"Mathematical physics consists of the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories." (from Journal of Mathematical Physics) This tag is intended for questions on methods used ...

learn more… | top users | synonyms (1)

10
votes
0answers
389 views

Nonabelian global symmetries, $SO(N)$ symmetric theory of $N$ scalar fields.

Consider an $SO(N)$ symmetric theory of $N$ real scalar fields,$$\mathcal{L} = {1\over2} \partial_\mu \Phi^a \partial^\mu \Phi^a - {1\over2} m^2 \Phi^a \Phi^a - {1\over4} \lambda (\Phi^a ...
7
votes
0answers
107 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 ...
7
votes
0answers
100 views

Existence of classical solution for equation $\Delta u + v(x) u = 0$

It is known that if $D$ is a bounded domain in $\mathbb R^n$, $n \geq 2$, with $\partial D \in C^2$ then the Dirichlet problem $$ \begin{array}{rl} \Delta u & = 0 \quad \text{in $D$}, ...
7
votes
0answers
352 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 ...
7
votes
0answers
314 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 ...
6
votes
0answers
80 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 ...
6
votes
0answers
192 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
votes
0answers
187 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 ...
6
votes
0answers
123 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 ...
5
votes
0answers
50 views

Affine geometry book for physicist

I'm looking for a textbook to help me with understanding the geometry of Galilean relativity and the Galilean group. The reason is that I tried going through V.I. Arnold's Mathematical Methods, but ...
5
votes
0answers
70 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 ...
5
votes
0answers
99 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 ...
5
votes
0answers
278 views

Explain this step in lecture notes

The bounty offered is for the person that explains me how the author gets from equation 3.19 to equation 3.20 in these lecture see here. Normally I would agree that copying the relevant equation would ...
5
votes
0answers
291 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 ...
5
votes
0answers
130 views

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
votes
0answers
63 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 ...
5
votes
0answers
96 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
votes
0answers
156 views

Can one reformulate tensor methods and young tableaux to account for spinor representations on $\operatorname{SO}(n)$?

Standard tensor methods and Young tableaux methods don't give you the spinor reps of $\operatorname{SO}(n)$. Is this because spinor representation are projective representations? If so, where does ...
4
votes
0answers
38 views

Physical meaning of Hawking's Singularity theorem

I'm studying O'Neill's "Semi-Riemannian Geometry with applications to Relativity". I know that the following theorems are related to the Big Bang, but I don't understand how. Let $M$ be a ...
4
votes
0answers
36 views

What is the mathematical understanding behind what physicists call a gauge fixing?

I'm learning fiber bundle from my poor physicist point of view. I understand that a gauge transformation (physicist language) corresponds to the transformation of the connections built from an ...
4
votes
0answers
40 views

Calculate the resistance between 2 adjacent nodes on a shape using graph theory

In shapes like regular octahedron or dodecahedron, how can Graph Theory be used to calculate the resistance between two adjacent vertices? All edges are assumed to have unit resistance. Is there ...
4
votes
0answers
60 views

Backgrounds of the p-Laplacian Operator

Motivation I encountered the following partial differential equation (PDE) in a mathematical paper $$\begin{array}{} u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta ...
4
votes
0answers
65 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 ...
4
votes
0answers
48 views

Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
4
votes
0answers
50 views

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$?

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$ ? To give context, this comes from: Dirac's Theory of General Relativity p19: http://imgur.com/mrkT5C7 I'm not comfortable with proofs regarding ...
4
votes
0answers
65 views

Defining the quantum group $U_q(\mathfrak{sl}_2)$

I've seen two defining relation for $U_q(\mathfrak{sl}_2)$ by the Serre relations $$[H,E]=E,\quad[H,F]=-F, \quad [E,F]=\frac{q^H-q^{-H}}{q-q^{-1}}, $$ or by taking $K=q^H$ $$KK^{-1}=K^{-1}K=1,\quad ...
4
votes
0answers
102 views

Green's function in a moving frame for a constant heat source

I am looking for the Green's function of the problem in two dimensions $r =(x,z)$, \begin{equation} \nabla^2g + \frac{v}{D}\frac{\partial g}{\partial z} = -\delta (r-r_0) \end{equation} Which ...
4
votes
0answers
74 views

Why is entropy = the Legendre transform?

