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

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Quantum Hamiltonian for an Inverse Cube Force Law

If you have a nonrelativistic quantum particle in $\mathbb{R}^3$ in an attractive inverse cube force, its Hamiltonian is $$ H = -\nabla^2 - \frac{c}{r^2} $$ where I'm keeping things simple by ...
4
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1answer
253 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}} ...
1
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0answers
33 views

Minkowski spacetime in Newman Penrose formalism

I have a rather basic question for which (surprisingly!) I cannot find a short and clear answer anywhere: I'm currently looking at the Newman Penrose (NP) formalism (I use primarily Chandrasekhar's ...
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55 views

Singular integral equation

Investigating a control problem for heat equation I stacked on solution of this integral equation which seems to be singular: $$ \int_0^1\mathcal{K}(\tau)u(\tau)d\tau=\mathcal{M}, $$ in which ...
9
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2answers
620 views

How should a mathematician approach the physics literature concerning percolation?

I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard ...
3
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1answer
169 views

Action of G_2 on certain 7x7 skew-symmetric matrices

I have been working with something related to Goldman bracket for $G_2$ gauge group. There I have something like "$\text{Tr}(M_{\gamma}O_i)$", where $M_{\gamma}$ is a monodromy which takes value in ...
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0answers
41 views

anomaly polynomial of generalized Hitchin system

I would like to ask about mathematical background of this object. So, I am trying to puzzle out with 4d $\mathcal{N}=2$ SQFT. As far as I can gather this theroy can be described in terms of ...
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108 views

Can the compactification of a (co)tangent bundle equipped with Saski metric be viewed as a “Wick rotation”?

We can equip the (co)tangent bundle of a Riemannian manifold (B,g) with a Saski metric $\hat{g}$ (see, for example, "On the geometry of tangent bundles" by Gudmunssun & Kappos) that looks like ...
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1answer
72 views

Existence and uniqueness for two-dimensional time-dependent Schrödinger equation

I currently have to deal with time-dependent Schrödinger equations in two variables on bounded domains and wanted to find out about uniqueness and existence of solutions. Unfortunately, I am a ...
5
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73 views

Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? : $V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). ...
10
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480 views

Witten's proof of Morse inequalities, question on eigenvalues?

See here. I present Theorem 6 and Corollary 7 as follows. Theorem 6. For large $s$, $\Delta_s$ and $H$ have the same number of eigenvalues in the interval $[0, s^{-2/5}]$. Corollary 7. $\dim ...
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396 views

Which journals publish applied mathematics with mostly pure mathematics content?

In the spirit of Which journals publish expository work? please advise: What consistently high quality journals$^1$ today publish results that would otherwise go to a pure mathematics journal if ...
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70 views

Petrov type D spacetime

I have a rather general question about the Petrov type D spacetime. The standard definition (also used in the official Wikipedia article) is that the Petrov classification is according to which Weyl ...
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1answer
117 views

A proof of the Ibragimov et al. commutation relation

Let $u_a(x),\,a=1,2,\ldots n$ be a $n$-component field in Minkowski spacetime $x^\mu,\,\mu=0,1,2,3$ and let $u_{a,\,\mu}=\frac{du_a}{dx^\mu}$. Let us introduce two operators (we use Einstein ...
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0answers
145 views

Proof of Arnold-Liouville theorem in classical mechanics [closed]

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
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53 views

When is the discrete logarithmic energy not approximable by its ostensibly more general counterparts?

In my answer at Maximum of the Vandermonde determinant / minimum of the logarithmic energy it is shown that that for each large enough natural $n$ there is some $a=(a_0,\dots,a_{n-1})$ with ...
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109 views

Gauge freedom in the tetrad

I'm reading the following paper about Petrov type D space times called "Type D vacuum metrics": http://scitation.aip.org/content/aip/journal/jmp/10/7/10.1063/1.1664958 by Kinnersley. I have a ...
3
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1answer
120 views

Stationary distribution of last passage percolation

Consider last passage percolation model on $\mathbb{Z}^2$. I am interested to know if there is any known result for the stationary distribution of passage times, given some distribution for the ...
2
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1answer
108 views

Conformally covariant distributions

In Conformal Field Theory (in $D$ dimensions) one considers (in particular) correlation functions of the form $$ \langle O(x)O(y)\rangle, $$ where $O$ is a scalar primary field. Scale covariance ...
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59 views

Extracting information from a differential equation if the zero eigenvalue eigen-function is known

Given the second order linear homogeneous differential equation $$ -\dfrac{d^2}{dx^2}\psi_m(x) + V(x)\psi_m(x)=E_m\psi_m(x) $$ with eigen-functions $\psi_m(x)$ and eigenvalues $E_m$, what information ...
6
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2answers
160 views

Gauss-Milgram formula for fermionic topological order?

For Bosonic topological order, a very useful formula was proved to be true: $\sum_a d_a^2 \theta_a=\mathcal{D} \exp(\frac{c_-}{8}2\pi i) $ (for more detail: $d_a$ is the quantum dimension of anyon ...
6
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3answers
736 views

Ambidexterity and Quantization

Here the nlab says about Hopkins-Lurie's ambidexterity paper: "The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in ...
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58 views

Kerr metric affine parameter

I'm going through the chapter about Kerr space-time of Chandrasekhar's "Mathematical theory of black holes", and have a question about the following transformation: the idea is, that one wants to ...
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60 views

Infinitesimal variation of spectrum of Schrödinger operator with changing domain

Suppose we have a Schrödinger operator $$-\frac{d^2}{dx^2}+V(x)$$ defined on $[a,b]$ with Dirichlet boundary conditions. I am interested in whether there are any results for the variation of the ...
4
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3answers
349 views

$A \wedge A \wedge A$ in Chern-Simons

I am confused with the wedging operations of Lie algebra valued differential forms. Especially, for instance, I have some problems with the Chern-Simons 3-form $$A \wedge dA + \frac{2}{3}A \wedge A ...
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3answers
276 views

Do non-normal states exist in the Solovay model?

Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator). Is this also true in the Solovay model ...
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71 views

How does one express a Lagrangian via differential forms? [duplicate]

I asked this question here on Physics.SE; and I accepted an answer, which thinking about it later I was dis-satisfied with; to save clicking on the link I'm reproducing the question below: In ...
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1answer
77 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C. I'd like a function μ:Cn×n→[0,∞) ...
4
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2answers
176 views

Find the expansion of the exact solution (beyond Taylor)

In a paper by Kitagawa & Ueda Squeezed spin states they give an argument that the minimum variance in one-axis twisting Hamiltonian scales like $V_{min} \propto S^{-2/3}$. I will shortly describe ...
5
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1answer
151 views

What's the relation between spin model for subfactors theory and physics?

In the sense of subfactor theory, a spin model is a commuting square of the form $$\begin{matrix} \Delta &\subset & M_n(\mathbb{C})\cr \cup &\ &\cup\cr \mathbb{C} &\subset ...
6
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2answers
370 views

Is there a nice “synthetic” way for doing differential geometry on infinite dimensional vector spaces?

If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing ...
5
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2answers
270 views

Weak solutions for a PDE of fourth order

I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side: $$ \Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial ...
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1answer
301 views

Physics that needs “new” math [closed]

Just curious: I can't think of a single example that a physicist did not had his mouth open in amazement when he learnt that all (OK, lets say the foundations) the math he needs for his brand-new ...
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116 views

General procedure to find the determinant of an operator?

I want to learn to find the determinant of an operator. I am given an operator like $\Sigma _{\alpha\beta}=-k^2g_{\alpha\beta}+i\theta\epsilon_{\alpha\beta\gamma} k^\gamma$ $k^2=k^μk_μ$, $g^{αβ}$ ...
3
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1answer
211 views

An integral equation

I have a Fredholm integral equation of second kind $$\frac{-1}{2\pi \omega'^2}+\int_{-\infty}^{\infty}\frac{1}{\pi ...
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92 views

Tetrad transformation

I have been reading an article about Type D gravitational fields, and came across the tetrad transformation that I cannot understand. The author evidently introduced a new coordinate, but just a ...
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1answer
141 views

Null geodesic congruence

I came across a statement in Chandrasekhar's "Mathematical Theory of Black Holes" that I don't understand (rather say disagree): Assume we have a Newman Penrose tetrad $\lbrace l, ...
2
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1answer
97 views

AKSZ sigma models for higher spin

The AKSZ framework constructs 2D sigma models in the BV formalism. Is there a generalization of the AKSZ approach to higher spin?
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0answers
128 views

On fundamental solutions to Poisson equation on 3-dimensional manifolds

I am interesting in solutions to Poisson equation $$\triangle \varphi = 4 \pi \rho \qquad (1)$$ defined on 3-dimensional oriented Riemannian manifold $(M,g)$, where $g$ is metric and ...
12
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2answers
314 views

Is there a relationship between Fourier transformations and cotangent spaces?

Maybe a trivial question but I can't seem to find it treated anywhere. Consider a smooth manifold $Q$ (configuration space) and its cotangent bundle $T^*Q$ (phase space). Any function ...
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0answers
70 views

Energy inequalities for Sobolev spaces of negative integer

I asked this question in mathematics stackexchange and couldn't get an answer. Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| ...
1
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1answer
87 views

Derivative of a time evolution operator w.r.t. a parameter

Let $N\geq1$ be an integer and let $H:[0,1]^2\to\mathbb C^{N\times N}$ be a pointwise hermitean matrix valued function. For $y\in[0,1]$ and $0\leq a\leq b\leq 1$, let $U_y(b,a)$ be the time evolution ...
3
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0answers
226 views

Unusual generalization of the law of large numbers

I have seen in physical literature an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...
27
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2answers
678 views

Volume of the unitary group

I saw a very remarkable asymptotic formula (or a conjecture?) for the volume of of the unitary group $ U(n)$ which is the following: $$\log[\mathrm{Volume}(U(n))] \sim_{n\rightarrow \infty} ...
5
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0answers
151 views

TQFT characterization of braiding statistics

In the TQFT language, quasiparticles correspond to Wilson loop operators. It is well-known that quasiparticles can have non-trivial braiding statistics. Take $2+1$ dimensional Abelian Chern-Simons ...
3
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2answers
181 views

Momentum a cotangent vector

Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities. Furthermore, many sources ...
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88 views

What is the symmetry group fixing norms of elements of a unitary matrix?

Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$. When exactly are two unitary matrices related in this ...
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1answer
81 views

Petrov classification/Weyl scalars

There is one calculation in Chandrasekhar's "Mathematical Theory of Black Holes" that I cannot understand. Here is the setup: We want to show that Petrov type D (i.e. two principal null directions) ...
3
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1answer
114 views

Reference of $\hbar$-differential operator from symplectic geometry perspective

I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian ...
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53 views

Orthogonality relation for associated Legendre functions

The associated Legendre Polynomials $P_l^m(x)$ obey orthogonality relations for fixed $m$ and fixed $l$: $$ \begin{align} \int_0^\pi P_k^m(\cos\theta)P_l^m(\cos\theta)\sin\theta d\theta ...