# Questions tagged [mp.mathematical-physics]

Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.

1,795
questions

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### Density of a graded algebra

I'm trying to prove the following proposition:
If $ v \in V $ and $ Y \in \mathfrak{so} (V) $ then $[\dot\mu(Y), B(v)] = B(Yv)$.
By definition
$[\dot\mu(Y), B(v)] = \dot\mu(Y)B(v) - B(v)\dot\mu(Y).$ I ...

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98 views

### Solving Fokker–Planck equation

Consider the Fokker–Planck equation:
$${\displaystyle {\frac {\partial }{\partial t}}p(x,t)=-{\frac {\partial }{\partial x}}\left[\mu (x,t)p(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}\left[D(...

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68 views

### Proof of the link between the Fokker–Planck equation and SDE?

I know the link between the Fokker–Planck equation and SDE given by the Feynman-Kac theorem is as follow:
$$d X_{t}=\mu\left(X_{t}, t\right) d t+\sigma\left(X_{t}, t\right) d W_{t}$$
$$\frac{\partial}{...

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**1**answer

272 views

### A detail in the proof of Schur's lemma: the closures of the $\mathcal{Ker}$ and $\mathcal{Im}$ of the intertwiner

$\renewcommand\Im{\operatorname{\mathcal{Im}}}\newcommand\Ker{\operatorname{\mathcal{Ker}}}$I was sure that this is a trivial question and placed it on Math Stackexchange
https://math.stackexchange....

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120 views

### Special zeta value and zeroes

Are there known relationships linking special values of the Riemann zeta function or MZV (multiple zeta values, i.e. $\zeta(n_1, \cdots n_k)$ with $n_i \in \mathbb Z^+$) to the nontrivial zeroes of ...

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188 views

### Reference for mathematical Palatini formalism of general relativity

I know that this is maybe not a research level question, but since the topic is quite special, I thought that the chance to get some reference is higher in this community.
(I already asked this ...

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54 views

### Does Anderson localisation occur if the potential are equal in pairs?

Consider the Anderson model given by the Hamiltonian $H \in B(l^2( \mathbb{Z}^d)) $ defined by $H = - \Delta + V$ where the potential $V$ acts on a unit vector $ \mid {x} \rangle \in l^2( \mathbb{Z}^...

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**1**answer

329 views

### Beta function, harmonic numbers, and integral values

A problem of Mathematical Physics that I am working on involves the computation of a certain integral. Part of the result reads:
$$
I_k:= [\beta_x( -1 - k, 0) + H_{-2 - k}]x^k
$$
where $\beta_x( -1 - ...

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**1**answer

183 views

### Initial conditions in the Klein-Gordon equation

I am interested in what are the largest families of initial conditions of the problem (in $\mathbb{R}^{4}$)
\begin{equation}\label{kg}
\left\lbrace
\begin{array}{ll}
(\square+m^2)F(x)=0\\
...

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270 views

### Energy levels of double well potential

Consider the (quantum) Hamiltonian on the real line
$$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$
Let us assume that the potential $V$ is an even smooth functions with exactly two non-degenerate ...

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42 views

### Implications for a simple deterministic chaos definition

Among many others, one definition of deterministic chaos terms "chaotic" a classical dynamical system that satisfies the following three properties:
sensitive dependence to initial ...

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71 views

### Dirac operator on a 5 dimensional tangent manifold with a $Spin(3)$-bundle

In p.3 of Witten paper from this Physics Letters B, Volume 117, Issue 5, 18 November 1982, Pages 324-328 Physics Letters B, 117(5), 324–328, he says that about the Dirac equation on a 5-dimensional
...

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112 views

### Discrete spectrum of Dirac operator

It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so
that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete.
For example at least for $d=4$, this ...

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54 views

### Formulas involving traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices

While working on the Atiyah problem on configurations of points, I came across formulas involving products of traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices.
To ...

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155 views

### Bound for the $\ell^3$ norm for the one-dimensional propagator

Problem: In Appendix (A.6) of Main paper is written
$$\lVert K(x; t_0, t_1, t_2, \frac{1}{2\pi}q_1,
\frac{1}{2\pi}q_2)\rVert_3 \leq \prod_{\nu=1}^{d} \lVert
p_{R^{\nu}}^{(d=1)}\rVert_3 \leq C
\...

