Questions tagged [mathematical-physics]

DO NOT USE THIS TAG for elemetary physical questions. This tag is intended for questions on modern mathematical methods used in quantum theory, general relativity, string theory, integrable system etc at an advanced undergraduate or graduate level.

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376 views

How are topological invariants obtained from TQFTs used in practice?

Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places: Atiyah, Topological quantum field theory Lurie, Topological Quantum Field Theory ...
16
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317 views

Was von Neumann's 1954 ICM address “Unsolved Problems in Mathematics” outdated?

I recently tried to "explain" the generalized probability theory aspect of quantum theory (as one common part of both quantum field theory and quantum mechanics), in the sense of motivations for the ...
12
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821 views

Learning roadmap to Topological Quantum Field Theories from a mathematics perspective

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
12
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0answers
325 views

What are D-branes (in a topological field theory)?

In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
11
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376 views

Connection between distributional and renormalized solutions for Boltzmann equation

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem ...
10
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1k views

Renormalization for mathematicians

Can someone explain to me the processes of renormalization and regularization used in quantum field theory and similar fields in a way that a pure mathematician might make sense of it? Is there a ...
9
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0answers
336 views

Open questions in Topological K-Theory

I am interested in knowing about current research in the Topological K-Theory, especially its interactions with String Theory. About one and a half decade back, there were some papers by Physicists (e....
8
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0answers
201 views

What are the mathematical foundations of the renormalisation group?

Briefly, RG refers to mathematical tools that allows systematic investigation of the changes of a physical system as viewed at different distance scales. These methods are very important in quantum ...
7
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176 views

Is this physical model exactly solvable?

There exists a popular model in the Physics of heavy quark bound systems, called the Cornell potential model, in which the inter-quark potential is modeled to vary with radial distance $r$ as $$V(r) ...
7
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130 views

Tautochrone and Brachistochrone problems are equivalent

It is a well known fact that the brachistochrone (the problem of finding the curve of quickest descent in a uniform gravitational field) and the tautochrone (the problem of finding a curve from which ...
7
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261 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
7
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245 views

A conjecture on Schatten 1-norm

I have a conjecture on Schatten 1-norm. Before presenting the conjecture, let us first specify the notions used here. A matrix $A$ is said to be a density operator if $A$ is positive semidefinite ...
7
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145 views

Implementation of a simulation of an incompressible Newtonian fluid with uniform density

Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I want to simulate an incompressible Newtonian fluid with uniform density $\rho$ and viscosity $\nu$. The evolution ...
7
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352 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ...
7
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0answers
294 views

Multivariable Integral, How to compute it?

Can anybody please tell me, how to evaluate a multivariate integral with a gaussian weight function. $$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm ...
7
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0answers
127 views

Cauchy representation and branch point order

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. The main application is for dispersion ...
7
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0answers
169 views

Percolation and number of phases in the 2D Ising model.

Update. As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive After a long time I came back to try to understand an article on the Ising model. ...
7
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0answers
804 views

The apex of parabolic motion forms an ellipse of constant ellipticity.

I am not sure how well-known this is idea is, but here is a .gif illustrating it: Basically, the set of highest points of parabolic motion at constant initial velocity forms an ellipse, with ...
6
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157 views

Heisenberg's inequality

The Heisenberg's inequality in $\Bbb{R}$ reads $$\|f\|_{L^2}^4\leq \int_{\Bbb{R}}x^2f(x)^2dx\int_{\Bbb{R}}\xi^2\hat{f}(\xi)^2d\xi$$ where by $\hat{f}$ we refer to the Fourier transform of $f$. The ...
6
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0answers
210 views

Application of SU(2) in physics

How can we interpret the representations of SU(2) and $\mathfrak{su}(2)$ in physics? I have studied a lot of mathematics including representation theory and differential geometry, so I understand SU(...
6
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0answers
205 views

Defining a trace-class operator with a Bochner integral

For complex numbers $z$ consider the system of $L^2(\mathbb{R})$-vectors with norm equal to $1$ \begin{equation} \psi_z=e^{-\frac{1}{2}|z|^2}\sum_{n=0}^{\infty}\frac{z^n}{\sqrt{n!}} |n\rangle, \end{...
6
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0answers
328 views

Approximating a discrete measure with a continuous one

In physics it is common to approximate distributions of point masses or charges with continuous distributions. To do this, one typically defines a density function by moving throughout the space a ...
6
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0answers
302 views

Integration by parts on manifold with a boundary

Suppose $C$ is a 3-form, and $G$ is a 4-form defined by $G = dC$. Also, $M_{11}$ is an 11-dimensional manifold (without a boundary), $W_{6}$ is a 6-dimensional submanifold of $M_{11}$ and $D_{\epsilon}...
6
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0answers
366 views

Could you explain the failure of the Hodge decomposition to exist for non-compact manifolds?

