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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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Finiteness of results in Connes-Kreimer approach

Remark: Although this is technically a physics-related post, the content heavily relies on pure mathematics, so I deemed it more appropriate here. When reading the papers by Connes and Kreimer (e.g. [...
NDewolf's user avatar
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1 vote
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27 views

How to expand $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ in $SU(2)$

I would like to calculate the following expression: $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ where $D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$ and $A_\mu^a$ are the components of a real $SU(2)$ ...
Hendriksdf5's user avatar
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38 views

A question about the definition of derivative in different coordinate systems

The definition of the derivative goes like this: If $x$ is an interior point of a set $E \subseteq {\Bbb R}^n$, then a function $f: {\Bbb R}^n \rightarrow {\Bbb R}^m$ is said to be differentiable at $...
WhyNót's user avatar
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Index of Callias operator and application in physics

In his article "Axial Anomalies and Index Theorems on Open Spaces" (https://link.springer.com/article/10.1007/BF01202525) C.Callias shows how the index of the Callias-type operator on $R^{n}$...
C1998's user avatar
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2 votes
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Can components of an eigenvector be deduced using linear algebra methods?

In the following equation $$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1\end{bmatrix} \psi = E \psi$$ by ...
James's user avatar
  • 802
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94 views

Why do numerical eigenvector computations turn out different compared to symbolic formulas?

The following $A \vec x=\lambda \vec x$ should have 4 eigenvector solutions of the form Setting $c=1, m=1, p_x=1, p_y=1, p_z=1$, the eigenvalues should be $$\lambda = \pm 2$$ while the corresponding ...
James's user avatar
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5 votes
1 answer
120 views

Prove the closed form of $\int_0^T \exp\left(\frac{ia}{T-\tau}+\frac{ib}{\tau}\right)\frac{d\tau}{\left[\sqrt{(T-\tau)\tau}\right]^3}$.

While working through the Dover book "Quantum Mechanics and Path Integrals", I stumbled across a problem requiring me to use the following identity given in the book's appendix. $$\int_0^T \...
Anne Jones's user avatar
1 vote
0 answers
44 views

Spherical Wave Equation With a Source and Time-dependent Velocity

In my research I am studying a type of soliton known as Q-balls. The standard equation of motion obtained for the spatial part of a Q-ball is (where $f = f(r)$) $$\frac{d^2f}{dr^2} + \frac{2}{r} \frac{...
Daniel Waters's user avatar
1 vote
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Find a solution of a conservation law

How can I prove that the (entropy) admissible solution of the equation \begin{equation} u_t + |u|_x = 0 \end{equation} is given by \begin{equation} u(t,x) = \begin{cases} -1,\quad\text{if}\quad x<-...
Paolo's user avatar
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1 answer
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Simplification of formula [closed]

I am trying to solve some plasma system with the governing equations (after normalization) $$ \frac{\partial n_s}{\partial t} + \frac{\partial}{\partial x}(n_s u_s) = 0 $$ $$ \frac{\partial u_p}{\...
Mariam Ibrahim's user avatar
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0 answers
23 views

Understanding the integral of a bounded operator on a Hilbert Space

In mathematical physics I've encountered the holomorphic functional calculus, which I've seen the defined as: Let $f$ be holomorphic in a neighbourhood of the spectrum of $T$, with $T$ being a bounded ...
AJE's user avatar
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0 votes
1 answer
102 views

The reason for curl free

I wonder about the reason for the idea of this, would you mind explain for me this can happen in mathematics. Thank you !
Đôn Trần's user avatar
1 vote
1 answer
68 views

Calculating deflection on a beam

This is for a hobby project, and to learn a little about elasticity along the way. I have a triangle wedge comb piece of decreasing width and angle for which the cross section is shown here: For each ...
vallev's user avatar
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1 vote
1 answer
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What does it mean to integrate with respect to the complex conjugate.

