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.

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Path integral methods

Are there detailed expositions of the path integral methods in (mathematical) physics other than Feynman-Hibbs and Glimm-Jaffe?
asv's user avatar
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Boundedness of Laplacian eigenfunctions

Let $A$ be a bounded domain in $\mathbb R^d$, $d>1$, and $\{u_k\}$ is the set of all $L^2$-normalized Laplacian eigenfunctions on $A$ with Dirichlet boundary condition (i.e., $\|u_k\|_2 = 1$). Is ...
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Book about fluid dynamics

Next Monday, I'll have an interview at Siemens for an internship where I have to know about fluid dynamics/computational fluid dynamics. I'm not a physicist, so does somebody have a suggestion for a ...
Barb's user avatar
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Is there a reasonable notion of spectral theorem on a pre-Hilbert space?

I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...
Sanchayan Dutta's user avatar
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4 answers
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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{\...
jonathan wolf's user avatar
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Weyl's Branching Rule for $SU(N)$-Setting

On the Wikipedia page for restricted representations https://en.wikipedia.org/wiki/Restricted_representation there is presented a number of explicit "branching rules". In particular, there is the ...
Nadia SUSY's user avatar
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Limit of a double integral

What is the $\varepsilon\to 0$ limit of the following double integral $$\int\limits_{-1}^1d\tau\;\sqrt{1-\tau^2}\;\tau\int\limits_0^\infty dq\;q^2e^{iq(\tau+i\varepsilon)}\;?$$ I was asked about this ...
Zurab Silagadze's user avatar
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Van Vleck-Morette Determinant

There seems to be something curious about the so-called Van-Vleck-Morette determinant, as I cannot find any source that properly defines it in terms of expressions previously defined in that source ...
Matthias Ludewig's user avatar
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Hard-sphere gases and the wave equation

I'm trying to bridge a basic gap in my own education: Where can I find a written discussion concerning the derivation of the wave equation (for the propagation of sound, say), assuming nothing (much?...
David Feldman's user avatar
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What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?

For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the Yang--Baxter equation if $$(R\otimes id)(id\otimes R)(R\otimes id) = (id\otimes R)(R\otimes ...
Jake Wetlock's user avatar
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Serre relations for Lie Superalgebras

Finite dimensional complex simple Lie algebras are classified using Cartan matrices. One of the main ingredients is Serre Relations. Lets call this Cartan-Killing theory. I have the following ...
GA316's user avatar
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Asymptotic $\int_M \mathrm{exp}[\mathbf{e}\left(n -\frac{t}{2\pi i}\right)] \left( 1 + \frac{5}{6} \mathbf{e}^2 \right)^{1/2} $ on quintic Calabi-Yau

Let $M = \{ G(x) = 0 \} \subseteq \mathbb{P}^4$ be a quintic Calabi-Yau and $\mathbf{e} \in H^2(M, \mathbb{Z})$ such that $\int_M \mathbf{e}^3 = 5$. Then as $t \gg 1$: $$ \int_M e^{n \mathbf{e}} e^...
john mangual's user avatar
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Harmonic Crystal using Random Walk

Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here. Basically, it is a ...
Amir Sagiv's user avatar
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"Uncertainty principle" for self-adjoint operators in a finite von Neumann algebra

Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a ...
Łukasz Grabowski's user avatar
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Monte Carlo simulations

I was wondering what were the models of statistical physics that are still considered difficult/slow to simulate (exactly, or approximately) with the current technology of Monte Carlo approaches. I ...
Alekk's user avatar
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Does it help to learn statistical mechanics in order to learn thermodynamic formalism?

Does it help to learn statistical mechanics or thermodynamics (as in physics or mathematical physics) in order to learn thermodynamic formalism: the study of equilibrium states, Gibbs measure, maximal ...
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In the rep theory of Quantum Double, why does the fusion of 2 "pure fluxes" yield a "pure charge"?

Physicist here, so my notation may be different from standard math notation. For the quantum double $D(G)$ of a group $G$, we may write representations of $D(G)$ in the following way: Consider a ...
pyroscepter's user avatar
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Initial conditions in the Klein-Gordon equation

I am interested in what are the largest families of initial conditions of the problem (in $\mathbb{R}^{4}$) \begin{equation}\label{kg} \left\lbrace \begin{array}{ll} (\square+m^2)F(x)=0\\ ...
Gabriel Palau's user avatar
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1 answer
529 views

Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories?

