Questions tagged [mathematical-physics]

DO NOT USE THIS TAG for elementary 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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How to set up Chern-Simons theory, compute its "topological gauge shift" term, and compute its quantum path integrals with Wilson lines?

Suppose we want to consider Chern-Simons theory on an (odd-dimensional) compact boundaryless smooth manifold $X$ for a Lie group (the "structure/gauge group") $G$. Is it possible to do so ...
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1 vote
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Verifying Legendre's equation

Use the following results: $$(l+1)P_{l+1}(x)-x(2l+1)P_l(x)+lP_{l-1}(x)=0,$$ $$P_l(x)+2xP'_l(x)=P'_{l+1}(x)+P'_{l-1}.$$ in order to show the following recurrence relations: $$(2l+1)P_l(x)=P'_{l+1}(x)-P'...
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-2 votes
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Why is it impossible to solve the heat diffusion/conduction equation as a function of time in 2D and 3D situations?

The current literature provides solutions for heat diffusion/conduct PDE only for a 1D geometric space plus time dimensions. Why there are no 2D and 3D spatial dimensions plus time. Are mathematical ...
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What is the most generic way to write a Lagrangian quadratic in velocities?

I'm trying to generalize the expression for a Lagrangian to a manifold that doesn't posses a metric. To be more clear, when the configurations space has a metric, we write the lagrangian quadratic in ...
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1 answer
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How to prove the restricted Lorentz Group is connected?

Many textbooks claim that $SO^{+}\left(1,n\right)$ is the identity component of $O\left(1,n\right)$. But how do we know that $SO^{+}\left(1,n\right)$ is connected itself?
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Mayer expansion: from product over i,j to sum over graph

I am studying the Mayer expansion used in Statistical Physics. We arrive at the following expression: $$ \prod_{i<j=1}^N (1+f_{ij}) $$ and then is find out that this expression is equivalent to the ...
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1 vote
1 answer
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Set $T^{\mathbb N}x$ dense in $\mathbb S^1$ (Poincaré recurrence theorem)

Let $Ω =\mathbb S^1$ be the unit circle in $\mathbb R^2 = \mathbb C$, and let $T : Ω → Ω$ be multiplication by $e^{i\alpha}$. For $α \notin π\mathbb Q$ and every $x ∈ Ω$, is the set $T^{\mathbb N}x$ ...
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Problem in Newton's Principia mentioned by V. I. Arnol'd

In an article named Conversation with Vladimir Igorevich Arnol'd, Arnol'd said I will state another problem mentioned there, this one variational, on the solid of revolution with least resistance ...
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Ergodicity on a finite set [closed]

Let $\Omega$ be a finite set ($ \#\Omega = n$), how many dynamical systems on $\Omega$ are ergodic?
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Confusions about function spectrum in Anderson duality

In appendix B of this paper https://arxiv.org/abs/math/0211216, Hopkins and Singer defined the Anderson dual $\tilde{I}(E)$ of a spectrum $E$ as the function spectrum of maps from $E$ to $\tilde{I}$, ...
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What are the steps to get that value integrating the given function?

How to calculate this integral $W=\int_0^{2\pi}\dfrac{6{\epsilon}{\mu}{\omega}{(R/C)^2}\cdot\left({\epsilon}\cos\left({\theta}\right)+2\right)\sin\left({\theta}\right)}{\left({\epsilon}^2+2\right)\...
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3 votes
3 answers
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Can a manifold be reconstructed from its charts?

I'm learning special relativity and I am having a confusion on this mathematical point. Whenever any sort of motion or non motion happens in the world, it can only be perceived by the scientist in a ...
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2 votes
1 answer
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Calculate the integral where $P_{n}$ and $P_{m}$ are Legendre Polynomials

Calculate the folowing integral: $$I_{k,m}=\int_{-1}^{1} x(1-x^2)P'_{n}(x)P'_{m} dx $$ So, my attempt to solve this consisted in: First, I thought of manipulating the folowing relations so i could get ...
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Question on the proof of the Spectral Theorem

In Nielsen and Chuang's book on quantum computing, the following proof for the spectral theorem (any normal matrix is diagonalizable with unitary matrices) is given. I'm unable to follow the ...
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1 answer
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Angle between two random unit vectors uniformly distributed

Consider $x, y \in S_{1}^{d-1}$ (the unit n-sphere in d dimensions) with $(x \cdot y)^2 = 1/d$. I need to compute the angle $\alpha$ between $x$ and $y$ for $d$ = $3$ and asymptotically for large $d$. ...
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Applications of Diophantine equations? [duplicate]

It had proved that there is no algorithm to solve Diophantine equations, for that reason I want to know what are the Diophantine equations that physicists or chemists need to solve? or any other ...
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time derivative of work energy theorem

so I stumbled upon a step in my textbook which I can't do by myself, hopefully someone can help me :). It goes as follows $${\frac{d \bf{p}}{dt} \cdot \bf{u}} = \frac{d}{dt}(\frac{m \bf{u}}{\sqrt{1-\...
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Defining free Dirac operator on $\big(L^2(\mathbb{R}^n)\big)^{n+1}$

So I have been considering free Dirac operator $D_0:(L^2(\mathbb{R}^3))^4 \supset (W^{2,1}(\mathbb{R}^3))^4 \to (L^2(\mathbb{R}^3))^4$ given by formula \begin{equation} D_0 = \left[\begin{matrix} ...
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1 vote
1 answer
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Are differential equations beyond coordinates?

