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
0
votes
0answers
40 views
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 ...
0
votes
0answers
98 views
Solving Fokker–Planck equation
Consider the Fokker–Planck equation:
$${\displaystyle {\frac {\partial }{\partial t}}p(x,t)=-{\frac {\partial }{\partial x}}\left[\mu (x,t)p(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}\left[D(...
1
vote
0answers
68 views
Proof of the link between the Fokker–Planck equation and SDE?
I know the link between the Fokker–Planck equation and SDE given by the Feynman-Kac theorem is as follow:
$$d X_{t}=\mu\left(X_{t}, t\right) d t+\sigma\left(X_{t}, t\right) d W_{t}$$
$$\frac{\partial}{...
9
votes
1answer
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....
2
votes
0answers
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 ...
3
votes
2answers
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 ...
1
vote
0answers
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}^...
4
votes
1answer
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 - ...
5
votes
1answer
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\\
...
7
votes
2answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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
...
3
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
155 views
Bound for the $\ell^3$ norm for the one-dimensional propagator
Problem: In Appendix (A.6) of Main paper is written
$$\lVert K(x; t_0, t_1, t_2, \frac{1}{2\pi}q_1,
\frac{1}{2\pi}q_2)\rVert_3 \leq \prod_{\nu=1}^{d} \lVert
p_{R^{\nu}}^{(d=1)}\rVert_3 \leq C
\...
2
votes
0answers
48 views
Reflection positivity on weighted $L^2$-spaces
Denote by $(t, x_{1}, \ldots, x_{d-1})$ the coordinates of $x \in \mathbb{R}^{d}$ and set
$$\mathbb{R}^{d}_{+}=\left\{t, x_{1}, \ldots, x_{d-1} \in \mathbb{R}^{d}|t > 0\right\}. $$
Write $\theta$ ...
2
votes
0answers
172 views
Maxwell $U(1)$ gauge theory's electric and magnetic sources turned on simultaneously in the classical differential geometry
Question:
How do we couple $U(1)$ electric (E) and magnetic (M) sources simultaneously in the classical differential geometry language, in a $U(1)$ gauge theory based on $U(1)$ gauge bundle and its $U(...
2
votes
1answer
247 views
Gamma matrices are irreducible
For $\mu=0,1,2,3$, let $\gamma^{\mu}$ the set of Dirac gamma matrices. What does it mean to say that $\{\gamma^{\mu}: \hspace{0.1cm} \mu=0,1,2,3\}$ is irreducible?
From my previous question, I know ...
2
votes
1answer
161 views
Computing (relative) cohomology classes on quotient (vector) space via Hodge theorem
I am working on a graded vector space $V = \bigoplus_{i\in \mathbb{N}}V_i$ (which is a parabolic Verma module in the sense of [1], but let's ignore such specifics) with a positive definite inner ...
3
votes
0answers
76 views
Linearized NLS/GP around a soliton and the spectrum of the evolution operator
I apologize if this has been asked before but so far I haven't found it anywhere.
Consider the Nonlinear Schrödinger equation with a potential (i.e. Gross- Pitaevskii) in $\mathbb{R}^{d}$
$$i\Psi_{t} =...
3
votes
0answers
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{...
3
votes
1answer
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, ...
0
votes
1answer
120 views
Reference for action-angle coordinates [closed]
Does anyone know a good reference to start studying Action-Angle coordinates?
Thank you in advance !
3
votes
1answer
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 ...
12
votes
1answer
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 ...
2
votes
1answer
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$ ...
0
votes
1answer
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 ...
1
vote
0answers
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 ...
4
votes
3answers
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\...
1
vote
1answer
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 ...
3
votes
0answers
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 ...
6
votes
0answers
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,...
0
votes
0answers
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}...
3
votes
0answers
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 ...
3
votes
1answer
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 ...
7
votes
2answers
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 ...
2
votes
0answers
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,...
3
votes
2answers
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)...
2
votes
0answers
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 "...
2
votes
0answers
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....
2
votes
0answers
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 ...
20
votes
4answers
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 ...
2
votes
0answers
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 ...
1
vote
0answers
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 (...
10
votes
1answer
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 ...
3
votes
1answer
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 ...
0
votes
0answers
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:
$$
y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6)
$$
...
6
votes
1answer
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.
...
1
vote
1answer
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 ...
6
votes
0answers
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....
