Questions tagged [physics]

For questions about mathematical problems arising from physics, the natural science studying general properties of matter, radiation and energy.

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Hilbert's sixth problem and QFT description

The Wikipedia entry on Hilbert's sixth problem about QFT description is “Since the 1960s, following the work of Arthur Wightman and Rudolf Haag, modern quantum field theory can also be considered ...
XL _At_Here_There's user avatar
33 votes
8 answers
3k views

Motivation and physical interpretation of the Laplace transform

Concerning the one-sided Laplace transform, $$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$ what is a motivation to come up with that formula? I am particularly interested in "physical&...
AlpinistKitten's user avatar
-2 votes
1 answer
122 views

Interpretation and validity of modified Heisenberg uncertainty principle in a metric context? [closed]

Considering the Heisenberg uncertainty principle, which states $\Delta x \cdot \Delta p \geq h$, I've explored a modified version by computing $(\Delta x + 1)(\Delta p + 1) \geq \Delta x \cdot \Delta ...
mathoverflowUser's user avatar
1 vote
1 answer
208 views

Qualitative values between two electrons in an atom or how to interpret these values?

This question is a little bit trying to understand physics through geometry of simplex: Let $E_{i,j}$ be the ionization energy in times the number of hydrogen ionization energy for an element with ...
mathoverflowUser's user avatar
0 votes
1 answer
557 views

What is this three dimensional curve that looks like an infinity sign called?

What is this three dimensional curve that looks like an infinity sign called? (Is there a known parametric equation for it?) It was generated with this Sagemath - script, where you can zoom in 3d in ...
mathoverflowUser's user avatar
1 vote
1 answer
258 views

Temporal evolution of a globally hyperbolic spacetime

Any globally hyperbolic spacetime can be assigned a global function of time as Hawking has demonstrated for stably causal spacetime. (Any globally hyperbolic spacetime is also stably causal). For ...
Bastam Tajik's user avatar
0 votes
1 answer
588 views

A RKHS interpretation of the Rydberg formula for hydrogen and an application for physics?

I was thinking if it is possible to define an inner product between two small physical objects with a positive definite kernel and was led to look at the Rydberg formula: The Rydberg formula for ...
mathoverflowUser's user avatar
1 vote
0 answers
66 views

Biot-Savart-like integral for a toroidal helix

The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon. Let us consider a toroidal helix parametrized as follows: $$ x=(R+r\cos(n\phi))\cos(\phi)...
AndreaPaco's user avatar
1 vote
1 answer
115 views

calculating a double limit

We have the following term: $$ (e^{-a h}+e^{-b h})^n / 2^n$$ Now we take the limit: $$ h\to 0, n\to \infty $$ What relation of $h$ and $n$ must be satisfied for the following limit to hold? $$\lim_{h\...
Lili Si's user avatar
  • 95
3 votes
1 answer
200 views

Interesting question about the Thomson problem for arbitrary number of electrons

This question is crossposted from here I believe this is a pretty hard question and so I decided to repost the question in the Math Overflow forum. If there is something wrong with doing this, I am ...
Rodrigo's user avatar
  • 51
3 votes
1 answer
126 views

Applications of maximal surfaces in Lorentz spaces

I have been working on minimal surfaces, only recently learnt about maximal surfaces and "maxfaces" in Lorentz spaces. I can clearly see the mathematical motivations. But I wonder if zero-...
Hao Chen's user avatar
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-4 votes
2 answers
461 views

Inverse square-law as a positive definite kernel?

