# Questions tagged [physics]

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

160
questions

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### 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 ...

1
vote

1
answer

172
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### 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 ...

16
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1
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### 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 = \...

19
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4
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### 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 ...

3
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1
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### 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 ...

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### 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 ...

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### 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 ...

0
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1
answer

248
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### 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 $\...

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1
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164
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### 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\...

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### 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 ...

2
votes

1
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### 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-...

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### 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 ...

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### 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 ...

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1
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### 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{...

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2
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### 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 ...

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### 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 ...

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0
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### 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(...

2
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1
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### 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 ...

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4
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576
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### 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 ...

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### 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) ...

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1
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### 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 ...

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### 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)...

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2
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### 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 ...

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### 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 ...

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### Deriving integral in Gaiotto-Tommasiello theory

I was looking at a paper by Takao Suyama on GT theory, and I couldn't figure out how he derived his formula (3.59):
$$\frac{1}{\pi}\int_a^bdx\frac{1}{z-x}\frac{\sqrt{(z-a)(z-b)}}{\sqrt{|(x-a)(x-b)|}}\...

3
votes

1
answer

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### Monotile that tiles when you apply a rubber band

My (non-mathematician) friend asked me a physics/tilings question that maybe someone here is interested in dissecting, or can point to the literature if this problem has been studied.
Does there ...

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1
answer

52
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### Set of eigenvalues of the boundary problem

I'm looking for the results about the set of eigenvalues of boundary problem for differential equation
\begin{equation}
\bigl(p(x) u'(x; \lambda) \bigr)' + q(x) u(x; \lambda) = -\lambda w(x) u(x; \...

2
votes

0
answers

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### geometrical or physical interpretation of second Chern classes of Calabi-Yau threefold

It's my first post.
Consider Calabi-Yau threefold $M$ and its tangent bundle $TM$. I know $c_1(TM)=0$ means metric on $M$ is a solution of vacuum Einstein equation. Then my question is "are there any ...

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### Applications of Generalized Geometry to Theoretical Physics [closed]

I'm looking for some topics on Generalized Geometry applied to Physics for a master thesis. I took an introductory course last year, and I have a degree in both Mathematics and Physics. I would ...

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votes

1
answer

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### Gadgets as primality tests

From the literature, showed below, I know two gadgets that provide a way to know if a positive integer (a positive quantity of units) is composite or a prime number. I would like to know if in the ...

2
votes

1
answer

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### Sufficient conditions for unitarity of a representation of a Lie Superalgebra

Suppose we have a Lie superalgebra with triangular decomposition:
\begin{equation}
\mathfrak{g} = \mathfrak{g}^{+} \oplus \mathfrak{g}^{0} \oplus \mathfrak{g}^{-}
\end{equation}
I've seen it stated ...

69
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### The Planck constant for mathematicians

The questions
Q1. What are simple ways to think mathematically about the physical meanings of the Planck constant?
Q2. How does the Planck constant appear in mathematics of quantum mechanics? In ...

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### Cardinal Invariants and Physics

There are many applications of topology to physics, but I wonder if there is a known application of cardinal invariants to physics.

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### What are the topological phases of quantum Hall systems?

(Fractional) quantum Hall systems are $2+1$-dimensional models which are said to possess topological order. One (maybe even complete) set of invariants of topological phases in $2+1$ dimensions is ...

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### Difference Between Total Least Squares Plane and Plane Orthogonal to Principal Axis of Inertia Tensor

Given a finite set $P$ of points in $\mathbb{E}^3$ , one can calculate an approximating plane either as the solution of a Total Least Squares problem or by interpreting the problem physically, ...

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### Does Dijkgraaf-Witten theory have a time-reversal symmetry?

By having a time-reversal symmetry I mean that there is a local anti-unitary symmetry (representing the non-trivial element of $Z_2$) of the state-sum construction (or, if you want, of the associated ...

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1
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### Importance of the principal bundle in Chern-Simons theory

This is a very basic beginners question about Chern-Simons theory. The configurations that we sum over to get the partition function are given by a Lie-algebra valued 1-form $A$ on a topological 3-...

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1
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### Fully extended TQFT and lattice models

I often read that fully extended TQFTs are supposed to classify topological phases of matter. So I would like to understand the formal nature of fully extended TQFTs on a more direct physical level (...

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### Energy-minimizing set of discrete points in a bounded domain

Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain.
Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize
$$
\sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|}
$$
over all ...

4
votes

0
answers

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### List of Replica Symmetry results for different models?

Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have?
I am aware of some of the more famous results, e....

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0
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### Questions about using mathematical methods to prove the Caratheodory's Concept of Temperature

Caratheodory's Concept of Temperature is not Carathéodory's theorem.
I have tried,but I found nothing about this question by searching online.
This is what I have seen in a thermodynamics textbook; ...

3
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0
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### Does there exist a compactly supported integrable function with infinite Coulomb energy?

The title of the question pretty much says it all. I am looking for a function $f\in L^1(\Omega)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain, such that
$$
E[f] = \iint\limits_{\Omega\...

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1
answer

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### Localization of solutions for time-dependent Schroedinger equation

I've been playing around with numerical solutions to the Schroedinger equation and I came across something that feels very natural, but I was not able to prove it with the math I know.
The ...

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0
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### 1D Schrödinger Equation with Measure-Valued Coefficients

I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following:
$$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}...

3
votes

2
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### Legendre equation: An interpretation [closed]

I am a student of physics and, especially in quantum mechanics, we are presented with the Legendre equation:
\begin{eqnarray}
(1-x^2)y''-2xy'+l(l+1)y=0.
\end{eqnarray}
Doing some calculations, we ...

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votes

2
answers

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### Runner's High (Speed)

I find the following mind-boggling.
Suppose that runner $R_1$ runs distance $[0,d_1]$ with average speed $v_1$. Runner $R_2$ runs $[0,d_2]$ with $d_2>d_1$ and with average speed $v_2 > v_1$. I ...

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### Does current follow the path(s) of least (total) resistance?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (...

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votes

0
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### Partial Liouville equation

In my master's thesis, I worked on mathematical multi-scale models for muscle tissue. Now after finishing it, I would like to find out if one direction could be a research topic for my PhD.
At one ...

2
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1
answer

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### PDE’s whose solutions can be presented using path integrals

It is well known that solutions of the Schroedinger equation and of the heat equation can be presented using path integrals:
$$\psi(x,t)=\int K(x,t;y,0)\psi(y,0)dy,$$
where the kernel $K(x,t;y,0)$ is ...

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votes

1
answer

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### Quantum tunneling on the line with non-symmetric double well potential

Consider the Schroedinger equation on the line
$$i\frac{\partial \Psi(x,t)}{\partial t}=[-\frac{d^2}{dx^2}+V(x)]\Psi(x,t),$$
where one assumes that $V(x)\to +\infty$ as $|x|\to +\infty$, and $V$ has ...