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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Yang-Mills theory v.s. Kaluza–Klein theory: Classical actions

In general Yang-Mills theory [1] seems to be different from the dimensional reduced Kaluza–Klein theory. However, the historical account was that people tried to trace back the origin of non-Abelian ...
wonderich's user avatar
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4 votes
1 answer
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Is this Hermite polynomial identity known?

In some physics related problem, I found out the curious identity $$\sum\limits_{n_1+n_2+n_3=n}\frac{n!}{n_1!\,n_2!\,n_3!}\,H_{2n_1}(x)\,H_{2n_2}(y)\,H_{2n_3}(z)=\frac{H_{2n+1}(r)}{2r},$$ where $H_n(x)...
Zurab Silagadze's user avatar
11 votes
3 answers
1k views

Navier-Stokes fluid dynamics, Einstein gravity and holography

There was some activity a while ago, like 10 years ago, string theoreists try to relate the fluid dynamics, for example, governed by Navier-Stokes equation, to the Einstein gravity, and its ...
wonderich's user avatar
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Anderson localization for fractional Laplacians

There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(\mathbb{Z}^d)$ such as $$ -\Delta+\lambda V $$ where $\Delta$ is the discrete ...
Abdelmalek Abdesselam's user avatar
7 votes
3 answers
584 views

Physical Applications of Locally Symmetric Spaces

Locally Symmetric Spaces are the basis of the Langlands program—a set of ambitious and interconnected conjectures connecting representation theory to number theory, firstly proposed in 1967 by Robert ...
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Yang-Mills theory with non-compact gauge groups G

Physicists are familiar working with Yang-Mills theory with compact and semi-simple gauge groups $G$ (Lie groups). However, it is not entirely clear the formulation of Yang-Mills theory with non-...
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GSO projection and $H^d(M, \mathbb{Z}_2)$

This follows up the comment which suggests that asking the later 2nd part of subquestion in "GSO (Gliozzi-Scherk-Olive) projection and its Mathematics" as a new different question GSO (...
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GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?

GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...
wonderich's user avatar
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8 votes
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Lifting a diffeomorphism into a spinor bundle automorphism

I know several papers that treat this, but it seems that most of these papers do things very differently with quite different conclusions, so I am confused. Basically, when one tries to do classical ...
Bence Racskó's user avatar
9 votes
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Harmonic oscillator in spherical coordinates

It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry. More precisely, the operator $$-\frac{d^2}{dx^2}+x^2$$ can be ...
ErwinSchr's user avatar
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Holomorphic map, Instantons of Complex Projective Space and Loop Group

It seems that holomorphic (or rational) maps play a crucial role to relate the following data: Instanton in 1-dimensional complex Projective Space $$\mathbb{P}^1$$ in a 2 dimensional (2d) spacetime. ...
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Chern-Simons theory with non-compact gauge groups G

This is related to a previous question, where a nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. There we ask the general ...
wonderich's user avatar
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Minkowski isometries

Consider theorem 1.7 from chapter III of 'Elementary differential geometry' by O'Neill. It says that: Theorem 1.7: If $\phi$ is an isometry of $E^3 $, then there exists a unique translation $T$ and a ...
Soennecken's user avatar
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Does current follow the path(s) of least (total) resistance?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (...
badmf's user avatar
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Witten zeta function v.s. Riemann zeta function

From a talk, we learned that The symplectic volume of the space $M$ of gauge equivalence classes of flat G connections is given by the “Witten zeta function”: where we sum over irreducible ...
wonderich's user avatar
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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 ...
wonderich's user avatar
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Conserved positive charge for a PDE

Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$: \begin{equation} \frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...
Maurizio Barbato's user avatar
7 votes
1 answer
1k views

Moduli space of flat connections over a Riemann surface

If I understand correctly, in the Refs below: We can see that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\...
wonderich's user avatar
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3 votes
1 answer
148 views

Reference request for quantum Teichmuller space

I would like to ask for some detailed reference for quantum Teichmuller theory, better in a mathematical taste. I read a little bit on Kashaev's or Chekhov and Fock's, but find that I need to fill ...
Peter Liu's user avatar
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1 answer
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The Precise Meaning of the Moduli Space of Flat Connections?