So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle $...
Alex Zorn's user avatar
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Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization?

The question is in the title. The form of the condition looks like the Bohr-Sommerfeld quantization formula of angular momentum, is there a link between the two formulas?
Issam Ibnouhsein's user avatar
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2 answers
698 views

Generalized basis

In quantum mechanics, people introduce the notion of "continuous basis" (I actually don't know the mathematical denomination of it). It is not a Schauder basis. I would like to know what could be a ...
Noix07's user avatar
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Vandermonde-type identity for Jacobi theta functions?

My question concerns an application in physics. By Vandermonde identity I refer to the following statement: take $f_j (z)=z^j$, where $z=x+iy$ is a complex coordinate and $j$ an integer. Make an $N\...
user15775's user avatar
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1 answer
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Is the Wikipedia depiction of the ergosphere of a Kerr black hole a Cassini oval?

Disclaimer: this a cross post from MSE, where this question was asked on November 4th 2019 and has so far received no upvote, no comment and no answer whatsoever. Glancing at https://en.wikipedia.org/...
Sylvain JULIEN's user avatar
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1 answer
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Turing machines and Ising model

I have currently started a new research line aiming to prove a mapping between a 2-symbol Turing machine and the one dimensional Ising model. The connection is seen by recognizing that a set of ...
Jon's user avatar
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From Sato grassmannian to spectral curve

Assume a tau-function of the KP integrable hierarchy is fixed by the point of the Sato grassmannian (that is by a semi-infinite set of Laurant series $\varphi_k(z)=\sum_{m>0}a_{km}z^{-k+m}$). Can ...
Sasha's user avatar
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738 views

In what precise sense is quantum (i.e., non-commutative) probability not expressable in terms of classical probability?

The quantum set-up has many settings, so let's fix some definitions. I will be taking the Hilbert space approach with a minor modification that I will make explicit. We begin with a Hilbert space $\...
Mehmet Coen's user avatar
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2 answers
390 views

Most general conditions for (weak or classical) solutions to Poisson's equation

I thought I knew this but have found it surprisingly difficult to find good references. I am interested in solving $$ \left\{ \begin{align} & \Delta \psi = - \rho & & \mbox{in } \mathbb{...
Ben Ciotti's user avatar
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1 answer
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Intuition behind the "Lapse Function"

I came across the following definite of the Lapse Function: $N=\sqrt{\frac{1}{2}g(L,\overline{L})}$ where $L,\overline{L}$ are the null geodesic vector fields. Further, I have been looking at this ...
GregVoit's user avatar
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General solution to null-divergence equation

I am interested in whether each component of a divergenceless vector can itself be written as a divergence. My motivation for this question is the characterization of so-called trivial conservation ...
user254433's user avatar
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2 answers
867 views

Permuting Racked Pool Balls with a Single Break

Given reasonable physical assumptions (on friction, collisions, etc.), would it be possible to "break" in a pool game such that when all the balls come to rest, the only difference is that the racked ...
Benjamin Dickman's user avatar
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2 answers
4k views

(Anti)Canonical divisor of a blow up

This question may be utterly trivial, or not, but as someone with hardly any knowledge of algebraic geometry I thought there could be a chance I get lucky. Let X be a rational surface obtained by n ...
philiph's user avatar
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1 answer
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Does BQP^P = BQP ? ... and what proof machinery is available?

Update #3: Over on TCS StackExchange, I have rated as "accepted" an ingenious construction by Luca Trevisan, which answers a two-part question (as reframed by Tsuyoshi Ito) that is in essence "Do ...
John Sidles's user avatar
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2 answers
645 views

What does it mean to extend a 2d (topological) conformal field theory to Deligne-Mumford space?

For 2D (topological) conformal field theory the corresponding moduli space is the space of Riemann surface with boundary, right? What does it mean by extending the theory to Deligne-Mumford space? How ...
Hao's user avatar
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2 answers
663 views

Dependence of trace norm on matrix size for smooth vs. random matrices.

