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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Lie Algebra Homomorphism for Fundamental Vector Field
This question is based on the exercise 10.1 (b) in "Geometry, Topology, and Physics" by Nakahara. Let $G\rightarrow P \rightarrow M$ be a principal bundle. Given an element of the Lie ...
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Book on $SL(2,C)$
Is there a book, which treats $SL(2,C)$ in detail as a group, Lie group, its Lie algebra, geometry of its subgroups etc.? It is often seen as an example in Lie Algebra/Group books but it always ...
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What will happen if we change all the resistances in an infinite resistance ladder to their double value? [closed]
Let's have a circuit as below:
Circuit 1
If the equivalent resistance of above circuit is $Z$ and if we have another circuit as below:
Circuit 2
And if the equivalent resistance of above circuit is $R$...
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Help me with this math problem [closed]
Is there an exponential density that satisfies the following condition $p(x\le 2)= \frac{2}{3} p(x\le 3)$? If so, what is the value of delta?
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Why is the solution of the Klein-Gordon PDE a distribution?
I've also posted this question on physics SE in case it is more appropiate there.
Consider the Klein-Gordon equation:
$$(\square + m^2)\phi = (\partial_t^2 - \Delta + m^2)\phi = 0 \tag{1}$$
The ...
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Fourier Series of a non periodic function
In our textbook the given example-question is as follows (written in bold):
Find a fourier series to represent $x-x^2$ from $x= -\pi$ to $x= \pi$
But the function given $x-x^2$ is non periodic, ...
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Tannaka–Krein duality in non-compact case
The Tannaka–Krein duality provides a way to reconstruct a group (up to isomorphisms) from the category of linear representations of that group.
In physics, this duality is sometimes used as a ...
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Equivalent definition of Hawking quasi-local mass
Recently, I came across a strange definition of Hawking quasi-local mass, which states that given a surface $S$ in the spacetime, the Hawking mass of $S$ is defined as
$$m(S)=\sqrt{\frac{\mathrm{Area}(...
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Applications of category theory in Computer Science and Mathematical Physics. [closed]
Could someone name some examples where we make use of category theory in Computer Science or Physics, specifically in Mathematical Physics?
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odd function conclusion in D'Alembert's formula
here I have a particular question of D' Alembert's formula for the homogeneous wave equation in 1D:
\begin{align}\frac{\partial^2 u}{\partial t^2} - C^{2}.\frac{\partial^2 u}{\partial x^2} =0\end{...
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Show that for associated Legendre function [closed]
Show that $P_n^m(-x)=(-1)^n(-1)^mP_n^m(x)$
I have tried to use the associated Legendre function. I also used Rodrigue's formula for $P_n^m$ and plug the value back in the associated Legendre function. ...
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Christoffel symbol on $T^*M$
I tried to prove the form of the Christoffel symbol on the contangent space given in the book "Elements of Noncommutative Geometry".
The Christoffel symbols $\Gamma^k_{ij}$ of the Levi-...
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Relation of connections on $TM$ and $T^*M$
I have troubles following the book "Elements of Noncommutative Geometry".
Let $E\to M$ be a vector bundle. Then a connection on $E$ is a linear map $\nabla:\Gamma^\infty(M,E)\to\Gamma^\infty(...
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Definition of Feynman diagram
I'm reading Costello's book "Renormalization and effective field theories" (preliminary PDF, p. 35). I am stuck on his discussion of Feynman diagrams.
He considers a finite-dimensional super ...
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Doubt on the general solution of three dimensional Laplace equation in Fourierspace [closed]
I am finding difficulty in deriving the general solution of laplace equation in fourierspace. The result alone was presented in paper by Faxen 1921( onlinelibrary.wiley.com/doi/10.1002/andp....
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Seeking Correct 2x2 Matrix Representation for CL(1,1) in Clifford Algebra
I'm working with the Clifford algebra $ \text{CL}(1,1) $ and attempting to find an appropriate $ 2 \times 2 $ matrix representation. This algebra corresponds to a space with the metric signature $ (1, ...
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Mathematical theory of plasma
I am working on a heavily mathematically project about plasma. In particular, I want to find references that treat the problem from microscopic models that include relativistic and magnetic effects (...
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Show that, in a normal coordinate chart, $\Gamma^{i}_{(x)(jk)}(p)=0$
I am currently working through the following proof and have a question regarding the computations done by the author:
Theorem: Let $(M, \mathcal{O}, \mathcal{A}, \nabla )$ be an arbitrary affine ...
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Is there any mathematically rigorous definition of deriving a matrix valued function with respect to one of its matrix argument?
