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Questions tagged [mathematical-physics]

This tag is intended for questions on mathematical methods used in quantum theory, general relativity, string theory, integrable system etc at an advanced undergraduate or graduate level. For questions about more basic mathematical problems originating from physics use the [physics] tag instead.

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Is the inverse of the position operator compact?

Let $X$ be the position operator on $L^2(\mathbb{R})$, defined by $(Xf)(x)=xf(x)$ on a suitable domain of definition. We can then obtain an operator $\frac{1}{|X|}$ by two (equivalent, I think) ways:...
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$u_{xx}-u_{yy} + \frac{4}{x}u_x+\frac{2}{x^2}u=0$

I have some problems with solving PDEs. \begin{cases} \ u_{xx}-u_{yy} + \frac{4}{x}u_x+\frac{2}{x^2}u=0 \\[2ex] u(x,x)=1,\quad u(1,y)=y \end{cases} What I've done: $$u(x,y)=\frac{1}{x^2}v(x,...
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Book on tetrads formalism and tetradic formulation of General Relativity

Could anyone give me some references for mathematicians (coordinates free notation, formalism of fiber bundles etc.) about tetrads, Palatini-Cartan theory, stuff about formulation of GR with tetrads? ...
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Introductory books on Complex Geometry for Theoretical Physics audience

I am looking for some good introductory books on Complex Geometry for Theoretical Physics audience. My background is General Relativity and Riemannian geometry. I am looking for a book with proper ...
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28 views

Convergence of scalar product in a hilbert space

Background: From this paper I'm trying to understand why the OPE in conformal field theory has a finite radius of convergence. The authors make the claim that the scalar product of two states ...
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2answers
38 views

Going from one notation to another in Yang-Mills

In many books on Yang-Mills theories, written from a physicist's point of view, the curvature tensor is written as: $$F_{\mu \nu}^a = \partial_\mu A_\nu^a-\partial_\nu A_\mu^a+gf^{abc}A_\mu^bA_\nu^c$$...
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Retrieving a function of many complex variables from its manifold of zeros

Physical background: In the Bargmann-Segal representation, the states of bosonic quantum systems are holomorphic functions of $N$ complex variables where $N$ is the number of degrees of freedom. The ...
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1answer
8 views

Asymptotic behaviour of Volterra integrodifferential equation

For an equation of the form, $$\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{y}(t) = \int_0^t \mathbf{K}(t-\tau)\mathbf{y}(\tau)d\tau,$$ Can it be shown that in the long time, $t\to\infty$, limit, the ...
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1answer
28 views

Use Milne Thomson circle theorem to show complex potential for this flow

I was wondering if anyone could help me with the following problem, as I'm unsure on how to begin. Any suggestions would be appreciated, thanks for reading this. I just don't understand how to apply ...
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1answer
56 views

An integral resulting from perturbation theory

After a calculation with perturbation theory in a many-body problem, I end up with the following integral which I cannot compute: $$I=\int_{-D}^{E}\int_{E}^{D}\int_{-D}^{E} dx_1 dx_2 dx_3 \dfrac{1}{x-...
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Particle in a 3-D Box

Reading through my quantum mechanics book I've stumbled on a question any help would be great. Suppose we had a 3-D square well i.e. $$V(x,y,z)=\begin{Bmatrix} 0 \ \text{if}\ 0\leq x \leq a \ , \ 0\...
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Curvilinear Coordinates Transform matrix

In Kusse's Mathematical Physics, equation gives the transformation matrix between 2 curvilinear systems as $q'_i=a_{ij}q_j$. Equation 4.100 lists $a$ as: $$a_{ij}=\frac{h'_i}{h_j}\frac{\partial q_i'}{\...
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47 views

Deriving the inequalities in potential theory

I am reading the following pages from my book on "foundations of potential theory" page 150/151: I understand up to inequality $(4)$. I can't derive inequality $(5)$ from $(4)$. Please derive ...
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1answer
34 views

Systematic way of obtaining conservation laws in dynamical systems

Motivation Consider a point particle of mass $m$ moving in $\mathbb{R}^3$ under the influence of some force field $\vec{F}(\vec{r},t)$. The fundamental equation governing the dynamics of this system ...
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1answer
31 views

How to find the matrix generators of higher dimensional irreducible representations of $\operatorname{SU}(2)$?