I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while). Specifically what I'm (pragmatically)...
6
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0answers
300 views

Hawking's and Ellis' derivation of the form of Einstein's field equations

On pages 72-73 of the book "The large scale structure of space-time" Hawking and Ellis show while determining the form of the field equations of general relativity that there is a relation of the form ...
6
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217 views

Reference for Dimensional Regularization

I am studying a bit of theoretical physics (QFT and string theory), and I obviously stumbled upon dimensional regularization. I have been told that this technique has in fact a solid mathematical ...
5
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45 views

Using abstract Hilbert spaces to solve differential equations

There are techniques for solving PDE's, such as Fock-Schwinger method in physics, which involve translating the problem from the language of distributions to the language of the abstract Hilbert ...
5
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0answers
154 views

Proving an equality for Schrödinger equation

Let $\phi\in \mathcal S(\mathbb R)$ and consider the uni-dimensional global Cauchy problem for the Schrödinger equation of a free particle, that is $$(*)\qquad iu_t+u_{xx}=0\quad \text{and}\quad u(x, ...
5
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0answers
61 views

References for finding coordinates in which some natural Hamiltonian becomes separable

Given a natural Hamiltonian, $$H = \frac{1}{2} g^{\mu \nu}(x) p_{\mu} p_{\nu} + V(x)$$ and a general potential $V(x)$ I want to be able to find the coordinates, given an expression for $V(x)$, such ...
5
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0answers
110 views

Reference for rigorous treatment of the representation theory of the Lorentz group

When I studied representation theory for the first time it was only focused on finite groups. It was the second half of a one semester course in group theory, and the book employed was "Representation ...
5
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0answers
155 views

Mathieu function coefficients, how to express $\gamma_m,\delta_m$in terms of $\alpha_m,\beta_m$?

Suppose $1>e>0, E>0$ are constants, and $f(x,y)$ and $g(x,y)$ satisfies the PDE: \begin{align} \sqrt{\frac{1+e}{1-e}}\frac{\partial f}{\partial y}+\frac{\partial g}{\partial x}=\frac{ieE}{\...
5
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0answers
83 views

Use of the word “topology” in physics

I don't know so much about topological order or topological phase transition or topological state of matter or other similar expressions from physics and mathematical-physics, but I can't see a direct ...
5
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0answers
547 views

Understanding twisted differential forms

I'm trying to understand twisted differential forms. I do know that they are like regular differential forms but under coordinate transformations they pick up an extra factor of the sign of the ...
5
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0answers
105 views

The Virasoro-Bott group and the KdV equations

The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group. For the famous $KdV$ equations these equations are given on the Virasoro-Bott ...
5
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0answers
287 views

Short pedagogical introduction to Young-tableaux and weight diagrams?

I am looking for a short pedagogical introduction to Young-tableaux and weight diagrams and the relationship between them. Hopefully one which would contain many detailed and worked out examples of ...
5
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0answers
82 views

Green's function the way George Green defined it

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x-x')$ function as their source on the RHS. But ...
5
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0answers
64 views

C. Neumann passage in Latin from *Annali di Matematica Pura ed Applicata*

Neumann, Carl. “Theoria nova phaenomenis electricis applicanda.” Annali di Matematica Pura ed Applicata 2, no. 1 (August 1868): 120–128. doi:10.1007/BF02419606. p. 121: Nova introducitur suppositio,...
5
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0answers
82 views

Deconvolution of distribution of diffraction reflexes

I'm a chemist stuck in a mathematical problem. Please bear with me as I'm trying to express myself in Math language. Let me explain in short terms the experimental method I'm using: X-ray diffraction....
5
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0answers
111 views

Second law of thermodynamics as a theorem about state space evolution

I once saw a mathematical explanation of the second law of thermodynamics. The statement was something like this: there is a mapping $f$ from the set of thermodynamic states $S$ to itself, and a ...
5
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0answers
350 views

Is there a relation between Super Riemannian manifolds and Kähler manifolds?

(This question has a physics motivation). Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kähler manifolds, or at ...
5
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0answers
337 views

Rewriting the advection-diffusion equation

This is mostly a reference request question, although I certainly appreciate any insights and/or comments. Let us assume $p:R^n×(0,∞)\to \mathbb R$ is a scalar concentration, $u\in R^n$ is the ...
5
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0answers
553 views

Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
4
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0answers
141 views

Is Leibniz integral rule (basic form) allowed in this (physics) improper integral? Why?

Electric potential at a point inside the charge distribution is: $\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'-\delta} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$ ...
4
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0answers
84 views

Solve wave equation with non-constant wave speed using method of characterstics?

I am trying to get a better understanding of wave pulses in a domain with a non-constant wave speed. I am trying to solve either one of the two equations: $$\frac{\partial^2u}{\partial t^2}-c(x)^2\...
4
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0answers
59 views

How can I solve this system of three PDEs?

I'm trying to solve this system of PDEs. $$\begin{cases}\dfrac AxG^2=\partial_x F-k\cos\theta\\\dfrac AxG(\partial_\theta G)=-\dfrac 1x\partial_\theta F-k\sin\theta\\h(\theta)=\displaystyle\frac1{x_0l}...
4
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0answers
119 views

Stability criterion for leapfrog in relativistic physics.

I am doing a 2D MD simulations of charge carriers in graphene using the Leapfrog algorithm. I am trying to prove that, in some specific cases (when distance between particles is small), the method is ...
4
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0answers
195 views

What kind of flow minimizes resistance?

Consider a domain $D$, where $\sigma(x)$ is the spatially dependent "conductivity". On the boundary we have $2$ "electrodes" $E_1$ and $E_2$ where matter flows in and out. The rest of the boundary is ...
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0answers
69 views

Domain issues with Weyl quantization

Most algebras of observables from quantum mechanics are closed. For example, fix a separable Hilbert space $H$, and consider the algebra of bounded operators on it. This is a Banach space. In ...
4
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0answers
119 views

Logic for physics

I'm looking for references on applications of logic (e.g. intuitionistic logic, toposes) to physics (especially quantum physics, as this is the area that I'm aware can be helped by logical ...
4
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0answers
108 views

Lie algebra-valued differential forms, exactness, closedness, and the Moyal product

This is my attempt to prove something (I'm not even sure if it's true to begin with) using somewhat loose arguments. I present here all the steps and ideas and I would be extremely grateful if someone ...