I am an undergraduate mathematical physics student doing a summer project. I am very familiar with "mathematicicans complex analysis", but am having difficulty with how fast and loose with ...
Jack's user avatar
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Dirac Points: Triangular vs. Honeycomb Lattices

I'm reading the paper 'Honeycomb Lattice Potentials and Dirac Points' by Fefferman&Weinstein. To my understanding they claim that the existence of Dirac Cones at K/K' points is entirely determined ...
Julian's user avatar
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2 votes
1 answer
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Yang-Mills action is invariant under conformal change of the metric

Let $(M,g)$ be a pseudo-Riemannian 4-manifold with a principal bundle $P\to M$. Prove that the Yang–Mills action $S_{YM}[A]$ is invariant under a conformal change of the metric $g$: $\quad g'=\mathrm{...
Siyuan Yin's user avatar
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0 answers
20 views

Restrictions on a set to be the spectrum of a 1D (discrete) Schrödinger operator.

What restrictions are there on a compact set $E\subset\mathbb{R}$ for $E$ to be the spectrum of a bounded (discrete) Schrödinger operator on $l^2(\mathbb{Z})$? Is there a known necessary and ...
Mathmo's user avatar
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1 answer
32 views

How to verify positive definitiveness of the given Kinetic term?

I was going through this paper on QCD chaos, where in Appendix B (page 10), for equation B12: $$\frac{\mathcal{S}}{\mathcal{T}}= \int dt\sum _{n=0,1} \left(\dot{c}_n^2-c_n^2 \omega _n^2\right)+11.3c_0^...
codebpr's user avatar
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Understanding the Application of Fourier Transform and ODE in a Specific Context

I am currently studying a mathematical concept and I have come across an equation from this post that I am having trouble understanding. Some key information from @Liding Yao is as follows: $$u(y)=e^{-...
Sym Sym's user avatar
0 votes
0 answers
85 views

Second Order Differential Equation- Underdamped Spring-Mass System [duplicate]

I have a spring with a some mass, m oscillating up and down, hooked to a ceiling with a circular disc attached to the bottom. Using newtons second law, $F = F_d + F_s$, where $F=ma$, $F_s = -kx$ (...
Eshwar Kolli's user avatar
-1 votes
1 answer
114 views

General relativity [closed]

I was reading a physic text, and I don’t understand how to transform their non rigorous argument. The part where I don’t get it is when they write $d\theta=d\phi=0$. It means nothing as $d\theta$ and $...
Maxime's user avatar
  • 174
2 votes
0 answers
72 views

Chaotic one-dimensional system

Why can the solution of a one-dimensional equation of the form $$m\ddot{x}=F(x)$$ not be chaotic if $F$ is not explicitly time-dependent? Multiplying by $\dot{x}$ and integrating with respect to time, ...
Diger's user avatar
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0 answers
23 views

Regarding 1D Asymmetric Simple Exclusion Processes

I have been trying to decipher a paper on Asymmetric Simple Exclusion Processes in 1D by B. Derria: "An exactly soluble non-equilibrium system: The asymmetric simple exclusion process". ...
mathphyguy's user avatar
2 votes
1 answer
42 views

Equivalence of Kraus operators on a single element

Let's say that I'm working in a Hilbert space $\mathcal{H}$,suppose I have two bounded operators $A.B\in B(\mathcal{H})$ and a positive semidefinite operator $x$, for example take $x$ to be a density ...
ana's user avatar
  • 75
2 votes
0 answers
70 views

Stiefel-Whitney Classes for Physicists

I am a theoretical physicist and i am trying to understand better how spin structure works. I understand fairly decently Riemannian geometry but have little to zero knowledge of algebraic/differential ...
LolloBoldo's user avatar
0 votes
1 answer
49 views

Solving a heat equation, with puzzling passages.

I have the heat equation $$ u_t=\frac{1}{2}u_{xx},\quad x\in \mathbb{R}\\ u(0,x)=x^2 $$ I tried to solve it with these passages: $$u(t,x)=\int_{-\infty}^{+\infty} G(t,x-y) \phi(y) dy$$ $$u(t,x)=\frac{...
Clyde A. Jansen's user avatar
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50 views

$C^*$ algebra and deformation quantization

I heard that (from ref ) Classical observables : The set of observables $\mathcal{O}$ of a classical systems are exactly the self-adjoint elements of a separable commutative unital $C^*$-algebra. ...
phy_math's user avatar
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1 vote
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79 views

Disagreement between two methods of computing the determinant of a differential operator