Questions: I would like to have a precise description of the meanings of the Moduli Space of Flat Connections, such that it is understandable by mathematical physicists and physicists. For 3d Chern-...
wonderich's user avatar
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12 votes
1 answer
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Coleman–Mandula theorem and a mathematical proof

Coleman–Mandula theorem (by Sidney Coleman and Jeffrey Mandula) [1] is a no-go theorem in theoretical physics. It states that "space-time and internal symmetries cannot be combined in any but a ...
wonderich's user avatar
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2 votes
1 answer
113 views

stationary measure for linear cocycle(random transformation matrices)

Let $(M,\mathcal B, \mu)$ be a probability space which $M=\{A_{1},A_{2},...,A_{N}\}^{\mathbb{N}}$ ($A_{i} \in GL(d ,\mathbb{R})$) and $\mu=p^{\mathbb{N}}$. Let $F:M\times \mathbb R^d\to M\times \...
Michal's user avatar
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1 answer
423 views

Further Developments of Lieb-Schultz-Mattis theorem in Mathematics

The Lieb-Schultz-Mattis theorem [1] and its higher-dimensional generalizations [2] says that a translation-invariant lattice model of spin-1/2's cannot allow a non-degenerate ground state preserving ...
wonderich's user avatar
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4 votes
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Complex projective algebraic variety, moduli space of flat connections, and instantons

In Looijenga's work below, if I understand correctly, it shows that Statement 1: At an algebraic variety, the moduli space of SU($N$) flat connections on a 2-torus $T^2$ is given by the space of ...
wonderich's user avatar
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372 views

Dimensions of the instanton moduli space from Atiyah-Hitchin-Singer

Atiyah-Hitchin-Singer Ref 1 states that the number of virtual dimensions of the instanton moduli space for SU(N) Yang-Mills theory with topological charge $\mathcal{Q}$ over a manifold $X$ is given ...
wonderich's user avatar
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14 votes
1 answer
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Borel-Écalle re-summation and resurgence: criteria and results

This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below. This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
wonderich's user avatar
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11 votes
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588 views

The Grassmannian Gr(2,8) and an E7 surprise

Are there any mathematical explanations for the following surprising facts? $$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$ and $$\int_{Gr(2,6)} c_{\text{top}}...
Richard Eager's user avatar
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Is there another quantum deformation of sl(2)?

By looking at defining relations of standard deformation of $\mathfrak{sl}_2$, which are: $$ [E,F] = \frac{q^{H}-q^{-H}}{q-q^{-1}}, \quad [H,E] = 2E, \quad \text{ and } \quad [H,F] = -2F, $$ some ...
Jedy's user avatar
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3 votes
2 answers
2k views

Casimir operator of a given Lie algebra and relation with its matrix representation

I'm following Gilmore's recipe to compute the abstract Casimir operator of a given algebra (in this example, I refer to algebra su(2)). This recipe bring up a matrix representation of the algebra and ...
AndreaPaco's user avatar
19 votes
1 answer
1k views

How is Chern-Simons theory related to Floer homology?

Chern-Simons theory (say, with gauge group $G$) is the quantum theory of the Chern-Simons functional $$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$ ...
Henry's user avatar
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5 votes
0 answers
269 views

Have complex manifolds with dual number structure on the holomorphic tangent bundle been studied?

If $M$ denotes a $2n$-dimensional real smooth manifold, then $M$ together with $J\in\operatorname{End}(TM)$ with $J^2 =\mathrm{id}$ (also called almost product structure on $M$) have been surveyed in ...
M.G.'s user avatar
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32 votes
1 answer
2k views

Limiting representation theory of quantum groups at roots of unity and $SL(2,\mathbb{C})$

Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
John Pardon's user avatar
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4 votes
1 answer
250 views

The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators

Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$). How is the embedding $\mathfrak{g}...
Christoph Mark's user avatar
6 votes
1 answer
415 views

Non-Hamiltonian actions in physics

I was reading the following article when I came across the interesting sentence "non-Hamiltonian [symplectic group] actions also occur in physics" I took a cursory look at the article cited but ...
R Mary's user avatar
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32 votes
0 answers
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Vertex coloring inherited from perfect matchings (motivated by quantum physics)

Added (19.01.2021): Dustin Mixon wrote a blog post about the question where he reformulated and generalized the question. Added (25.12.2020): I made a youtube video to explain the question in detail. ...
Mario Krenn's user avatar
6 votes
2 answers
711 views

Gauge integral versus path integral

According to this paper, "The gauge integral [a.k.a. Henstock-Kurzweil integral] provides the only formal framework that is close to the original development of the Feynman path integral", and also "...
Nik Weaver's user avatar
1 vote
1 answer
65 views

Existence of bisolutions for wave operators on globally hyperbolic Lorentzian manifolds

