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.
2,124
questions
5
votes
3
answers
831
views
Path integral methods
Are there detailed expositions of the path integral methods in (mathematical) physics other than Feynman-Hibbs and Glimm-Jaffe?
5
votes
3
answers
948
views
Boundedness of Laplacian eigenfunctions
Let $A$ be a bounded domain in $\mathbb R^d$, $d>1$, and $\{u_k\}$ is the set of all $L^2$-normalized Laplacian eigenfunctions on $A$ with Dirichlet boundary condition (i.e., $\|u_k\|_2 = 1$).
Is ...
5
votes
3
answers
1k
views
Book about fluid dynamics
Next Monday, I'll have an interview at Siemens for an internship where I have to know about fluid dynamics/computational fluid dynamics. I'm not a physicist, so does somebody have a suggestion for a ...
5
votes
2
answers
452
views
Is there a reasonable notion of spectral theorem on a pre-Hilbert space?
I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...
5
votes
4
answers
813
views
Limit of an integral vs limit of the integrand
I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral
$$
I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
5
votes
3
answers
807
views
Weyl's Branching Rule for $SU(N)$-Setting
On the Wikipedia page for restricted representations
https://en.wikipedia.org/wiki/Restricted_representation
there is presented a number of explicit "branching rules". In particular, there is the ...
5
votes
2
answers
416
views
Limit of a double integral
What is the $\varepsilon\to 0$ limit of the following double integral
$$\int\limits_{-1}^1d\tau\;\sqrt{1-\tau^2}\;\tau\int\limits_0^\infty dq\;q^2e^{iq(\tau+i\varepsilon)}\;?$$
I was asked about this ...
5
votes
2
answers
3k
views
Van Vleck-Morette Determinant
There seems to be something curious about the so-called Van-Vleck-Morette determinant, as I cannot find any source that properly defines it in terms of expressions previously defined in that source ...
5
votes
3
answers
474
views
Hard-sphere gases and the wave equation
I'm trying to bridge a basic gap in my own education:
Where can I find a written discussion concerning the derivation of the wave
equation (for the propagation of sound, say), assuming nothing (much?...
5
votes
1
answer
688
views
What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?
For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the Yang--Baxter equation if
$$(R\otimes id)(id\otimes R)(R\otimes id)
= (id\otimes R)(R\otimes ...
5
votes
2
answers
599
views
Serre relations for Lie Superalgebras
Finite dimensional complex simple Lie algebras are classified using Cartan matrices. One of the main ingredients is Serre Relations. Lets call this Cartan-Killing theory.
I have the following ...
5
votes
1
answer
542
views
Asymptotic $\int_M \mathrm{exp}[\mathbf{e}\left(n -\frac{t}{2\pi i}\right)] \left( 1 + \frac{5}{6} \mathbf{e}^2 \right)^{1/2} $ on quintic Calabi-Yau
Let $M = \{ G(x) = 0 \} \subseteq \mathbb{P}^4$ be a quintic Calabi-Yau and $\mathbf{e} \in H^2(M, \mathbb{Z})$ such that $\int_M \mathbf{e}^3 = 5$. Then as $t \gg 1$:
$$
\int_M
e^{n \mathbf{e}}
e^...
5
votes
1
answer
681
views
Harmonic Crystal using Random Walk
Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here.
Basically, it is a ...
5
votes
2
answers
578
views
"Uncertainty principle" for self-adjoint operators in a finite von Neumann algebra
Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a ...
5
votes
3
answers
593
views
Monte Carlo simulations
I was wondering what were the models of statistical physics that are still considered difficult/slow to simulate (exactly, or approximately) with the current technology of Monte Carlo approaches. I ...
5
votes
4
answers
764
views
Does it help to learn statistical mechanics in order to learn thermodynamic formalism?
Does it help to learn statistical mechanics or thermodynamics (as in physics or mathematical physics) in order to learn thermodynamic formalism: the study of equilibrium states, Gibbs measure, maximal ...
5
votes
1
answer
177
views
In the rep theory of Quantum Double, why does the fusion of 2 "pure fluxes" yield a "pure charge"?