Can someone give me a mathematician's explanation (and not a physicist's) as to why $$\int_{\Omega}\Psi^*(b(u(t))$$ is called the entropy where $\Psi^*$ is the Legengre transform of ...
4
votes
0answers
49 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 ...
4
votes
0answers
312 views

Contour Integration - Quantum field theory

I am a physics student. In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk ...
4
votes
0answers
88 views

The Theory of Probabilistic Sets

${P}_m := \{ \Phi_{1m}, \Phi_{2m}, \ldots , \Phi_{nm} \}$ where $$|\Phi_{im} \rangle = \wp_{1m} | \phi_1 \rangle + \wp_{2m} | \phi_{2m} \rangle + \cdots + \wp_{km} | \phi_{km} \rangle$$ such that ...
4
votes
0answers
50 views

Solution of a particular PDE in 4 variables with non-constant coefficients

I have come across the following equation while reading about the Unruh Effect in Black Hole Physics. . K is a function of $x,y,\rho,t$ i.e $K=K(x,y,\tau, \rho)$. $\omega, k,m$ are constants. ...
4
votes
0answers
220 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 ...
4
votes
0answers
591 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 ...
4
votes
0answers
601 views

Curvature of Hyperbolic Space

I'm trying to prove that hyperbolic space has constant sectional curvature $-1$, but keep running into difficulties. Could someone show me a way out? I've been given the metric ...
4
votes
0answers
385 views

Solving inhomogenous bessel equation

I have the following differential equation to be solved $\dfrac{d^2\psi}{dr^2}+\dfrac{d\psi}{rdr}+4\left(\omega^2-k_0^2-\dfrac{n^2}{r^2}\right)\psi=AJ_n^2(kr)+\dfrac{k}{r}J_n(kr)J_{n+1}(kr)-\omega ...
3
votes
0answers
40 views

Can I split this integral to a sum over three contours?

I have the following integral $$ Z = \frac{1}{2\pi i} \int dx \, \frac{1}{(x-a_1)(x-a_2)(x-a_3)}\times \frac{1}{(x+\epsilon - a_1)(x + \epsilon - a_2)(x+ \epsilon - a_3)} $$ and this integral has ...
3
votes
0answers
93 views

Heat equation in 2d Circle polar coordinates

I was presented this problem in PDE class involving heat equation on unit circle in polar coordinates using separation of variables, giving the following heat equation problem: $ u_t = 9\Delta u ...
3
votes
0answers
21 views

Constructing a coset representative of $SO(n,4)/(SO(n) \times SO(4))$.

In $\mathcal N = 2$ Supergravity the scalar components of Hypermultiplets form a quaternionic Kaehler manifold. Only isometries of this so-called target manifold can be gauged. I am interested in ...
3
votes
0answers
104 views

Upper bound on the Lipschitz constant of entanglement entropy

I'm looking for an upper bound for the Lipschitz constant of entanglement entropy between two subsystems with respet to the standard distance measure of pure states in the Hilbert space of the full ...
3
votes
0answers
42 views

Solvability of an integral equation

Is the following integral equation solvable ? $$ F(x)-\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\frac{\sin \gamma(x-y)}{\pi(x-y)}$$ and $$f(x)=e^{i\gamma x}$$ and $\gamma$ is a parameter.
3
votes
0answers
51 views

Notation for Christoffel symbols used by Gödel in “An example of a new type of cosmological solution of Einstein field equations of gravitation”

I have difficult to understand the meaning of the notation used by Gödel in the article cited in the title of this post. You can find it here: http://www.lygeros.org/10552b.pdf In the second page ...
3
votes
0answers
101 views

Regge symmetry and outer automorphisms of Dynkin diagrams

Quantum $6j$-symbols are the coefficients of the change of basis matrix in the central extension of Temperley-Lieb algebra(see the book by Kauffman and Lins). It is my understanding that Ocneanu has ...
3
votes
0answers
157 views

Wave equation for a string nonuniform (PDE)

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman , but I have been impossible these paragraphs. The displacement $u$ of a nonuniform string ...
3
votes
0answers
21 views

Is compatibility with a gauge sufficient to turn a parallel transport into a connection?

As the title suggests, is compatibility with a gauge sufficient to turn a parallel transport into a connection?
3
votes
0answers
214 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 ...
3
votes
0answers
78 views

Perturbation of Laplacian

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian $$-\Delta+V(x)$$ is self-adjoint on $H^2(\mathbb{R}^3)$. My idea is to use Kato-Rellich theorem; ...
3
votes
0answers
96 views

Functional derivatives on manifolds

This might be more of a physics question, but it is mathematics-related, I hope I am not out of place with this. Let $(M,\mathcal{S},g)$ be a smooth, $n$-dimensional manifold equipped with a Riemann ...
3
votes
0answers
203 views

Numerical scheme to 1D advection equation

I am trying to numerically solve a system of equations which model the early universe in 1D. The equations I am stuck on are; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 ...
3
votes
0answers
59 views

Simply-connected Lorentzian manifold and event horizon

Can a simply connected Lorentzian manifold admit an event horizon? Or does the event horizon makes it non-simply connected?