I want to compute the determinant of the following operator over the interval $[0,L]$: \begin{equation} A=-\frac{d^2}{dx^2}+\omega^2. \end{equation} I imposed the boundary conditions $\phi(0)=\phi(L)=...
Ervand's user avatar
  • 11
2 votes
1 answer
46 views

Decomposition of function into products

Given a single variable function $f(x)$, is there a way of decomposing it into the product of a family of function. Something similar to, $$f(x) = \prod_n p^{a_n}_n(x)$$ I am trying to find the ...
PRITIPRIYA DASBEHERA's user avatar
0 votes
1 answer
43 views

Question about Cartan's Theory of Spinors, Section 53 a spinor is a Euclidean tensor

Context I'm studying spinors in detail as part of research project. I'm working through Cartan's Theory of Spinors [1]. In section 53, A spinor is a Euclidean tensor, Cartan asks us to, "Consider ...
Michael Levy's user avatar
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3 votes
1 answer
71 views

How to evaluate this definite integral in terms of Bessel functions.

In the context of Green's functions for the Free Klein-Gordon field, the following integral occurs: $$\int_m^{\infty}{\rho e^{-\rho r}\over\sqrt{\rho^2-m^2}}\; d\rho.$$ Here $m$, is a positive ...
Albertus Magnus's user avatar
2 votes
0 answers
29 views

linear elasticity equation and spectral equivalence

Consider $$\nabla \cdot \sigma = f \quad \text{in} \quad \Omega \\ \sigma = 2\mu\varepsilon(u) + \lambda ( \nabla \cdot u ) I \quad \text{in} \quad \Omega \\ u = 0 \quad \text{over} \quad \Gamma_D \\...
XYZ's user avatar
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0 votes
0 answers
66 views

How to solve this system of algebraic equations?

I was solving a Physics Problem and after going on solving I arrived at these equations $$(D_1u+1)\left[{D_2}\cdot\dfrac{(D_1-L)u+1}u+1\right] =\dfrac 1{M_1} $$ $$(D_2u+1)\left[{D_1}\cdot\dfrac{(D_2-L)...
user1318878's user avatar
2 votes
0 answers
78 views

How to find Riemann invariants of a system

I'm trying to find the Riemann invariants of the system \begin{array}{l} \frac{\partial \alpha }{\partial t} +( \alpha +V)\frac{\partial \alpha }{\partial x} -\alpha \frac{\partial V}{\partial x} =0\\ ...
Kris's user avatar
  • 21
-2 votes
1 answer
57 views

What does it mean for a spring to have negative stiffness? [closed]

When working with 2nd order, linear, homogeneous, constant coefficient ODEs of the form my'' + by' + ky = 0, k is indicative of spring stiffness. The stiffer the spring, the more force it exerts ...
The Math Potato's user avatar
0 votes
0 answers
27 views

Properties of Harmonic Functions and Monotonicity

I have a general question surrounding certain harmonic functions. I was able to solve the Laplace equation $\Delta f$ = 0 in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, ...
HtmlProg's user avatar
4 votes
1 answer
118 views

How are polynomials of tempered distributions defined?

I am reading a textbook on rigorous quantum mechanics and quantum field theory in which there often appears statements such as Let $A(\phi)$ be a polynomial function defined on $\mathcal{S}'(\mathbb{...
CBBAM's user avatar
  • 6,255
7 votes
2 answers
185 views

Is time-orientability a condition on the metric, smooth or topological structure of a manifold?

I recently asked a question on Physics Stack Exchange about orientability and time-orientability of a manifold in the language of fiber bundles. This new question is related to, but independent, of ...
Níckolas Alves's user avatar
4 votes
2 answers
97 views

Convert an equation to its elliptic form

I am failing to convert $L = \frac{\sqrt{2}}{2}\int_{0}^{\phi_0} \frac{\sin(\phi)}{\sqrt{\sin \phi_0 - \sin \phi}} d\phi$ into $L=\int_{\theta_1}^{\pi/2}\frac{2k^2\sin^2(\theta)-1 }{\sqrt{1-k^2\sin^2 \...
victor's user avatar
  • 155
1 vote
1 answer
51 views

Uniqueness and stability of equilibrium

Two points $P_1,P_2$ of mass $m$ move constrained respectively to $y=x^2, y=x^2-1$ in a vertical plane $xy$. Points are connected with a spring with constant $k>0$, and rest's length equal zero. ...
Turquoise Tilt's user avatar
2 votes
2 answers
89 views

How to prove $\int_{-\infty}^{+\infty}\langle f(t)f(t+s)\rangle\cos(a s)ds=a^2\int_{-\infty}^{+\infty}\langle\dot{f}(t)\dot{f}(t+s)\rangle\cos(as)ds$?