Using the most general form of a $2\times 2$ unitary matrix $U$ of determinant $+1$ and using the formula $$T^a=-i\frac{\partial U}{\partial\theta^a}|_{\{\theta_a\}=0} ~{\rm with}~ a=1,2,3. $$ In this ...
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Generating function of cuts of some graph

Consider the graph $G$, each edge of which has weight $T$. Also consider some cut of $G$. We call the weight of the cut - product of all edges included in the cut. Generating function of cuts of $G$ ...
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37 views

How general is the tennis racket theorem?

My question is related to this post from physics SE, which deals with the tennis racket theorem There are surely a thousand ways to prove the result described in that question, i.e. that rotation ...
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1answer
36 views

Evaluation of generalized Laguerre function integrals using orthogonality relations

(NB - I am not asking to be spoon-fed with complete solutions, just pointing out any useful transformations, or giving general pointers would suffice.) The orthogonality relation for generalized ...
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14 views

Refractive index as a harmonic function of y

Given a refractive index of the form $n(y) = n_0 \cos(ky)$ , where $n_0$ and $k$ are positive constants. Is it possible to determine the trajectory of a ray of light, in the x-y plane, traveling ...
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23 views

Two equivalent definitions of fibration structure on toric CY $3$-fold

I've been trying to reconcile two seemingly different definitions of what a toric space is, specifically a toric Calabi-Yau 3-fold. The first definition is from the paper Branes and Toric Geometry, by ...
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19 views

Geodesics of null surface and induced connection

Although this is a question related to GR this is purely a math question, so I think this is the right place to ask it. Let $(M,g)$ be a Lorentzian manifold. Let $\phi : \Sigma\to M$ be one null ...
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Fourier Transform of Poisson's equation, then taking it back to real space.

I'm a bit stuck on a homework problem and could use some guidance. The problem asks to use a specific potential in Poisson's equation ($ \nabla^2\Phi = -\rho/\epsilon_0 $), Fourier transform it, ...
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What is meant by “collective behavior” in the definition of plasma?

"Plasmas are many-body systems, with enough mobile charged particles to cause some collective behavior ." [M.S. Murillo and J.C.Weisheit Physics Reports 302, 1-65 (1998)]. In the above definition ...
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32 views

equilibrium point in an inverted pendulum

The unstable upright position of an Inverted Pendulum on a cart does this corresponds to a Hyperbolic equilibrium point or a non- hyperbolic equilibrium point? please give me a lucid explanation on ...
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1answer
66 views

Homogeneous Fredholm integral equation with asymmetric logarithmic kernel

In the study of few-particle quantum systems, I have come upon the following integral equation: \begin{equation} f(x) = \lim_{\Lambda\rightarrow \infty}\frac{3\sqrt{3}}{\ln(4x^2+6\lambda) + 2\gamma} \...
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35 views

Is it possible to localise the renormalisation group?

The renormalisation group allows you to consider what happens to a model when considered on different spatial scales. It is not a group, reflecting the fact that you can always move to a larger ...
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2answers
38 views

Elastic collision between a circle and a point

In a 2D environment, I have a circle with velocity v, a stationary point (infinite mass), and I am trying to calculate the velocity of the circle after a perfectly elastic collision with the point. ...
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24 views

Thomas method for Crank–Nicolson scheme with a central finite analog. How to get the boundary values of the coefficients?

The crank nicolson scheme: $$\begin{cases}\Large \frac{w_i^k - w_i^{k-1}}{h_{t}} = \frac{m}{2c} (\frac{w_{i+1}^k - 2w_i^k + w_{i-1}^k}{h_x^2} + \frac{w_{i+1}^{k-1} - 2w_i^{k-1} + w_{i-1}^{k-1}}{h_x^2}...
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1answer
48 views

How do you solve this physics problem?

A $3000\,\rm kg$ car travels at a velocity of $6.00\,\rm m/s$ due north then accelerates at $3.00\,\rm m/s^2$ due south. What is the velocity of the car after $1.50$ seconds?
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How to prove those Integrals formulae

I come across the following two integral formulae The first integral formula is \begin{equation} \int_C d^2z |z|^{2a}|z-x|^{2c}|z-1|^{2b} = \frac{S(a)S(c)}{S(a+c)}|I_{0x}|^2+\frac{S(b)S(a+b+c)}{S(...
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Prove two Integrals formulae in two dimension

