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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Follow Up Question: Witten's explanation of Feynman diagrams
This is a follow-up to a recent question of mine Witten's proof of Wick Formula of QFT. The background of this question can be found there if needed, but I feel my question is simple enough ...
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The mathematical sense of a physical quantity
I am not a good student, so, perhaps, my question is stupid, however...
What is the mathematical sense of a physical quantity? For example, if we consider the mass, can we say it is a surjection (...
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What are the statistical methods to analyze the solutions of a Diophantine equation?
How can I statistically analyze the many but finite integer solutions of a cubic multivariable homogeneous Diophantine equation in a given interval?
Suppose that I can find the integer solutions using ...
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Let there be a circle on a vertical plane
Let $A$ be a heavy moving point without friction. $A$ is tied with a flexible and inextensible rope which passes through B without friction with a mass Q. Determine the position of equilibrium of $A$.
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Given the streamline equation as $\frac yx = C$ ($C$ is constant), find the complex potential.
Given the streamline family equation as $\frac yx = C$ ($C$ is constant), find the complex potential.
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Examples of relativistic equations
I am posting this question here because it is just reference request and I do not need a fully detailed answer.
Attending my physics class, we introduced two relativistic equations:
$$ \frac{d}{dt}\...
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Finding the growth rate of a wave
I have an equation (from the dispersion relation) for $w$ (frequency) in terms of $a$ (plasma frequency), $k$ (wave vector) and $v$ (velocity of the particles):
$$w^4-(a^2+2k^2v^2)w^2+k^2v^2(k^2v^2-a^...
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Finding a suitable form factor for given conditions
This is basically a physics problem but I will try my best to highlight the mathematics behind it.
Suppose I have two functions:
$$T(z,B)=\frac{\text{z}^3 e^{-3 A(\text{z})-B^2 \text{z}^2}}{4 \pi \...
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how to solve single nonlinear algebraic equation in two variables?
(I am not a mathematician; I am having physics background.)
How to solve a single nonlinear algebraic equation in two variables, $x$ and $y$?
(I know that - if there are two variables, one needs two ...
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How to calculate the convolution product of $(H_0 e^{\alpha t}) * (H_0 e^{-\alpha t})$?
everyone!
I am doing an exercise concerning the resolution of differential eaqtion in the sense of distributions.
Let $\alpha \in \mathbb{R}_+^*$. Let be the differential equation, for $f \in \mathrm{...
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What is meant by the cross section of an orbit?
In Folland's quantum field theory book he says:
Let $\{\sigma_t : t \in \mathbb{R}\}$ be a one-parameter group of measure-preserving diffeomorphisms of $\mathbb{R}^n$ whose orbits are (generically) ...
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inclusion of von Neumann algebras implies reversing inequality of its modular operators
I'm working with von Neumann algebras and I stumbled with this statement in a work of Borchers (1999)
where we must think $\mathcal{N}$ and $\mathcal{M}$ as concrete von Neumann algebras over the ...
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Proof of $A\psi = \lambda \psi \Rightarrow h(A)\psi = h(\lambda)\psi$ on Borel functional calculus
Let $A: D(A) \to \mathscr{H}$ be a densely defined unbounded self-adjoint operator on a separable Hilbert space $\mathscr{H}$. By the Spectral Theorem, there exists some finite measure space $(M,\mu)$,...
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Is there any references for solving inverse Ising problem w.r.t. some objective functions other than MaxLikelihood
I am trying to formulate an inverse Ising problem that optimizes some defined objective functions other than maximum likelihood. I am pretty new to this field (only some background on Markov random ...
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Witten's proof of Wick Formula of QFT
Let $\mathcal{S}$ be a finite dimensional real vector space with a positive definite summetric bilinear form $B$. Let $dv$ be a Lebesque measure on $\mathcal{S}$ such that
$$\int_{\mathcal{S}}e^{-B(v,...
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Relation between inverse square root divergence and delta-function?
Dirac $\delta$-function is widely used in science and engineering. Its Poisson kernel representation is of the form $$\eta_\epsilon(x)=\frac{1}{\pi}\mathrm{Im}\frac{1}{x-i\epsilon}=\frac{1}{\pi}\frac{\...
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Is the wave behind a duck a parabola or a hyperbola?
I tried to solve this problem when I saw some ducks in a lake. Suppose a duck moves along a straight line with a constant speed. Is the wave behind it a parabola or half of a hyperbola. I checked the ...
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Direct sum decomposition of Hilbert spaces
Let $A$ be a self-adjoint operator on a Hilbert space $\mathscr{H}$ with dense domain $D(A)$. From the continuous functional calculus, for every continuous function $f \in C(\sigma(A))$ we can ...