In Physics when we write down Newton's second law, the differential equation we have is coordinate agnostic. Meaning, we can put in any coordinates into the equation and get a second order DE which ...
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How to derive the following approximation for a set of coupled equations?

Background I have the following set of coupled equations $\frac{d}{dt}\rho_{ab} = -i\omega A - \frac{1}{2}\Lambda B$ with $\omega, \Lambda$ parameters that are real and $A, B$ 4x4 matrices that are ...
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0 answers
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what are vassiliev invariants and how are they related to Chern-Simmons?

Possibly this question is very open for this platform and a little diffuse but I am a physicist and my knowledge in tides is not that high but I do not understand graphically what the vassiliev ...
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1 vote
1 answer
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2n independent variables and DOF = n in Hamiltonian Mechanics

I’m studying Lagrangian and Hamiltonian mechanics for the first time. In the Lagrangian approach, that is the one I’ve studied first, a fundamental point is the definition of the degree of freedom of ...
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What does it mean for the momentum operator P to satisfy P|n>=hn|n>?

Using periodic boundary conditions to analyze momentum eigenstates on the circle $S^1$, we have the momentum operator $P=-i\hbar\frac{d}{d\phi}$ which satisfies $P|n>=\hbar n|n>$ (for clarity, |...
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Relation between normal matrices of matrix Lie algebras and their Cartan subalgebras

Math question: Let $\phi = \phi_A X^A \in \mathfrak g$ where $\phi_A \in \mathbb C$ for all $A =1,\cdots, \dim \mathfrak g$, and $X^A$ form a basis of some semi-simple Lie algebra $\mathfrak g$ over $\...
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Are tensors constructed such that one forms "act" on some complex vector field?

I am a physics student, trying to learn differential geometry. I have some confusion understanding the motivation in constructing tensors (or tensor fields). On a differentiable manifold $\mathcal{M}$ ...
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1 answer
55 views

How to calculate the new position at time t of bodies with variable acceleration?

Intro For an N-body simulation of calestial bodies I need to calculate on the one hand the accelerations of the calestial bodies based on the received gravitational forces [done] and on the other ...
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0 answers
39 views

An interesting case of regular family of regular functions.

I found this family of functions denoted by $W_{g,n}(z_1 , z_2, \ldots , z_n)$ which are regular appearances in the variable $a$ but I want to find a proof or argument for why this is happening. If ...
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2 votes
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What is the lower $n$ such that $\mathbb{HP}^2$ can be embedded in $\mathbb{R}^n$?

Applying basic cohomological calculus it is possible to derive that $\mathbb{HP}^2$ cannot be embedded in a Euclidean space of dimension 11 or less. From other side, applying spinorial cohomological ...
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Irreps of the Lorentz Group

On Wikipedia and elsewhere, one finds claims that the exhaustive list for the irreps of the (proper, ie. connected-to-identity component of the) Lorentz group are labelled by two half-integer labels $(...
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1 vote
0 answers
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Integral measure quantum fields

In many physics books (especially QFT), the integral measure for a quantum field is written down as \begin{equation} \phi(x)=\int\frac{d^{3}p}{(2\pi)^{3}}.... \end{equation} It is a summation over the ...
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  • 51
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1 answer
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A boundary value problem of a Harmonic potential

A 2D electrostatic (i.e. harmonic potential) boundary value problem is shown in the figure. The solid lines are conductors (all are parallel), the two conductors with potential $V$ are infinitely long,...
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  • 686
0 votes
1 answer
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Error in Exponential Sum Formula

I have noticed that, for the exponential sum formula: $$\sum_{n=0}^{N-1} r^n = \frac{1-r^N}{1-r}.$$ the formula is not correct (left side $\neq$ right side) when N has a factional part (not just ...
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3 votes
0 answers
29 views

Time evolution and matrix ODE

Let $M$ be a linear operator on a finite-dimensional complex Hilbert space (thus, $M$ is just a matrix). Assume that $\text{Tr}M = 1$, $M$ is self-adjoint and $M \ge 0$ (positive semi-definite). In ...
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  • 1,451
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0 answers
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Why does the solution to pendulum problem with the geometric approach of Jacobi metric does not correspond to the solution with Lagrangian approach?