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48 views

### Reflection positivity on weighted $L^2$-spaces

Denote by $(t, x_{1}, \ldots, x_{d-1})$ the coordinates of $x \in \mathbb{R}^{d}$ and set
$$\mathbb{R}^{d}_{+}=\left\{t, x_{1}, \ldots, x_{d-1} \in \mathbb{R}^{d}|t > 0\right\}. $$
Write $\theta$ ...

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172 views

### Maxwell $U(1)$ gauge theory's electric and magnetic sources turned on simultaneously in the classical differential geometry

Question:
How do we couple $U(1)$ electric (E) and magnetic (M) sources simultaneously in the classical differential geometry language, in a $U(1)$ gauge theory based on $U(1)$ gauge bundle and its $U(...

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247 views

### Gamma matrices are irreducible

For $\mu=0,1,2,3$, let $\gamma^{\mu}$ the set of Dirac gamma matrices. What does it mean to say that $\{\gamma^{\mu}: \hspace{0.1cm} \mu=0,1,2,3\}$ is irreducible?
From my previous question, I know ...

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**1**answer

161 views

### Computing (relative) cohomology classes on quotient (vector) space via Hodge theorem

I am working on a graded vector space $V = \bigoplus_{i\in \mathbb{N}}V_i$ (which is a parabolic Verma module in the sense of [1], but let's ignore such specifics) with a positive definite inner ...

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76 views

### Linearized NLS/GP around a soliton and the spectrum of the evolution operator

I apologize if this has been asked before but so far I haven't found it anywhere.
Consider the Nonlinear Schrödinger equation with a potential (i.e. Gross- Pitaevskii) in $\mathbb{R}^{d}$
$$i\Psi_{t} =...

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91 views

### Multiplication of two Pauli string

Given a Pauli string $P_i \in \{ I,X,Y,Z\}^{\otimes n} $
Example: $P_0 = XXYIZ = X \otimes X \otimes Y \otimes I \otimes Z $.
Here $I,X,Y,Z$ are Pauli matrices defined explicitly as:
$$
I = \begin{...

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**1**answer

200 views

### Representations of the Lorentz group

The first few lines of this post is based on this lecture notes, but similar expositions can be found in other physics books such as Peskin & Schroeder's book.
On chapter 8 of the linked notes, ...

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**1**answer

120 views

### Reference for action-angle coordinates [closed]

Does anyone know a good reference to start studying Action-Angle coordinates?
Thank you in advance !

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236 views

### What is the relationship between the Dirac algebra and the Clifford algebra?

While I'm still trying to understand the issues raised on my previous question, I decided to first address the Clifford algebra used on formulating the famous Dirac equation. In this context, what is ...

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334 views

### Is there a straightforward generalization of min(x,y) to positive-semidefinite Hermitian matrices?

This is an open-ended question I have. Is there a function of two positive-semidefinite hermitian operators $\min(A,B)$ returning another positive-semidefinite Hermitian operator such that:
If A and ...

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205 views

### Free field rigorous quantization - possibly a misunderstanding?

I'm sorry if this is not the right place to ask this question but I've been struggling with this for days now (and I think this is too technical/specific for math stack).
Notation: A conjugation $C$ ...

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156 views

### Hamilton equations-Symplectic scheme [closed]

We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta ...

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25 views

### Nonintegrable classical dynamical systems and deterministic chaos

I'm trying to delineate a minimal (and informal) "taxonomy" for classical continuous dynamical systems that could be interested by the phenomenon of "chaos" - unfortunately the ...

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**3**answers

166 views

### Topology on Minkowski space $\mathbb{R}^{4}$ and Lorentz invariant measure

Let us consider the Minkowski space $(\mathbb{R}^{4},\eta)$ and the mass shell $H_{m}$, $m\ge 0$, given by:
\begin{eqnarray}
H_{m}:=\{x=(x_{0},x_{1},x_{2},x_{3}) \in \mathbb{R}^{4}: \hspace{0.1cm} x\...

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55 views

### What does the “scaling invariant” Serrin condition mean?

What does it mean that the function space $L^q_tL^p_x$ is invariant under the (3D) Navier-Stokes scaling $u(x,t) \mapsto \lambda u(\lambda x, \lambda^2 t)$ if $2/q + 3/p = 1$?
Does it mean that you ...

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53 views

### Schrodinger equations on foliated spaces

Schroedinger type equations have been constantly explored in mathematics. I would like to know if is it possible to make sense of physical interpretation in the following setting:
One has a closed ...

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120 views

### What is known about “dimension two” vertex algebras?

In the paper Chiral Koszul duality, Gaitsgory and Francis develop a notion of a chiral algebra living on an arbitrary variety $X$. When $X=\mathbf{A}^1$ and the chiral algebra is translation invariant,...