On my way of satudying Heisenberg matrix mechanics, I get blocked by formulas engaging derivations with respect to a matrix arguments. My question is the following : Is there any mathematically ...
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What are the invariant polynomials of representations of the Lorentz group $SO^+(3,1)$ and $SL(2,\mathbb{C})$?
The Lie Groups $SO^+(3,1)$ and $SL(2,\mathbb{C})$ occupy a particular, unique place in physics. I am interested in the following problem: suppose I have a finite-dimensional representation (not ...
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How to optimize the thruster direction on a rocket with infinite fuel.
So suppose we have a rocket that can produce a finite acceleration $a$ in any direction. Given the rocket's current position $\mathbf p$ and velocity $\mathbf v$, we want to know what direction to ...
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Seeking 4x4 Real Matrix Representation of Generators for Clifford Algebra Cl(3,1)
I am currently delving into the study of Clifford algebras, particularly $ \text{Cl}(3,1) $, in the context of theoretical physics and am seeking clarity on a specific representation issue.
As I ...
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How to calculate the relative error of the wave propagation speed in a string?
So as a part of a laboratory assignment, I am trying to determine the relative error of the speed of wave propagation a string. The string is in tension using a weight.
What variables I have thus far:
...
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Can there be a finite closed form for the one dimensional heat kernel $e^{\frac{d^2}{dx^2}}$ in operator calculus?
In this question we manage to show the existence of a closed form for arbitrary $e^{a(x) \frac{d}{dx} + b(x)I}$ as a single term of the form $k_1(x) f(k_2(x))$ where $k_1, k_2$ obey an interesting ...
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Standing wave operator?
Everyone knows the d'Alembert wave operator acting on a scalar function $\Psi(\boldsymbol{r},t)$ as:
$$ ( \partial_t^2/c^2 - \Delta ) \Psi(\boldsymbol{r},t) = 0 $$
The homogeneous solution in 3 ...
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Why is a relationship between $d\mathbf{s}$ and $d\mathbf{S}$ true?
I'm reading the tutorial of MPM in physic based simulation (https://www.math.ucla.edu/~cffjiang/research/mpmcourse/mpmcourse.pdf). I encountered some mathematical problem when reading the section 5 ...
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Will branch cut choice affect branches?
Will brach cut affect branches? Eg. $z^{\frac{1}{3}}$ there are three branches for $-\pi$ to $\pi$ branch cut: $r^{\frac{1}{3}} e^i \frac{\theta}{3}$, $r^{\frac{1}{3}} e^i \frac{\theta}{3}+i2\frac{\...
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Definition of $a_{0}$ - what is wrong with my calculations?
Consider the following problem, which was extracted from page 13 of these lecture notes. Consider $w$ to be radially symmetric ($w(x) = w(|x|)$), compactly supported and nonnegative. Let $f$ be ...
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Partition function of a QFT.
There is a YouTube lecture by Robert Dijkgraaf titled:"Introduction to Topological and Conformal Field Theory (1 of 2)."
https://www.youtube.com/watch?v=jEEQO-tcyHc&t=2977s
At one point ...
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How is the Hodge star operator defined for vector-valued forms?
Let $M$ be an oriented Riemannian manifold of dimension $n$. For any $\omega \in \Omega^k(M)$, we define the Hodge star operator $\star$ of a $\omega$ as the unique $n-k$ form $\star\omega$ that ...
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Why does this trick make the oscillating exponential integral converge?
I have the following integral:
$$\int_0^{+\infty} e^{iEt} dt,$$ where $E$ is a real constant.
I know this integral does not converge. However, I have seen the following trick which makes it converge:
\...
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Discrepancy in Results with Self-Adjoint Operator on a Special Hilbert Space in 2D Geometric Algebra
I am exploring the behavior of multivectors in 2D geometric algebra, specifically examining the product $\mathbf{u}^\ddagger \mathbf{u}$, where $\mathbf{u}=a+xe_1+ye_2+be_{12}$ and its Clifford ...
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When are the eigenvalues of a Schrödinger operator linear in $n$?
It is well known that if one takes the Schrödinger operator
$$H = - \frac{1}{2} \frac{d^2}{dx^2} + \frac{1}{2} x^2$$
acting on some dense subspace of $L^2(\mathbb{R})$ then this has discrete spectrum $...
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Understanding's Wikipedia's definition of a spinor
I am trying to understand spinors from a mathematical view. I've seen similar questions on this website but I'm still unclear on what they are exactly. On Wikipedia they state:
Although spinors can ...