Newtons law for gravity states that: $$F_{12} = \frac{G m_1 m_2} {|x_1-x_2|^2}$$ The function : $$k(x,y):=\exp(-| x-y|^2)$$ is known to be a positive definite function, called the RBF-kernel. It ...
mathoverflowUser's user avatar
0 votes
0 answers
26 views

Generating a proper finite difference scheme

I have recently started studying the finite difference schemes for numerical analysis. While I can now calculate difference schemes fairly easily for simple equations, I've recently come across a ...
Syed Ali Mohsin Bukhari's user avatar
2 votes
1 answer
206 views

inclusion of von Neumann algebras implies reversing inequality of its modular operators

I'm working with von Neumann algebras and I stumbled with this statement in a work of Borchers (1999) Since $\mathcal N \subseteq \mathcal M$, it follows by standard arguments that $\Delta_{\mathcal ...
Gabriel Palau's user avatar
3 votes
2 answers
350 views

Mathematical difference between solitons and traveling waves for a non-linear dispersive PDE

I see many mathematicians conflating the definitions of traveling waves and solitons, and I am unable to understand, from a mathematical point of view, the differences between these two types of ...
Niser's user avatar
  • 73
1 vote
0 answers
84 views

How to smoothly interpolate gravitational field between trajectories in high dimension?

I'm looking for the adequate numerical interpolation technique to solve the following problem. This is probably trivial for physicists who study gravitational fields, but I didn't find clear answers ...
Youcef's user avatar
  • 19
11 votes
1 answer
451 views

D'Alembert's Principle: rigorous formulation using notions from modern differential geometry

Is there a rigorous definition of D'Alembert's principle of virtual dynamic work in the language of differential geometry? Some questions I'm hoping to answer are: How to view the configuration space ...
mcmathy's user avatar
  • 111
1 vote
0 answers
136 views

A question about Roger Penrose's spin networks and mathematical formalization?

Let $a,b,c$ be "units" in the spin network. Then there are there are the following three requirements to fulfill (according to the relevant Wikipedia entry): $a,b,c \in \mathbb{N}$ Triangle ...
mathoverflowUser's user avatar
1 vote
0 answers
144 views

polynomial approximation of hypergeometric function 2F1

I have the following function $T(k_1,k_2)$ resulting from multiphoton transition matrix elements calculations: $T(k_1,k_2)=\gamma^{-k_2}\sum_{j=0}^{k_1}(j+2)_{l+1}\binom{k_1}{j}(k_1+1)_3(\gamma-1)^{j}{...
Omer Amit's user avatar
0 votes
1 answer
266 views

Mathematical characterization of gravitational geons as reference request, and their properties as main question

I've edited (ten days ago) a question on Physics Stack Exchange, this Mathematical characterization of gravitational geons, post with identifier 726281 the users of the site were kind adding in the ...
user142929's user avatar
5 votes
0 answers
125 views

Particles sent into the same direction with uniformly distributed speed

Fix a positive integer $n$. Every second, a particle is sent along a straight line from a fixed position in a fixed direction, at a random integer speed chosen uniformly in $\{1,\ldots, n\}$ meters ...
Dominic van der Zypen's user avatar
18 votes
9 answers
5k views

How does a Masters student of math learn physics by self?

I am a Masters student of math interested in physics. When I was an undergraduate, I took the introductory course of physics, but it is just slightly harder than high school physics course. To be ...
LZB's user avatar
  • 183
2 votes
1 answer
346 views

Examples of ODEs with complex constant coefficients and applications to physics?

This question is asked on stackexchange: Are there examples for ODEs with complex coefficients with applications in physics? but received no answers. I am reposting it here on the hope that it catches ...
Medo's user avatar
  • 676
3 votes
0 answers
191 views

How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials

From numerical experiments in Mathematica, I have found the following expression for the integral: $$ \int_{-1}^{1}h_{n}^{(1)}\left(\sqrt{a^{2}+b^{2}+2ab\tau}\right)P_{n}^{m}\left(\frac{a\tau+b}{\sqrt{...
Chris's user avatar
  • 31
9 votes
1 answer
373 views

Why are discreteness and smoothness in physics inversed with respect to geometry?

