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

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103 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 ...
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0answers
42 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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0answers
257 views

Vafa's semi-Ricci flat metric

Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and noncompact Calabi-Yau manifolds. Nuclear Physics ...
2
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0answers
91 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 ...
1
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1answer
61 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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45 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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3answers
673 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 ...
4
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1answer
80 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 ...
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54 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 ...
2
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1answer
249 views

Is the structure constant additive on connected components?

This is the reanimation of a question which already got an answer, that I did not fully understand. Coming back to it, after let it sit in a corner for some time, I keep not getting the point. I would ...
5
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1answer
561 views

Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello! Let's say I have $(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$ with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$). Now separate the Amplitude and Phase of the solution: ...
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180 views

The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$: ...
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0answers
53 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 ...
7
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3answers
262 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 ...
24
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1answer
1k views

Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...
4
votes
1answer
249 views

Subset of causal spacetime+Imprisonment Condition+Compact Closure -> Stably Causal spacetime?

My question arose after studying the article "John K. Beem: Conformal Changes and Geodesic Completeness". (http://projecteuclid.org/euclid.cmp/1103899983) One of the results there is: Let $(M,g)$ ...
4
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3answers
328 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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5answers
1k views

Spectral theorem for self-adjoint differential operator on Hilbert space

I need a reference concerning a theorem that shows the following result, stated very roughly: Given a self-adjoint differential operator densely defined on a Hilbert space, then the given Hilbert ...
3
votes
3answers
323 views

Positivity of the Coulomb energy in two dimensions

In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{\|\cdot\|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...
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0answers
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
76 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
170 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
143 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 ...
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2answers
262 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 ...
6
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2answers
348 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 ...
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0answers
87 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
272 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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0answers
113 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
207 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 ...
12
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2answers
284 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 ...
3
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0answers
119 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 ...
1
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1answer
109 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, ...
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0answers
74 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?
10
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1answer
826 views

Vanishing of $\hat{A}$ genus and positive scalar curvature

Classicly, for a spin Riemannian manifold $M$, the $\hat{A}(M)$ genus will be $0$, if the scalar curvature is positive. The proof is to use the Lichnerowicz formula. we have the index of the Dirac ...
3
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1answer
163 views

higher order Noether identities

Noether's second variational theorem gives a correspondence between symmetries of a Lagrangian and Noether identities, which are relations among the Euler–Lagrange equations. How about relations ...
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0answers
143 views

Deformation of Noether's first theorem

Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...
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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 \| ...
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1answer
83 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 ...
27
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2answers
658 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} ...
3
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0answers
217 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 ...
21
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1answer
2k views

What is about nonassociative geometry?

At the end of a conference given by Alain Connes in 2000 (here is a video in French), a member of the audience asked a question. I transcribed and translated it for you below: Audience: You showed ...
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0answers
165 views

unfolding as resolution

Has anyone described 'unfolding' as used in mathematical physics (e.g. on-shell AND off-shell) as analogous to a resolution in algebra - higher derivatives are unfolded in terms of new variables?
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3answers
3k views

Quantum mechanics formalism and C*-algebras

Many authors (e.g Landsman, Gleason) have stated that in quantum mechanics, the observables of a system can be taken to be the self-adjoint elements of an appropriate C*-algebra. However, many ...
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0answers
128 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
168 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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0answers
83 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
60 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
109 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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43 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 ...
0
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1answer
129 views

Positivity of logarithmic energy of certain measures

Let $\Gamma$ be a smooth closed curve in the complex plane (for all practical purposes). Assume $f$ is a real-valued continuous function defined on $\Gamma$ and let $d\mu=fdm$, where $dm$ is the ...