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.

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Representation theory of spinors - Understanding how $\mathrm{SO}_3$ acts in particle physics

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David ...
Andrea's user avatar
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Predicting the peak "amplitude" of a damped sine wave in the frequency spectrum with FFT

In one line: Given an exponentially decaying sine wave $x(t)$, how can we predict the amplitude of the resulting peak in frequency spectrum using discrete Fourier transform. In nuclear magnetic ...
AChem's user avatar
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Algebra/Algebraic geometry in statistical mechanics

This is a soft question. I am currently studying statistical mechanics and I found this one by chance: Algebraic statistical mechanics And I also found some workshops on interactions between ...
FFjet's user avatar
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N dimensional, not-locally Euclidean, non-Hausdorff topological space

Take a topological space $(M, \tau) $ where $\tau$ is the collection of open sets of $M$. Suppose: the Lebesgue covering dimension of this space is $N \geq 1$ Non-Hausdorff Not locally Euclidean The ...
Bastam Tajik's user avatar
14 votes
1 answer
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Hilbert's sixth problem and QFT description

The Wikipedia entry on Hilbert's sixth problem about QFT description is “Since the 1960s, following the work of Arthur Wightman and Rudolf Haag, modern quantum field theory can also be considered ...
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Discrete spectrum of $A \otimes 1+ 1 \otimes B$ [migrated]

Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$, with both non-empty discrete spectra. Let us say, for instance, $\inf \, \sigma(A) = \lambda_1^A$ and $...
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Symplectic structure of Higgs branch

I've been reading Kamnitzer's survey Symplectic resolutions, symplectic duality, and Coulomb branches. Here the Higgs branch is defined as a projective GIT quotient, but I couldn't figure out how the ...
Ji Woong Park's user avatar
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On the linearized evolution equations in general relativity

The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
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Perturbation method for time-periodic singular system of ODEs

I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form: $$ \begin{cases} \dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
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Why does deformation quantisation have anything to do with $\mathbf{E}_2$/little disks?

Kontsevich proved that any Poisson manifold $M$ has a quantisation $\mathcal{O}_\hbar(M)$: an associative algebra recovering the $\mathcal{O}(M)$ with its Poisson bracket by taking $\hbar=0$. Later he ...
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Recommendation to understand mean field theorem

I am studying Rodnianski and Schlein - Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Everything was clear for me and I reproved everything before inequality (3.22) (except ...
Mr. Proof's user avatar
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What is the Lebesgue covering dimension of this topological space?

Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal. Take the induced topology defined by the Lorentzian metric called Alexandrov topology. This topology matches ...
Bastam Tajik's user avatar
7 votes
1 answer
604 views

What are double groups mathematically?

In physics and chemistry, there is the concept of double groups. These are double covers of the usual point groups, obtained by "adding an element $R$, which represents rotation by $2\pi$" ...
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Definition of second quantization

The standard textbook for second quantization is Reed & Simon. However, I am a bit confused with their notation. They write: Let $\mathscr{H}$ be a Hilbert space, $\mathcal{F}(\mathscr{H})$ the ...
MathMath's user avatar
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7-sphere x 4-sphere manifold and its physical significance [closed]

I am looking for sources about this manifold 7-sphere*4-sphere and its relations to theoretical physics. Where to go to read about 7-sphereX4-sphere manifold and its physical significance?
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What is the dual of a hyperbolic configuration of points?

Let $C_n$ denote the configuration space of $n$ distinct points in hyperbolic $3$-space $\mathcal{H}$. If $\mathbf{x} := (\mathbf{x}_1, \dots, \mathbf{x}_n) \in C_n$, where $\mathbf{x}_i \in \mathcal{...
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Why computing $n$-point correlations?

I am trying to find a sufficiently convincing answer to this question, but it has been taking so much of my time and I can't get anywhere. It also follows my previous question on PSE. In axiomatic QFT,...
MathMath's user avatar
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4 votes
2 answers
191 views

Reference for rigorous interacting many-body quantum mechanics

Are there good references for (both zero and finite time) interacting systems of quantum many-body theory? More precisely, I would be interested in references discussing the following topics: Second ...
MathMath's user avatar
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Mixing for a gas of hard spheres

The gas of hard spheres is a model for a gas in a container, where each particle is a sphere of radius $\epsilon$. The spheres interact with each other and with the container with elastic collisions. ...
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Is the Gödel universe Wick rotatable?