In a closed (say differentiable) Riemannian manifold you see only continuous features when looking at small neighbourhoods of points. From afar, discrete features appear ((co)homology, closed ...
Roland Bacher's user avatar
3 votes
0 answers
107 views

Are square configurations the only critical points of the energy on the circle?

$\newcommand{\S}{\mathbb{S}^1}$ $\newcommand{\la}{\lambda}$Let$$M=\{(x_1,x_2,x_3,x_4) \in (\S)^4\,\, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$ Define $E:M \to \mathbb{R}$ by $$E(x_1,x_2,...
Asaf Shachar's user avatar
  • 6,611
4 votes
0 answers
100 views

Superspace derivation of supersymmetric non-linear sigma model in Supersolutions by Deligne and Freed

I am having a little trouble understanding passage from the linear to the non-linear sigma model in Section 4.1 of Supersolutions by Deligne and Freed. Most of my confusion comes down to the ...
user avatar
3 votes
1 answer
495 views

Why are solenoidal fields called solenoidal?

A solenoidal tangent field, mathematically speaking, is one whose divergence vanishes. They are also called incompressible. I understand why they are called incompressible — a fluid flow is called ...
Mozibur Ullah's user avatar
16 votes
1 answer
731 views

From a physicist: How do I show certain superelliptic curves are also hyperelliptic?

As the title suggests, I am a physicist and have a question about how to show certain superelliptic curves are also hyperelliptic. The superelliptic Riemann surfaces in question has the form $$w^n = \...
Kestrel's user avatar
  • 163
19 votes
4 answers
2k views

Applications of complex exponential

In calculus we learn about many applications of real exponentials like $e^x$ for bacteria growth, radioactive decay, compound interest, etc. These are very simple and direct applications. My question ...
Max's user avatar
  • 199
4 votes
1 answer
381 views

How are spatial coordinate systems in physics defined?

Grothendieck once asked "What is a meter?" (https://golem.ph.utexas.edu/category/2006/08/letter_from_grothendieck.html). This innocent sounding question, made me to think about how ...
mathoverflowUser's user avatar
2 votes
0 answers
164 views

Is there an example Hamiltonian that is uncomputable?

In a paper from 2015 Toby S. Cubitt et al showed that the problem of determining the existence of a band gap in the excitation spectrum of a quantum many-body system, was undecidable. This result ...
user400188's user avatar
2 votes
4 answers
313 views

EM-wave equation in matter from Lagrangian

Note I am not sure if this post is of relevance for this platform, but I already asked the question in Physics Stack Exchange and in Mathematics Stack Exchange without success. Setup Let's suppose a ...
Chopin's user avatar
  • 61
-1 votes
1 answer
401 views

Harmonic function in infinite domain in $\mathbb{R}^3$, constant on the boundary and decaying as $1/r^2$

EDIT: Let $\Omega\subset \mathbb{R}^3$ be a bounded domain with smooth connected boundary. Let $f\colon \mathbb{R}^3\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\...
asv's user avatar
  • 21.1k
0 votes
1 answer
372 views

Harmonic functions in infinite domain in Euclidean space

EDIT: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\...
asv's user avatar
  • 21.1k
0 votes
0 answers
83 views

The specific connection between the Hecke operator and the t'Hooft Operator

As I was reading some articles concern about the Selberg trace formula and its general form, I have noticed that the Selberg trace formula and its general form can be understand as the energy spectrum ...
loveimissyou123's user avatar
2 votes
1 answer
160 views

Vacuum state generating functional

In Theorem 1 of this paper Segal stablish a relation between states and generating functionals. He assert that in order to $\mu$ be a generating functional must satisfy $$ \sum_{j,k\in F} \mu (z_j-...
Gabriel Palau's user avatar
8 votes
0 answers
1k views

Is there any physics theory which is similar to these analogies?