Physicist here, so my notation may be different from standard math notation.
For the quantum double $D(G)$ of a group $G$, we may write representations of $D(G)$ in the following way: Consider a ...
5
votes
1
answer
511
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\\
...
5
votes
1
answer
529
views
Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories?
So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle $...
5
votes
2
answers
520
views
Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization?
The question is in the title. The form of the condition looks like the Bohr-Sommerfeld quantization formula of angular momentum, is there a link between the two formulas?
5
votes
2
answers
698
views
Generalized basis
In quantum mechanics, people introduce the notion of "continuous basis" (I actually don't know the mathematical denomination of it). It is not a Schauder basis. I would like to know what could be a ...
5
votes
1
answer
622
views
Vandermonde-type identity for Jacobi theta functions?
My question concerns an application in physics. By Vandermonde identity I refer to the following statement: take $f_j (z)=z^j$, where $z=x+iy$ is a complex coordinate and $j$ an integer. Make an $N\...
5
votes
1
answer
327
views
Is the Wikipedia depiction of the ergosphere of a Kerr black hole a Cassini oval?
Disclaimer: this a cross post from MSE, where this question was asked on November 4th 2019 and has so far received no upvote, no comment and no answer whatsoever.
Glancing at https://en.wikipedia.org/...
5
votes
1
answer
811
views
Turing machines and Ising model
I have currently started a new research line aiming to prove a mapping between a 2-symbol Turing machine and the one dimensional Ising model. The connection is seen by recognizing that a set of ...
5
votes
1
answer
1k
views
From Sato grassmannian to spectral curve
Assume a tau-function of the KP integrable hierarchy is fixed by the point of the Sato grassmannian (that is by a semi-infinite set of Laurant series $\varphi_k(z)=\sum_{m>0}a_{km}z^{-k+m}$). Can ...
5
votes
2
answers
738
views
In what precise sense is quantum (i.e., non-commutative) probability not expressable in terms of classical probability?
The quantum set-up has many settings, so let's fix some definitions. I will be taking the Hilbert space approach with a minor modification that I will make explicit.
We begin with a Hilbert space $\...
5
votes
2
answers
390
views
Most general conditions for (weak or classical) solutions to Poisson's equation
I thought I knew this but have found it surprisingly difficult to find good references. I am interested in solving
$$
\left\{
\begin{align}
& \Delta \psi = - \rho & & \mbox{in } \mathbb{...
5
votes
1
answer
3k
views
Intuition behind the "Lapse Function"
I came across the following definite of the Lapse Function:
$N=\sqrt{\frac{1}{2}g(L,\overline{L})}$
where $L,\overline{L}$ are the null geodesic vector fields. Further, I have been looking at this ...
5
votes
1
answer
526
views
General solution to null-divergence equation
I am interested in whether each component of a divergenceless vector can itself be written as a divergence. My motivation for this question is the characterization of so-called trivial conservation ...
5
votes
2
answers
867
views
Permuting Racked Pool Balls with a Single Break
Given reasonable physical assumptions (on friction, collisions, etc.), would it be possible to "break" in a pool game such that when all the balls come to rest, the only difference is that the racked ...
5
votes
2
answers
4k
views
(Anti)Canonical divisor of a blow up
This question may be utterly trivial, or not, but as someone with hardly any knowledge of algebraic geometry I thought there could be a chance I get lucky.
Let X be a rational surface obtained by n ...
5
votes
1
answer
1k
views
Does BQP^P = BQP ? ... and what proof machinery is available?
Update #3:
Over on TCS StackExchange, I have rated as "accepted" an ingenious construction by Luca Trevisan, which answers a two-part question (as reframed by Tsuyoshi Ito) that is in essence "Do ...
5
votes
2
answers
645
views
What does it mean to extend a 2d (topological) conformal field theory to Deligne-Mumford space?
For 2D (topological) conformal field theory the corresponding moduli space is the space of Riemann surface with boundary, right? What does it mean by extending the theory to Deligne-Mumford space? How ...