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66 views

### Definition of tensor product of dense subspaces of Hilbert spaces

Let $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ be Hilbert spaces. If $\psi_{1}\in \mathscr{H}_{1}$ and $\psi_{2}\in \mathscr{H}_{2}$, define $\psi_{1}\otimes \psi_{2}$ to be a function on $\mathscr{H}_{1}...

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116 views

### What should I study to approach the frontier of integrable probability research?

In terms of math, I know measure theory, measure theory based probability, differentiable manifolds, galois theory, some algebraic topology, and some representation theory. I have only physics 101 ...

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**1**answer

381 views

### Arnold's book on classical mechanics [duplicate]

Arnold's book “Mathematical methods of classical mechanics” develops the standard material on mechanics (e.g. the 3 Newton’s laws and the gravity law etc.). But what differs it from all other ...

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695 views

### Young-Fibonacci version of Nekrasov-Okounkov

This question addresses a hierarchy of linear recurrences
which arise from an attempt to generalize the Nekrasov-Okounkov
formula to the Young-Fibonacci setting.
A related posting
extensions of the ...

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42 views

### Stability in coefficients for the rescaled radiative tranport equation

One form of the radiative transport equation is as follows:
$$ v\cdot \nabla_x u + \left(\epsilon \sigma_a(x) + \frac{1}{\epsilon}\sigma_s(x)\right) u - \frac{1}{\epsilon}\sigma_a(x)\int_{S^{n-1}} p(v,...

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203 views

### On some inequality (upper bound) on a function of two variables

There is a problem (of physical origin) which needs an analytical solution or a hint. Let us consider the following real-valued function of two variables
$y (t,a) = 4 \left(1 + \frac{t}{x(t,a)}\right)...

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90 views

### Topological implications of curvature singularities

In popular articles on astronomy/physics, singularities are typically described as "holes or rips in the fabric of space".
Now algebraic topology has a lot of methods for detecting "...

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62 views

### How to compute the functional derivative of the following functional encountered in electrical impedance tomography?

Note:
I have raised this question in Mathematical stack exchange but it received no attention. That is why I proceed to here to ask this question. Please tell me if this is not appropriate, thank you....

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128 views

### Are Weyl sequences polynomially bounded?

Look at the Hilbert space $l^2( \mathbb{Z}) $ and let $A$ be a translation invariant band operator. I.e. if $\{ e_n \}_{n \in \mathbb Z} $ is the standard basis for $l^2( \mathbb{Z}) $ then it holds ...

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1k views

### Rigged Hilbert spaces and the spectral theory in quantum mechanics

I'm trying to learn some quantum mechanics by myself, and because of my mathematics background, I'm trying to understand it in a rigorous way. Since then, I've been intrigued by the use of rigged ...

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139 views

### Dynkin diagrams in Yang-Mills theory

I was searching for applications of Dynkin diagrams in physics and came across this site: https://ncatlab.org/nlab/show/McKay+correspondence but I can't quite understand how useful Dynkin diagrams are ...

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33 views

### Entropy per site of quantum spin chain

It’s fork lore that von Neumann entropy (and free energy) grows linearly with respect to the size of a quantum system. Is there a rigorous demonstration in the toy model of a quantum spin chain with (...

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342 views

### Alternative approaches to topological QFTs

A while ago I read the paper 'Quantum Field Theory and the Jones Polynomial' by Edward Witten. This article uses a lot of concepts from physics like BRST symmetry and the Chern-Simons action which ...

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167 views

### Can one define a degree of a period?

In the context of Feynman integrals, certain values of the zeta function appear at certain loop orders. Given that the tree-level amplitudes are rational, and assuming the zeta values to be ...

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42 views

### Can the Bessel functions tend to a plane wave?

Can the Bessel functions tend to a plane wave?
If I have this function:
$$
y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6)
$$
...

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211 views

### Stabilizer groups of Yang-Mills connections

Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface.
...

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221 views

### Is there a Hilbert space approach to commutative probability theory on locally compact spaces?

I was recently made aware (thanks to the answers on Why does Riesz's Representation Theorem apply in quantum mechanics?) that the $C^*$ algebra approach and the Hilbert space approach to quantum ...

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104 views

### What is Ryu-Takayanagi Entanglement Entropy?

I have a question about how to think about the Ryu-Takayanagi entanglement entropy mathematically.
For simplicity, let's work in the simplified setting of a time-symmetric slice of $AdS_4$ space -- i....