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Legendre polynomials: computing $\int_0^1 xP_n(x)\,dx$
I am trying to find the integration:
$$
\int_{0}^{1}{xP_n\left(x\right)}\,dx
$$
I know that I should split Rodrigues's formula up into two parts $P_{2n}\left(x\right)$ for even terms and $P_{2n+1}\...
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What are the strategies to deal with intractable integrals encountered when solving ODE using variational method.?
I am trying to solve a nonlinear ODE: $$u_{xx}+\tan^2(x)u+gu^2=0$$ using the variational method, and I encountered an intractable integral. What are the strategies to deal with intractable integrals ...
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Dipole-dipole interaction type of integral in Mathematica
I've been trying to solve the following integral
$$ \text{Int} = \iiint \left( \frac{1}{r^3} - 3\frac{r_z^2}{r^5} \right) dr_x dr_y dr_z$$
where $ r \equiv \sqrt{r_x^2 + r_y^2 + r_z^2 }$.
The integral ...
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Does the arbitrary polynomials of the N-dimensional irreducible representation of SU(2) generate U(N)?
Assuming I have the $N$-dimensional irreducible representation of $SU(2)$ (The $N$-dimensional spin matrices $Sx$,$Sy$ and $Sz$) and I can consruct arbitrary polynomials from them of the form:
$$\sum{...
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Sand pile - container problem
Imagine that there's a box container shown in the graph. (You may consider it as a 2-dimensional rectangle.)
A division plate is in the middle of the container.
A sand pile is on the division plate, ...
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Jones polynomial of a knot in terms of its Seifert matrix
It is well known that the Alexander polynomial of a knot can be written in terms of the Seifert matrix of the knot by a simple relationship $$\Delta(t)=\det(V^T-tV),$$ where $t$ is a formal variable ...
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Calculate distance travelled by particle with positive velocity and negative acceleration in fixed time
Let's assume a ball is thrown upwards with an initial velocity V and gravitational acceleration -A is acting on it in downward direction
At some height H the particle with reach its max height and ...
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Why is angular velocity in the $z$-direction?
My physics professor (at a T10 university) was unable to explain this. He mentioned something about a bivector, but fundamentally did not say why $w_{z} = \frac{d\theta}{dt}$, and that is what my ...
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Compactly supported sections of a vector bundle form a cosheaf
I am trying to understand why the compactly supported sections of a vector bundle form a cosheaf. I have proven that they form a precosheaf: simply extending the compactly supported section by zero ...
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Reference request: well-posedness of nonlinear Schrödinger and relativistic Schrödinger equations?
I have been trying to find a book that covers the well-posedness of non-linear equations that appear in theoretical physics such as the Schrödinger, Klein-Gordon, and Dirac equations without much ...
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Why is relative entropy negative in this computation?
I am running some tests using relative entropy on physical systems in equilibrium, and I am seeing some strange results. I wonder if this is an issue in Mathematica itself, but here goes.
I have two ...
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Inequalities for Extensions of Unbounded Operators
Background
I am trying to understand the (non-)tension between the following definitions/propositions. I will always assume I am working with states in a Hilbert space H and operators on that Hilbert ...
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Approximation scheme for potentials in Schrodinger Equation- A Request
Setup: The Time Independent Schrodinger Equation(Eigenvalue problem):
$(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$
When dealing with computing bound states/bound state energies of say an electron in ...
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How does the sum of all polynomial roots gives the result of $-\frac{a_{n-1}}{a_{n}}$?
I recently read "Mathematical Methods for Physics and Engineering: A Comprehensive Guide" by Riley, Hobson, Bence. Chapter 1, subsection 1.1.3 "Property of Roots" says we can state ...
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Why is $\infty$ always a singularity of a differential equation?
Consider Legendre's Equation:$\frac{d^2y}{dx^2}-\frac{2x}{1-x^2}\frac{dy}{dx}+\frac{l(l+1)}{1-x^2}y=0$.
It is quite evident that the coefficients of $\frac{dy}{dx}$ and $y$ do not diverge at $x \to \...
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Name for the representation on function space induced by a group action on the domain
I'm learning some representation theory in the context of a mathematical physics class. It seems that the main motivation for studying representation theory in physics is the following result: if $G$ ...
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Do Kepler's laws (really) solve the Kepler problem? And should the terms of the central force problem be treated as dependent variables?
See the bottom for an added definition and discussion.
Context
Note: I've never had a course in differential equations, so my question may demonstrate gross ignorance.