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Expanding the product of two dot products
I have an expression:
$$
W = \sum_{k,k'}(\mathbf{q}\otimes\mathbf{q})(\gamma^k\gamma^{k'})
$$
And $\gamma$ is:
$$
\gamma^{k} = \mathbf{f^{k}} \cdot \mathbf{Ae}
$$
Basically, expression W is a product ...
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Inverse Proportionality
Inversely proportional: $ab = K$ where $a$ and $b$ are variables and $K$ is the constant of inverse proportionality (?). So, if $a$ increases/decreases, $b$ decreases/increases. Neat!
$x + y = K$. $x$...
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Asymptotic rate of decay of the integrals
For each $n$ define the sequence $y_n= \int_{0}^\infty (1-e^{-\frac{x^2}{4}})^n e^{-x}dx$. By dominated convergence theorem, it is clear that $y_n$ converges to $0$. I am interested in finding the ...
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Evaluating the integral $\int_C |\Bbb dz|/z$
I have to evaluate the integral $$\int_C \frac{|\Bbb dz|}{z}$$ where $C$ is the arc from $z = 1$ to $z = -1-i$.
So first of all I made the parameterization $z_1(1-t)+tz_2 = (1-t)x_1+tx_2+i((1-t)y_1+...
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How to calculate the length of the image of the curve $\Gamma$? [closed]
How to calculate the length of the image of the curve $\Gamma= \{z(t) :z (t) =at, t \in [0,1]\}$ under the mapping $w = z^n$, where $n \in \mathbb{N}, a \in \mathbb{C}$?
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What would be the new Equation of motion if the magnetic field's origin is shifted from the origin of a co-rotating spherical polar coordinates?
The equation of motions due to the dipole magnetic force of a planet in a frame corotating with the planet and origin at the centre of planet assumed to be sphere components wise are given as below:
\...
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Analytically solving PDEs on irregular domains in Physics
In many Physics courses you solve PDEs like heat or wave on square, circular, or spherical domains with separation of variables. Are there ways to solve PDEs and Boundary value problems on irregular ...
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Show that the real K-G equation, $(\Box + m^2)\phi=0$ is the EOM for the action $S=\frac12\int d^4x(\partial^\mu{\phi}\partial_\mu{\phi}-m^2\phi^2)$
This question concerns a real scalar field.
Show that the real Klein-Gordon equation, $(\Box + m^2)\phi=0$ is the equation of motion, $\delta S[\phi(x)]/\delta\phi(x)=0$, for the action
$$S=\frac12\...
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The algebra $\Gamma^\infty(\mathbb{C}l(M))$ is generated by $\Omega^1(M)$
I'm reading "Elements of Noncommutative Geometry" by Garcia-Bondía. There it was mentioned that the algebra $\Gamma^\infty(\mathbb{C}l(M))$ is generated by $\Omega^1(M)$.
Here $\Omega^1(M)$ ...
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Optimization problem involving discrete and continuous variables
In my mechanics class I am to solve an optimization problem that is basically reduced to finding a set of points $x_i,y(x_i)$ such that the following functional is minimized:
$$\sum\limits_{i}^{} \...
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System of nonlinear first-order PDEs
The following system of nonlinear first-order PDEs describes the
one-dimensional incompressible flow of an ideal fluid in an open long channel
$$h_t+(hv)_x=0,$$ $$v_t+vv_x+gh_x=0,$$ where $h = h(x, t)$...
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How to know if a set of equations are Lorentz Invariant Spinors
I'm currently busy with a course in QFT and am completely baffled by Spinors. In particular there are two parts, that while I mostly understand the theory, struggle to show mathematically (especially ...
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On Schoen and Yau's proof of the positive mass theorem: extracting a minimal surface in the limit
I'm reading Schoen and Yau's 1979 paper on the Positive Mass theorem. I'm having trouble understanding the proof of how they extracted a minimal surface as the limit of solutions to the Plateau ...
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I am trying to find the time for different paths for a traversing ball between two points under the influence of gravity? But, I am getting infinity?
I am trying to find the time for different paths (functions) for a traversing ball between two points under the influence of gravity, but I am often getting strange answers where the integrand tends ...
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Calculate Electric Field on the Z-axis from a finite charge wire
I've been trying to find the electric Field on the Z-axis from a non-uniform charge density line charge. The wire is placed on the z-axis from $z=0$ to $z=1$, $E=?$ at $z>1$ and $z<0$
$$
\rho =...
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Verification: Eigenfunction expansion using products of bessel and spherical harmonics
Preface
This is a can you please check my work post. Maybe these are frowned upon. However, I have been working on this for a couple of days now. I can use some help.
Context
I am solving a physics ...
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