Denoting the time derivative of $f$ by by $\dot{f}$, I want to prove equation (2.96) given here: $$\int_{-\infty}^{+\infty} \langle f(t) f(t+s) \rangle \cos(a s) ds = a^2 ~\int_{-\infty}^{+\infty} \...
Mike's user avatar
  • 83
4 votes
1 answer
47 views

How to show that $\left[D_\mu, F^{\mu\nu}\right]=\left(\partial_\mu \delta_{ae}-gf^{bae}A_\mu^b\right)F^{\mu\nu a}t^e$?

I'm trying to show that $$\left[D_\mu, F^{\mu\nu}\right]=\left(\partial_\mu \delta_{ae}-gf^{bae}A_\mu^b\right)F^{\mu\nu a}t^e\tag{1}$$ where the covariant derivative, $D_\mu=\partial_\mu+ig A_\mu$, ...
Sirius Black's user avatar
0 votes
0 answers
80 views

Conjugate Representation is equivalent to Dual Representation

I was trying to solve the following problem, but I don't know under what "formal" hypothesis it holds, and what is really being asked: Let $D: G \to Aut(V)$ be a linear unitary G-...
piug's user avatar
  • 43
1 vote
1 answer
60 views

Asymptotic expansion of an integral by expanding series first

Question: as $s\to\infty$ use the substitution $u = {\rm e}^{-x}\,$ to obtain the first 2 terms in asymptotic expansion in the integral: $$ \operatorname{B}\left(s,t\right) = \int_{0}^{1}u^{s - 1}\,\,\...
vegetandy's user avatar
  • 305
0 votes
1 answer
39 views

Green's Formula for vector fields in the Navier Stokes Weak Formulation

I am currently studying the weak formulation of the Navier-Stokes equations and came across the following equation: \begin{equation} \int_{\Omega} \mathbf{v} \cdot \Delta \mathbf{u} \, dx = -\int_{\...
Luigi's user avatar
  • 75
3 votes
0 answers
130 views

Non-autonomous system of two nonlinear ordinary differential equations with conditions

Consider the ODE system: $$ \frac{df}{dx}= -\sqrt{g},\tag{1} $$ $$ \frac{dg}{dx}= -\sqrt{x}f,\tag{2} $$ where $f=f\left(x\right)$ and $g=g\left(x\right)$ are the functions on the interval $x\in\left[0,...
Khristo Mikhail's user avatar
1 vote
1 answer
62 views

Moller Operator and the Determination of Bound States in Quantum Scattering Theory

I am trying to understand the Moller operator in quantum scattering theory. An important result concerning this operator is the following useful property that can be used to determine bound states. ...
Debbie's user avatar
  • 854
2 votes
0 answers
81 views

What is a transformation (physical) in precise terms?

I’m reading some quantum field theory from physics (David Tong’s lecture note), but I don’t quite understand what it means to perform some transformation $x \rightarrow x’$. I’m trying to view it in ...
Frozer Clark's user avatar
0 votes
0 answers
73 views

Solve a trick differential partial equation

So basically I have came across a partial equation in the $AdS_{d+1}$ spacetime, $$z_0^{d+1} (z_0^{-d+1} \phi_{,0})_{,0} + z_0^2 \phi_{,jj} = \delta^{d+1}(x-v)$$ Where $,k$ means derivative wrt $k$. ...
Lac's user avatar
  • 761
4 votes
1 answer
126 views

What can semigroup theory do better in the study of PDEs compared to alternative methods?

I've recently come across semigroup theory in my mathematical physics class and while the theory itself feels nice to work with, I have not yet understood what does the theory offer for the study of ...
Cartesian Bear's user avatar

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