Let $(M,g)$ be a globally hyperbolic Lorentzian manifold with Lorentzian volume density $dV$ and $\Sigma$ a Cauchy hypersurface, i.e. each inextendible causal curve hits $\Sigma$ exactly once. ...
megggs's user avatar
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0 votes
1 answer
155 views

Nonlocal integral

I have a little problem with the next integral, $$ \int{d^3{\bf r^{\prime}}\left[\frac{exp(-ar^{\prime})}{r^\prime}\right]u({\bf r}-{\bf r^\prime}})=\int{4\pi r^\prime dr^{\prime}exp(-ar^{\...
Jhoan Perez's user avatar
1 vote
0 answers
195 views

Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices

Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm (http://mathworld.wolfram.com/OperatorNorm.html) of the $2\times 2$ ...
Paul B. Slater's user avatar
2 votes
1 answer
473 views

PDE’s whose solutions can be presented using path integrals

It is well known that solutions of the Schroedinger equation and of the heat equation can be presented using path integrals: $$\psi(x,t)=\int K(x,t;y,0)\psi(y,0)dy,$$ where the kernel $K(x,t;y,0)$ is ...
asv's user avatar
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2 votes
0 answers
51 views

What restrictions on the form of an integral equation have a unique solution f=0?

We're stuck on the following question in a problem relevant to a physics paper on AdS/CFT that we are working on. Given a Fredholm equation of second kind with the form $f$+$\int_D K f\,dx = 0$, where ...
Ning Bao's user avatar
8 votes
3 answers
1k views

Simple Subalgebras of Simple Lie Algebras

Given a complex simple Lie algebra $\mathfrak{g}$ of rank $n\in\mathbb{N}$ with $n$ sufficiently large (say $n\ge10$), is there a way to determine whether $\mathfrak{g}$ contains a simple subalgebra ...
Gregoire Rad's user avatar
17 votes
2 answers
1k views

What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?

Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$ superconformal field theories (Wikipedia, nLab, PhysicsOverflow) sometimes called Theory $...
Arun Debray's user avatar
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-5 votes
1 answer
305 views

Borromean rings, Condorcet's paradox and Quantum chromodynamics [closed]

In https://plus.google.com/108432079989441783124/posts/LHewqvcj5Xo T. Abderrahman explains what Borromean rings are. As I noticed in a comment, the underlying order structure is the same as in ...
Sylvain JULIEN's user avatar
2 votes
1 answer
145 views

$P(1)$ strange type classical Lie superalgebras

I am reading the book "Lie superalgebras and enveloping algebras" by Ian M. Musson. The strange type $P(n)$ series of Lie superalgebras are defined (§2.4.1, p. 17) only for $n \ge 2$ even though for $...
GA316's user avatar
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5 votes
0 answers
392 views

Mathematical proof of Regge symmetry

In the representation theory of the group $SU_2$ a big role is played by so-called $6j-$symbols. Let me sketch its definition (some other interpretations could be found here). Denote a representation ...
Daniil Rudenko's user avatar
5 votes
0 answers
201 views

About mod 2 Index of Dirac Operators in 3D on Non-Orientable Manifold

I was reading Witten's paper "Fermion Path Integrals and Topological Phases" (https://arxiv.org/abs/1508.04715). He claimed that Indeed, on an orientable 3-manifold, the eigenvalues of the Dirac ...
Weicheng Ye's user avatar
12 votes
0 answers
490 views

CohFT: Witten vs. Kontsevich and Manin

Is there any connection to CohFTs as defined by Witten in his 1988 paper (via topological twist) and the CohFTs as defined by Kontsevich and Manin (in the context of Gromov-Witten theory of course). ...
Gorbz's user avatar
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1 vote
0 answers
90 views

The hermitian Einstein manifolds

I take an hermitian manifold $(M,g,J)$ and I define from the riemannian curvature $R(X,Y)$ as a $2$-form: $$ Ricc(J)= \sum_i R(J e_i,e_i) $$ with $(e_i)$ an orthonormal basis of the tangent. $$ 2R(J)=...
A.Balan's user avatar
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2 votes
2 answers
308 views

Criteria for Schrödinger operator on real line to have simple spectrum

Consider a Schrödinger operator $H:=-\Delta+V$ on $\mathbb R$, where $V$ is such that $H$ has a purely discrete spectrum $-\infty<\lambda_1\leq\lambda_2\leq\cdots$ converging to $+\infty$. Do there ...
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