When we solve the pendulum problem with EL equation, we get to the differential equation $\ddot{q}+\frac{g}{l}\sin q=0$ but when I apply the substitution $t \rightarrow t\sqrt\frac{g}{l}$ and ...
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1 vote
0 answers
32 views

Change orientation of turns of a trajectory

Consider a robot following the dynamical system $R_B'=R_B\exp(\omega_B^{\times}), v_W'=R_B\alpha_B + g_W$, where $R_B\in SO(3), v, \alpha, \omega, g \in \mathbb{R}^3$, and $\omega^\times$ is the skew ...
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4 votes
1 answer
102 views

Do the "physicist common knowledge" that "solenoidal vector fields have closed integral curves" have any mathematical foundation?

I remember having heard some physicist claiming that the integral curves of the magnetic field have to be closed, or "closed at $\infty$", due to the fact that the magnetic field is ...
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Find $\max_V \text{Tr} \left((Z_2 (V \otimes I) Z_1 (V^\dagger \otimes I)\right)$

I am doing a quantum optimization where the final problem has the following form $$\max_V \text{Tr} \left((Z_2 (V \otimes I) Z_1 (V^\dagger \otimes I)\right),$$ where $V \in \mathbb{C}^{d\times d}$ is ...
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5 votes
0 answers
46 views

Is every bounded linear functional on a subspace of a Hilbert space given by a function?

This question comes from the Limiting Absorption Principle (LAP). I want to obtain the most general statement possible, so I proceed as follows. Let $M$ be a topological space, $(H, \langle \cdot, \...
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-1 votes
1 answer
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Projectile motion science 9 [closed]

A foot ball is kicked at a certain angle above the horizontal. The vertical component of its initial velocity is $40 \ m/s$ and the horizontal component is $50 \ m/s$. Determine: a. its time of ...
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1 vote
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How to set up Contour Integration

So I am evaluating the integral $\int_0^\infty\frac{z^2dz}{(z^2-4)(z^2+9)}$. The residues for the four poles at $\pm2,\,\pm3i$ give$$\operatorname{Res}_{\pm2}(f)=\pm\frac{1}{13},\,\operatorname{Res}_{\...
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1 vote
1 answer
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Why does the arclength functional take same form in rotated coordinate systems?

I'm going through the textbook "Emmy Noether's Wonderful Theorem" by Dwight Neuenschwander. Therein, the author defines the coordinate transformation (infinitesmal rotation of orthogonal ...
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1 vote
0 answers
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What mathematical background is there in topological materials?

many physicists talk about topological materials but I can't find the reason why they say they use topology, is there any paper that explains these materials from a mathematical point of view? I ...
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0 votes
1 answer
56 views

How to extract $\mathbf b$ from $M = a I + \mathbf b \cdot \mathbf S$?

Any $2 \times 2$ hermitian matrix $M$ can be written $$M = a I + \mathbf b \cdot \boldsymbol \sigma,\tag{1}$$ where $a \in \mathbb R$, $\mathbf b \in \mathbb R^3$, $\boldsymbol \sigma$ is the Pauli ...
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0 answers
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Why were hall cohomological algebras created or what is the motivation behind that algebra?

I can't find a paper that talks about motivation or how they are used in physics, it's that an article by calabi-yau mentions that algebra caught my attention and how I'm learning about topological ...
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0 votes
1 answer
43 views

How can I prove the triangle inequality and the given corollaries? [duplicate]

How can I prove the following 3 claims? Or, if possible, could you provide suggestions for how I could prove them? For real numbers $x$ and $y$: Claim 1. $|x| − |y| ≤ |x − y|$ Claim 2. $|x-y| ≤ |x|+|...
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1 vote
0 answers
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What kinds of questions are people asking and answering with 𝐸8 lie algebra? [closed]

I don't understand much because there are many papers about it but I don't know what the objectives are or what is sought with this group of lie? or where is the investigation going or what is it ...
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0 votes
2 answers
83 views

Calculate Projectile flight time based on Gravity

Hey I want to calculate the flight time of a Projectile in 3D Space based on Bullet's speed, Velocity, Acceleration and Gravity or a custom downward force. I already have a formula that calculates the ...
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0 votes
0 answers
24 views

How to determine the fractal dimension of a random walk in 2D?

I have to determine the fractal dimension of a random walk in 2D. How can I do that? I am first supposed to picture the random walk for 2D as in the following site. However, I am also asked to ...
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  • 113
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0 answers
9 views

Relative speed and size

Background: I'm researching Low Earth Orbit (LEO) satellites and want to include some mathematical modeling to show a comparison between the potential impact that a large (50,000+) LEO satellite ...
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4 votes
1 answer
62 views

Symplectic Reduction of 3-D Chern Simons Theory

So, I'm new to gauge theories and symplectic reduction and was trying to analyze the Chern Simons theory in three dimensions. I have a few questions regarding the steps towards reduction. First off, ...
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