Problem Consider two d x d complex matrices, R and S, whose entries lie in the unit disk: $\quad |R_{i,j}|<1 \quad$ and $\quad |S_{i,j}|<1 $. Say that R is constructed by randomly choosing ...
Jess Riedel's user avatar
5 votes
1 answer
409 views

Reference Request for a particular approach of (rigorous) statistical mechanics

I was reading Mathematical Aspects of Quantum Field Theory by. E. de Faria and W. de Melo, and the following caught my attention. In (Hamiltonian) mechanics, the states of a system are described by ...
IamWill's user avatar
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5 votes
1 answer
298 views

In search of a combinatorial proof on particular set of partitions

Given a partition $\lambda=(\lambda_1\geq\lambda_2\geq\dots)$, denote the conjugate partition by $\lambda'=(\lambda_1'\geq\lambda_2'\geq\dots)$. For example, if $\lambda=(4,2,2)$ then $\lambda'=(3,3,1,...
T. Amdeberhan's user avatar
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1 answer
253 views

approximate closed form for infinite series (no power series)

The electric potential of a charge between two infinite conducting planes can be expressed as an infinite series (of image charges) [Kellogg1929] as $$ V=\sum_{n=-\infty}^{\infty}\left( \frac{1}{\...
Christian Wagner's user avatar
5 votes
2 answers
488 views

connection between the distribution of energy levels and spacings between prime numbers [closed]

I'm a student in university, I'm new in this forum. My teacher told me to use it for my research. I came to ask for advice and help. The research is about the connection between prime numbers and ...
Ayoubayjx's user avatar
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1 answer
536 views

Question about the Aganagic-Vafa A-brane

According to Aganagic-Vafa (hep-th/0012041) and Fang-Liu (arXiv:1103.0693), for a semi-projective toric Calabi-Yau 3-manifold $X$, the Aganagic-Vafa A-brane $L_{AV}\subset X$ is defined by the ...
Acky's user avatar
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1 answer
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Absent 2nd order terms in deformation quantization of Poisson manifolds

I am reading Kontsevich' famous paper on deformation quantization of Poisson manifolds. In section 1.4.2 on page 4 he gives the general formula for the star product associated to a Poisson structure ...
miramo's user avatar
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1 answer
250 views

Well defined Tensoring of spectral triples

Hi, I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it. Question: In connes standard model he takes ...
SMF's user avatar
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3 answers
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Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way

I am interested in a characterization of the creation and annihilation operators that is in some sense invariant under $O(n)$ rotations of $\mathbb{R}^n$: Background The Harmonic Oscillator on $\...
Otis Chodosh's user avatar
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5 votes
2 answers
414 views

When can we factor out the time dimension?

First of all, my knowledge of both GR and differential geometry is quite weak, so forgive me if the physics here doesn't make much sense. Let $(M, g)$ be a smooth, connected Lorentzian manifold of ...
Daniel Litt's user avatar
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1 answer
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Can Fock spaces be replaced by arbitrary Hilbert spaces under some hypothesis to justify path integrals?

I was reading this post from PSE and it reminded me an old question of mine, in which the use of creation and annihilation operators were discussed. Both questions got answers which agreed on the fact ...
IamWill's user avatar
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1 answer
504 views

Riemann-Hilbert approach to Selberg integral

I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with ...
Marcel's user avatar
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1 answer
775 views

A set of questions on continuous Gaussian Free Fields (GFF)

As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
IamWill's user avatar
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5 votes
2 answers
366 views

Connections between two constructions of infinite dimensional Gaussian measures

Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space $l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} ...
IamWill's user avatar
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5 votes
1 answer
443 views

Scattering theory for Coulomb potential

Both physical and mathematical theories of quantum scattering seem to be well developed in the case when the potential (or a more general perturbation of the Laplacian) decays fast enough at infinity ...
asv's user avatar
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5 votes
1 answer
645 views

Gauge invariance of Chern-Simons functional integral for a 3-manifold with boundary

I am trying to understand how the functional integral for Chern-Simons theory for a possibly non-compact 3-manifold with boundary is made gauge invariant. For a compact 3-manifold, $M$, without ...
Mtheorist's user avatar
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314 views

Nonlinear sigma models with non-compact groups / target spaces

A nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. The target manifold T is equipped with a Riemannian metric g. Σ is ...
wonderich's user avatar
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