5
votes
2
answers
663
views
Dependence of trace norm on matrix size for smooth vs. random matrices.
Problem
Consider two d x d complex matrices, R and S, whose entries lie in the unit disk:
$\quad |R_{i,j}|<1 \quad$ and $\quad |S_{i,j}|<1 $.
Say that R is constructed by randomly choosing ...
5
votes
1
answer
409
views
Reference Request for a particular approach of (rigorous) statistical mechanics
I was reading Mathematical Aspects of Quantum Field Theory by. E. de Faria and W. de Melo, and the following caught my attention.
In (Hamiltonian) mechanics, the states of a system are described by ...
5
votes
1
answer
298
views
In search of a combinatorial proof on particular set of partitions
Given a partition $\lambda=(\lambda_1\geq\lambda_2\geq\dots)$, denote the conjugate partition by $\lambda'=(\lambda_1'\geq\lambda_2'\geq\dots)$. For example, if $\lambda=(4,2,2)$ then $\lambda'=(3,3,1,...
5
votes
1
answer
253
views
approximate closed form for infinite series (no power series)
The electric potential of a charge between two infinite conducting planes can be expressed as an infinite series (of image charges) [Kellogg1929] as
$$
V=\sum_{n=-\infty}^{\infty}\left(
\frac{1}{\...
5
votes
2
answers
488
views
connection between the distribution of energy levels and spacings between prime numbers [closed]
I'm a student in university, I'm new in this forum. My teacher told me to use it for my research. I came to ask for advice and help. The research is about the connection between prime numbers and ...
5
votes
1
answer
536
views
Question about the Aganagic-Vafa A-brane
According to Aganagic-Vafa (hep-th/0012041) and Fang-Liu (arXiv:1103.0693), for a semi-projective toric Calabi-Yau 3-manifold $X$, the Aganagic-Vafa A-brane $L_{AV}\subset X$ is defined by the ...
5
votes
1
answer
284
views
Absent 2nd order terms in deformation quantization of Poisson manifolds
I am reading Kontsevich' famous paper on deformation quantization of Poisson manifolds. In section 1.4.2 on page 4 he gives the general formula for the star product associated to a Poisson structure ...
5
votes
1
answer
250
views
Well defined Tensoring of spectral triples
Hi,
I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it.
Question: In connes standard model he takes ...
5
votes
3
answers
2k
views
Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way
I am interested in a characterization of the creation and annihilation operators that is in some sense invariant under $O(n)$ rotations of $\mathbb{R}^n$:
Background
The Harmonic Oscillator on $\...
5
votes
2
answers
414
views
When can we factor out the time dimension?
First of all, my knowledge of both GR and differential geometry is quite weak, so forgive me if the physics here doesn't make much sense.
Let $(M, g)$ be a smooth, connected Lorentzian manifold of ...
5
votes
1
answer
411
views
Can Fock spaces be replaced by arbitrary Hilbert spaces under some hypothesis to justify path integrals?
I was reading this post from PSE and it reminded me an old question of mine, in which the use of creation and annihilation operators were discussed. Both questions got answers which agreed on the fact ...
5
votes
1
answer
504
views
Riemann-Hilbert approach to Selberg integral
I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with ...
5
votes
1
answer
775
views
A set of questions on continuous Gaussian Free Fields (GFF)
As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
5
votes
2
answers
366
views
Connections between two constructions of infinite dimensional Gaussian measures
Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space $l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} ...
5
votes
1
answer
443
views
Scattering theory for Coulomb potential
Both physical and mathematical theories of quantum scattering seem to be well developed in the case when the potential (or a more general perturbation of the Laplacian) decays fast enough at infinity ...
5
votes
1
answer
645
views
Gauge invariance of Chern-Simons functional integral for a 3-manifold with boundary
I am trying to understand how the functional integral for Chern-Simons theory for a possibly non-compact 3-manifold with boundary is made gauge invariant.
For a compact 3-manifold, $M$, without ...
5
votes
1
answer
314
views
Nonlinear sigma models with non-compact groups / target spaces
A nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T.
The target manifold T is equipped with a Riemannian metric g. Σ is ...