Since I am doing this little "research" project on my spare time and in my physical neighborhood there are not many people to discuss these ideas, I wanted to share with you a small point of ...
mathoverflowUser's user avatar
11 votes
1 answer
1k views

State of rigorous effective quantum field theories

It's well-known that there are no rigorously constructed and physically relevant QFTs. There is, however, a lot of mathematical work on effective field theories and renormalization, such as the books ...
Pedro's user avatar
  • 269
4 votes
1 answer
494 views

Formula involving Wigner's 3j symbols and integration over irreducible representations of SU(2)

In some calculations, I saw the following formula $$\int_{\mathrm{SU}(2)}\,\mathrm{d}g\,D^{j_{1}}_{m_{1}n_{1}}(g)D^{j_{2}}_{m_{2}n_{2}}(g)D^{j_{3}}_{m_{3}n_{3}}(g)=(-1)^{j_{1}+j_{2}+j_{3}}\begin{...
B.Hueber's user avatar
  • 987
4 votes
2 answers
1k views

Reference for mathematical Palatini formalism of general relativity

I know that this is maybe not a research level question, but since the topic is quite special, I thought that the chance to get some reference is higher in this community. I am looking for a reference ...
B.Hueber's user avatar
  • 987
1 vote
0 answers
165 views

Angular velocity from rotation matrix difference [closed]

I am working on something for a game. I need to calculate the angular velocity, however in my situation I only have access to the previous rotation matrix and the current rotation matrix. My angular ...
Chryfi's user avatar
  • 111
1 vote
0 answers
70 views

Is there an analytic formula (or even a name...) for a plane curve with curvature inversely proportional to x?

I'm interested in plane curves with curvature inversely proportional to distance from the axis: $$\kappa(t) = \left(\frac{x'(t) y''(t) - y'(t)x''(t)}{(x'(t)^2 + y'(t)^2)^{3/2}} \right) = \frac{1}{a x(...
Jacob Schwartz's user avatar
3 votes
1 answer
321 views

What is the precise relationship between real Poisson algebras and commutative $C^*$ algebras?

I've been teaching myself quantum mechanics, and I realized that I'm missing something fundamental. Namely, there are two pictures that I don't know how to reconcile: Quantum Mechanics generalizes ...
Andrew NC's user avatar
  • 2,011
3 votes
4 answers
942 views

Applications of Hamiltonian formalism to classical mechanics

In many courses in theoretical classical mechanics Hamiltonian formalism takes an important place. However I did not see it applied to problems of classical mechanics (unless one expands the scope of ...
asv's user avatar
  • 21.1k
15 votes
6 answers
3k views

Maxwell equations as Euler-Lagrange equation without electromagnetic potential

In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) ...
asv's user avatar
  • 21.1k
1 vote
1 answer
109 views

Is there a Bell inequality for each of $2\times 2$, $3\times 1$, $2\times1\times1$ and $1\times1\times1\times1$ configurations?

There was no answer in https://physics.stackexchange.com/questions/600494/is-there-a-bell-inequality-for-2-times-2-and-1-times1-times1-times1-configur. Hence posting in mathoverflow on the possibility ...
Turbo's user avatar
  • 13.6k
9 votes
1 answer
778 views

Why the least action principle is always (?) used in this particular form?

The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order)...
asv's user avatar
  • 21.1k
3 votes
2 answers
380 views

Classification of Lagrangians with given Euler-Lagrange equations

In (mathematical) physics the equations of motion of a system of particles are often interpreted as Euler-Lagrange equations for appropriate Lagrangian $L=L(x,\dot x,t)$ where $x$ is a collection of ...
asv's user avatar
  • 21.1k
2 votes
0 answers
230 views

What is the relationship between Riemannian and sympletic musical isomorphisms on the cotangent bundle?

Let $M$ be a smooth manifold. Its cotangent bundle naturally has a symplectic structure, and this gives rise to musical isomorphisms. These musical isomorphisms are the ones from physics that relate ...
Tim Phalange's user avatar