When the distinction is clear, I ...
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Finding equation of motion for given Lagrangian with respect to metric
Given the following action in d dimensional (0,1,...,d-1) curved spacetime:
$$ S= \int d^dx\sqrt{-g}\mathscr{L}[\chi,\Phi,g^{\mu\nu}] $$
Where:
$$\mathscr{L}=e^{-2\Phi} \left(-\frac{1}{2\kappa^2}[R-2\...
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Closed and exact forms, and functions $f : \mathbb{R}^2 \setminus \{ 0 \} \to S^1$
I have a simple question on the classic example of a closed form not being exact. It is well-known that the one-form
$$
\omega = \frac{x \mathrm{d}y - y \mathrm{d}x}{x^2 + y^2}
$$
is closed on $\...
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small variations in math
Given, where all function are non linear functions,
\begin{align*}
\begin{cases}
f_1(x_1,x_2,x_3,x_4) = y_1
\\
f_2(x_1,x_2,x_3,x_4) = y_2
\\
f_3(x_1,x_2,x_3,x_4) = y_3
\\
f_4(x_1,x_2,x_3,x_4) = y_4
\...
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The differential equation $\dot{\mathbf{x}}(t)=M(t)\mathbf{x}(t)$, conditions for a specific type of matrix $M$ to commute at different times
I have some $N$x$N$ matrices $M_N(t)$ of the general form
$$\tag{1}
M_4(t)=\left[\begin{matrix}-f_0 & f_1 & 0 & 0 \\
0 & -f_1 & f_2 & 0 \\
0 & 0 & -f_2 & f_3 \\
...
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Solving a Linear Functional
Solve the Euler's equation for the functional:
$$J[y]=\int_1^2\frac{\sqrt{1+y'^2}}{x}dx$$
with $y(1)=0$ and $y(2)=1$
This would be the automatic approach:
$$\frac{d}{dx}\frac{\partial F}{\partial y'}-...
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Wentzel–Kramers–Brillouin Connection Formula Derivation
I'm reading 2nd edition of Merzbacher's Quantum Mechanics, and am having trouble following a step in his derivation of some formulas related to the Wentzel–Kramers–Brillouin approximation. On p. 118, ...
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Method to obtain a function out of its asymptotic behaviour
I'm dealing with a problem that involves solving a pretty nasty differential equation of the shape:
$\frac{\mathrm{d}a}{\mathrm{d}t}=f(a)$
Even though it can be solved through variable separation, the ...
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Is it necessary to antisymmetrize the products of components in a product of differential forms in order to get the components of the product form?
Hopefully all of the evil things Mathematica's LaTeX conversion tool did to my expressions. I should warn the reader that Mozilla is not rendering this correctly for me.
In an effort to understand ...
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Derivation of Generalised Delta Function
I'm going through "Generalised Functions in Mathematical Physics" by A.S. Demidov, where he justifies the delta function from physical ideas.
Starting from the idea that the definition of a ...
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What does it mean exactly that every finite dimensional vector space is isomorphic to its dual space?
For context, I'm trying to really understand bra-ket notation in QM. I tried a few years ago and IIRC, something like $\langle a|b\rangle$ is just the dot product of vectors $a$ and $b$. Technically,...
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Discretization of 1D Heat Equation with Temperature, Position Dependent Thermophysical Properties
I am attempting to create a finite difference solution for 1D Heat Diffusion through a composite wall with an abrupt material change and no thermal boundary resistance, assuming temperature- and ...
2
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How do we show that every orthogonal matrix of $SO\left(3\right)$ represents a rotation about an axis?
I am confident that, other than the identity matrix any $3\times3$ matrix $\mathfrak{T}$ of real numbers such that $\det \mathfrak{T}=1,$ and $\mathfrak{T}^{-1}=\mathfrak{T}^{T}$ expresses a rotation ...
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What's the divergence of Oseen tensor for the Zimm model?
While studying the Zimm model on the Doi & Edwards book "The Theory of Polymer Dynamics", I faced equation 4.41 which states that:
$$
\frac{\partial}{\partial\boldsymbol{R}_j}\cdot\...
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Neumann boundary condition for Maxwell equations?
For the Poisson equation, we have Dirichlet boundary condition, Neumann boundary condition and Robin boundary condition. But for the time-harmonic Maxwell equation, I have only seen two types of ...
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2
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acceleration of a particle moving along a streamline using tensor calculus
In a steady flow, the streamline coincides with the particle trajectory. In a book on MHD, I saw that if I pick a streamline $C$ and $s$ is a curvilinear coordinate measured along $C$, $V(s)$ is the ...
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