I come across the following two integral formulae The first integral formula is \begin{equation} \int_\Bbb C d^2z |z|^{2a}|z-x|^{2c}|z-1|^{2b} = \frac{S(a)S(c)}{S(a+c)}|I_{0x}|^2+\frac{S(b)S(a+b+c)}{...
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2answers
128 views

Proof that one can replace coordinate derivatives in coordinate formula for Lie derivative with covariant derivatives

How would we show that for a tensor of any rank we can replace the partial derivatives by co-variant (Levi-Civita) derivatives, I was reading this is a GR text where it was left to the reader as an ...
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23 views

Trajectory of points at infinity

If two points initially starts at zero and travel to infinity,what is the nature of the space if the trajectory where to: 1) converge 2) diverge & what will happen to the trajectory of two points ...
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22 views

Direction of Electric field vector

Why in an electromagnetic wave, the direction of electric field vector is considered normal to the direction of propogation?
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65 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\...
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1answer
25 views

Logarithmic non-linear system of equations

As part of an heat exchanger problem, we are given two unknowns $T_{cs}$ and $T_{fs}$ solving the following equations: $\forall (Q,U,F, m_c,m_f, C_c,C_f, \Sigma , T_{ce},T_{fe}) \in \mathbb{R_+^*}^{10}...
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1answer
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How to find the Tangential Speed? [closed]

This is the problem: An airplane accelerates at $47.5/s^2$ while turning through a loop $500$ m. Calculate the plane's tangential speed. I have tried up and down to solve this problem. The only ...
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Grasping the idea of Virasoro Algebras in 2D Conformal field theory

I have been trying to understand the connection between Virasoro algebras and CFT. After a course in string theory, I was under the impression that the Virasoro algebra was simply the Lie algebra of ...
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31 views

Pagerank indegree relation derived process

I read paper which derive the relation between indegree and pagerank. But I was confused about some steps. I confused the $NP(\mathbf{k})$ part which marked by Question in steps below. Notation and ...
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50 views

Shallow water equation entropy concept

I am a beginner in shallow water equation. I am interested in the equation $$h_t+(hu)_x=0$$ $$(hu)_t+(hu^2+\frac{1}{2}gh^2)_x=0$$ I have the following doubts 1)Weak solutions are not unique in ...
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51 views

Why a minus sign in Helmholtz Equation?

Combining Gauss's law and $\vec{E} = -\nabla \phi$ gives Poisson's equation \begin{equation} -\nabla^2 \phi = \frac{\rho}{\epsilon_0} \end{equation} for which the motivation is clear as to why the ...
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Quantum Info in math departments

I'm currently a graduate student in mathematics getting my master's degree. I am interested in a bit probability and partial differential equations, and, secondarily, a bit of mathematical physics; ...
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1answer
32 views

Where is it used that the symmetry conserves the symplectic form in noether's theorem.

For the proof of Noether's theorem, it seems like that the only thing that's important for the symmetry map $S_g : M \to M$ is that it conserves the Hamiltonian (which will then imply that the moment ...
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38 views

A treatment of mathematical foundations with motivation from physics

I am looking for a work that treats the foundations of mathematics (linear algebra, real analysis, vector analysis, complex analysis…) in particular insofar they find application in mathematical ...
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38 views

What is symmetry in physics in the mathematical sense, using (Lie) groups

In physics, we sometimes say that, for example, a certain classical system has a certain symmetry, which is given by some group. I don't feel like I understand this well enough. Are there some good ...
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Laplacian of $1/r$ in a tensor

As we know the $$\nabla^2(1/r) =- 4 \pi \delta^3(r).$$ However, I recently was readling an hydrodynamic book (An introduction to dynamics of colloids By J.K.G Dhont). The Oseen tensor is defined as: ...
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1answer
26 views

Elastic Scattering angle

I was reading Introduction to Nuclear Physics by Krane and stumbled on the following (page 47): In Elastic scattering, the initial electron wave function is of the form $e^{i k_i r}$ (free particle ...
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2answers
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Complex SUVAT help?

Two objects are dropped from the top of a cliff height $H.$ The second is dropped when the first has travelled a distance $D.$ Prove that the instant when the first object has reached bottom, the ...
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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) ...
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1answer
79 views

The axiomatic minimum required to have unique solutions to the Schrödinger equation

Let us consider the free non-relativistic Schrödinger equation $$i\partial_t \psi =-\frac{1}{2}\partial_x^2 \psi=:H\psi.$$ Adapting Fritz John's pathological solution to the heat equation, I find that ...