votes
4
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Is there a mathematical and information theoretic explanation for this cube packing phenomenon?
I saw this unintuitive result on dice packing:
A jumble of thousands of cubic dice, agitated by an oscillating
rotation, can rapidly become completely ordered, a result that is hard
to produce with ...
4
votes
1
answer
459
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an application of nth moment of Poisson distribution with stirling number
I was reading the paper on arixv.
I was confused the equation of nth moment of Poisson distribution.
The detail and partial paper as follow:
...
For large N, this connection probability takes ...
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How to Evaluate the ABJM partition function for N=2
This is the ABJM partition function on the 3-sphere,
$$ Z(2) = \int \frac{d^2\mu}{(2\pi)^2} \frac{d^2\nu}{(2\pi)^2}
\frac{\left[ 2 \sinh \frac{\mu_1 - \mu_2}{2}\right]^2\left[ 2 \sinh \frac{\nu_1 - \...
14
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1
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For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$
Let stable matrix (i.e., its eigenvalues have negative real parts) $B \in \mathbb R^{n \times n}$ and anti-symmetric matrix $T \in \mathbb R^{n \times n}$ satisfy
$$B^\top - T B^\top = B + B T$$
...
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0
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84
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Particle density in phase space normalization under proliferation
Consider $1,..,N$ indistinguishable particles in $\mathbb{R}^2$ and let them evolve according to a brownian motion and proliferation. Let $u: \mathbb{R}_+ \times \mathbb{R}^2 \rightarrow \mathbb{R}_0^+...
7
votes
1
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424
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Naturally occuring counting process with a 1/log asymptotics?
Besides prime numbers, is there another physically realizable counting process that exhibits a 1/log density ? The reason I am posting this question is that we are measuring the response of a quantum ...
47
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2
answers
4k
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The two ways Feynman diagrams appear in mathematics
I've heard about two ways mathematicians describe Feynman diagrams:
They can be seen as "string diagrams" describing various type of arrows (and/or compositions operations on them) in a monoidal ...