Take Wick rotatability being as the way defined in the following article by Helleland and Hervik: Christer Helleland, Sigbjørn Hervik, Wick rotations and real GIT, Journal of Geometry and Physics 123 ...
Bastam Tajik's user avatar
1 vote
1 answer
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Green's function for a linear PDE initial value problem

For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem $$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{...
Abhishek Halder's user avatar
1 vote
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How causal is a strongly causal purely electric spacetime?

Take a generic Lorentzian spacetime $(M, g)$ where $M$ is a time-oriented 4d manifold and $g$ is the Lorentzian metric that is strongly causal and purely electric. According to this answer: Is every ...
Bastam Tajik's user avatar
2 votes
0 answers
344 views

Is a Wick rotatable spacetime necessarily strongly causal?

There are a few viable ways to formulate Wick rotatability that preserve distinct features. One is mentioned in the post: Obtain Lorentzian manifolds from Riemannian ones by Wick rotation There's also ...
Bastam Tajik's user avatar
5 votes
5 answers
974 views

What are the local maxima and minima of $\frac{\sin(nx)}{\sin(x)}$

FYI: I asked this question here couple of days ago but got no answer yet. $n$ is an integer We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are ...
RajaKrishnappa's user avatar
7 votes
2 answers
511 views

Prove that the following function is positive

Consider the following function: $$K(x, y; t) = \sum_{n \geq 0} \frac{e^{-(2n+1)t}}{\sqrt{\pi} 2^n n!} H_n(x) H_n(y) \exp\left(-\frac{(x^2 + y^2)}{2}\right) $$ This is Mehler's kernel, and can be ...
matilda's user avatar
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Is every strongly causal spacetime purely electric?

Take a time 4-dimensional orinted Lorentzian manifold $(M,g)$. A spacetime is called Strongly Causal at point $p$ if and only if for every neighbourhood $U$ of the point $p$ there exists a ...
Bastam Tajik's user avatar
3 votes
0 answers
160 views

Proof of the existence of a mirror Calabi–Yau manifold

Let $X$ be a Calabi–Yau threefold. Here, Calabi–Yau is understood to a mean a smooth simply connected projective threefold with $h^1(\mathcal{O}_X) = h^2(\mathcal{O}_X)=0$ and holomorphically trivial ...
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AQFT from a Lagrangian

In physics, the fundamental description of physical theories frequently revolves around the concept of a Lagrangian. My expertise encompasses diverse algebraic formulations within the domain of ...
Gabriel Palau's user avatar
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Flat Maurer-Cartan connection iff flat Berry connection

I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$. The first is the canonical or $H$-...
Victor V Albert's user avatar
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133 views

Reference request: an introduction to nuclear spaces

I am looking for a short introduction to nuclear spaces and nuclear operators. I am interested in these spaces as they often arise in mathematically rigorous quantum field theories. I have read the ...
CBBAM's user avatar
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1 answer
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Overview resources for (rigorous) critical phenomena

I recently came across this overview which discusses some results in the theory of critical phenomena. It is already quite old and I would like to know if there are other (more recent) overviews in ...
MathMath's user avatar
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7 votes
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174 views

Complex cobordism and integrable systems

In Jack Morava's paper On the complex cobordism ring as a Fock representation, it was remarked right at the beginning that complex cobordism may play a role in the theory of integrable systems. In ...
user1271629's user avatar
6 votes
0 answers
182 views

Does a fusion ring with F and R-symbols uniquely determine a braided tensor category?

Background : In mathematical physics, 'anyons' in (2+1) dimensional systems are described by braided tensor categories. The anyon types correspond to the irreducible objects of the category. From such ...
Alex_Bols's user avatar
4 votes
0 answers
120 views

Coulomb branches which are not of cotangent type

To each $3d \, N=4$ supersymmetric quantum field theory $\mathcal{T}$, there is a related space called the Coulomb branch of this theory, $\mathcal{M}_C(\mathcal{T})$ (it is a piece of the moduli ...
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2 votes
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Can Coulomb branches have symplectic resolutions?

My question is about Coulomb branches of a $3D$ $\mathcal{N}=4$ supersymmetric gauge theory, in the sense of Bravermann, Finkelberg and Nakajima Towards a mathematical definition of Coulomb branches ...
jg1896's user avatar
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1 vote
0 answers
224 views

Is there any good book about mathematical physics? [closed]

Is there any book that generally introduces/talks about mathematical physics as a whole and that emphasizes on mathematics, not physics? Or is there no such single book because mathematical physics is ...
ale_7's user avatar
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11 votes
1 answer
579 views

Progress on Simon's 1984 problem of the proof of Universality

I am writing this post to inquire if any progress has been made in solving problem 8B (Proof of Universality) proposed by Barry Simon in 1984. The problem goes like this: Show that the critical ...
Leibniz's Alien's user avatar
3 votes
0 answers
185 views

Decomposition of forms on manifolds

Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold, i.e. $M=I\times\Sigma$ with $I\subset\mathbb{R}$ being an open interval and $\Sigma$ a spacelike smooth Cauchy hypersurface. The metric is of ...
G. Blaickner's user avatar
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2 votes
1 answer
154 views

Exact calculations with Moyal product by "Bopp Shift"

I'm now working on my Phd thesis on the area of deformation quantization and field theory. After doing all the "ground work" (definitions, motivations, basics of the theory etc) I have now ...
Diego Santos's user avatar
6 votes
1 answer
226 views

Question about a remark on quantization of Coulomb branches

I will follow the definition of Coulomb branches of $3d$ $\mathcal{N}=4$ gauge theories from the paper by Braverman, Finkelberg and Nakajima, Towards a mathematical definition of Coulomb branches of 3-...
jg1896's user avatar
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2 votes
0 answers
114 views

Rigorous QFT from integration over subspace

Many perturbative QFTs suffer from the lack of a rigorous definition of a "good enough" measure over the space of paths (or fields) $P$, $$\mathcal{Z} = \int_{{x \in P}} e^{iS(x)} Dx$$ There ...
Student's user avatar
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6 votes
2 answers
592 views

Explicit form of this unitary transformation

Disclaimer: This question has its motivation from physics. It is probably not entirely rigorous at the moment. I just want to clarify some steps and try to make the arguments rigorous afterwards, if ...
IamWill's user avatar
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4 votes
2 answers
255 views

Regularity of solution of $(-\Delta + w)f = 0$

I am studying the following Schrödinger equation: $$(-\Delta + w)f = 0$$ which represents a quantum state with zero energy. Here $w$ and $f$ are defined on $\mathbb{R}^{3}$. For simplicity, let us ...
MathMath's user avatar
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7 votes
0 answers
195 views

Rigorous treatment of Ostrogradsky's instability theorem?

The Ostrogradsky instability theorem says that if a Lagrangian depends on more than the position and velocity, the corresponding Hamiltonian is unbounded below. This has been suggested as a reason why ...
user479223's user avatar
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1 vote
0 answers
65 views

Are analytic solutions for the Navier-Stokes equations sufficient?

Generally, we ask for solutions of the Navier-Stokes equations, when the starting conditions are in the Schwartz space. However, I am wondering, whether it is possible to consider just analytic ...
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5 votes
1 answer
224 views

Rozansky-Witten invariants of hyperkahler manifolds and independence of complex structure

Recently I have been learning about Rozansky-Witten invariants, mainly through Hitchin-Sawon's paper "curvature and characteristic numbers of hyperkahler manifolds" and through Justin Sawon'...
BS math's user avatar
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3 votes
1 answer
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1D topological defects in $d>3$ spatial dimensions

I am trying to construct a 1D topological defect solution in 4 spatial dimensions, i.e., a solution to some PDE (likely the equations of motion of some Lagrangian) on $\mathbb{R}^{4}$ which is ...
math_lover's user avatar
2 votes
1 answer
135 views

Definition of average $\langle \langle \cdot \rangle \rangle$

I started reading the paper Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model and I would like to clarify the notation $\langle \langle \cdot \rangle\rangle$ the authors use in ...
IamWill's user avatar
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-2 votes
1 answer
194 views

Is there any case of remormalization in which we have to solve it by ways in two different systems? [closed]

In renormalization of physics, $$\sum_{j=1}^{\infty}j=-\frac{1}{12}$$ We may obtain the result in two ways: first we may redifine the sum so we have used two system of math with different definition ...
XL _At_Here_There's user avatar
2 votes
0 answers
53 views

Anderson localization for time-dependent noises

Anderson localization concerns the localization properties of the Schrödinger operator with a Hamiltonian of the form $$H=-\Delta+V(x),$$ where $V$ is a highly oscillatory random potential. A simple ...
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