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

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Perturbation theory of eigenvalues - Effects of degeneracy/ multiplicity

In Quantum mechanics Schrödinger's perturbation theory is very important (see Wikipedia) which deals with perturbation of the discrete spectrum of a self-adjoint operator. Where can I find a ...
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198 views

Causal+Imprisonment Condition+Compact Clausure -> Stably Causal?

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 $(M,g)$ ...
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448 views

Geodesics on $SU(4)$

Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find? In the adjoint representation, one can express the Killing form as a matrix and consider it as ...
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282 views

decomposition of Hilbert space into tensor product $L^2([0,\tfrac{1}{2}]) \otimes L^2([\tfrac{1}{2},1]) \simeq L^2([0,1])$

The definition of entanglement entropy in Quantum Field Theory involves decompositing a Hilbert space into a tensor product $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$. As an example, is it ...
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257 views

Reference request for instantons

I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...
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538 views

Multiplicative Identity for all elements in SU(n)

Let $\{P_i\}$ be a subset of $SU(n)$ such that for any $U$ in another subset (or perhaps subgroup) $H$ of $SU(n)$: $$P_1UP_2U\cdots P_mU=I$$ where $I$ is the identity element. Is there a sequence ...
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115 views

graded generalization of the Moyal–Weyl product

Has anyone written about the graded generalization of the Moyal–Weyl product/star product, that is, where the original algebra is already graded? Is it just a matter of signs?
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99 views

First return time in an interval for N particles rotating on the circle at constant random speeds

Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...
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64 views

Implicit/Explicit Time Dependence for Melnikov Functions

My question concerns an article by Koiller and Carvalho found here: http://link.springer.com/article/10.1007/BF01260390 On page 645, they parameterize the time variable $t$ in terms of one of the ...
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84 views

Complex structure on the set of prequantization line bundles

For geometric quantization, the set of equivalence classes of prequantization line bundles of a quantizable symplectic manifold $(M, ω)$ is parametrized by $H^1(M, S^1)$ which represents the ...
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145 views

The Cauchy Problem in General Relativity: Existence of a Hausdorff Development

This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were ...
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56 views

Finite differencing scheme for Hamilton's equation with planar linkages

I am trying to simulate the movement of a planar linkage in the plane whose position and momentum obey Hamilton's equations, which is to say that $${{dq}\over{dt}} = {{dH}\over{dp}}$$ and ...
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125 views

Error in trace of operator

Imagine you have a Schroedinger operator $H:=-\frac{d^2}{dx^2}+V$ with $V \in C[a,b]$ on some compact interval $[a,b]$. The boundary conditions are supposed to be taken in such a way that this ...
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68 views

Do principally polarized abelian varieties enjoy a genus expansion?

This is a vague question from an interested outsider: It is well known that abelian varieties which arise as Jacobian of a curve (or a bit more general as Prym variety) are distinguished by the fact ...
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442 views

Vector bundles, Higgs bundles and the Langlands program

This question is somewhat vaguely structured. But, I hope someone can make it more precise (or) it is indeed possible to answer it in the form that I am stating it.  Background : I recently chanced ...
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618 views

Some explanation about Dynin's formalism

I have seen this claim on the Wikipedia page for the Yang-Mills Millenium problem by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I ...
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220 views

How to evaluate the wiener measure of sets?

I would like to understand how the Wiener measure of some simple sets can be evaluated. I will sketch the construction of Wiener measure I have in mind: We denote the one point compactification of ...
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191 views

Self-adjoint extensions and delta potentials

Is there a self-adjoint extension of an operator that corresponds to a particle in a box $[a,b] \times [c,d] \subset \mathbb{R}^2$ with a delta potential, i.e., $-\Delta + \lambda \delta_y $ on ...
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173 views

Asymptotic behavior of Schrödinger operators

I am currently dealing with $1$ or at most $2$-dimensional Schrödinger operators on compact domains. A classical result of spectral theory is the Weyl approximations for this operator $H = -\Delta ...
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426 views

Free Boson Correlator $ \langle X(z)X(w) \rangle =- \ln |z - w| $

In physics papers, the massless free boson has a definition involving an action: $$ S(X) = \frac{1}{8\pi} \int d\sigma^2\, \partial X \overline{\partial X}$$ The random functions $X(z)$ are ...
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169 views

What to read for many-body problems in 3D Schrodinger equation

I am a graduate student just started learning dispersive PDE in MSRI's summer program. I roughly finished reading the paper by Klainerman and Machedon "ON THE UNIQUENESS OF SOLUTIONS TO THE ...
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223 views

Comparison of Different Types of QFT

As far as I can tell, there are a number of major types of quantum field theory. For example, Constructive QFT, which has two major branches (Algebraic/Axiomatic QFT and Functorial QFT). Topological ...
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156 views

Are there any explicit probability conserving solvers for Pauli equation?

I know that there exist probability conserving explicit solvers for time-dependent Schrödinger's equation, for example, Visscher's one. But when I tried to add spin into account in this scheme, it ...
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69 views

Smallest subalgebra of $\mathfrak{su}(4)$ arrising from a control problem on $SU(4)$

What is the smallest subalgebra of $\mathfrak{su}(4)$ containing the span of the set $A = \{A, B_1, B_2\}$ where: $A = i (J^x \sigma_x \otimes \sigma_x + J^y \sigma_y \otimes \sigma_y + J^z \sigma_z ...
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Review of Tim Maudlin's New Foundations for Physical Geometry [closed]

Tim Maudlin, a philosopher of science at NYU, has a book out called: New Foundations for Physical Geometry: The Theory of Linear Structures. The section on about the book says the following: ...
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132 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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113 views

Heat transfer: boundary conditions with fluid velocity

The following equation is considered: $$ \frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f. $$ I have difficulties in formulating boundary conditions for this equation. If ...
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121 views

Equivalence of the construction of the Lagrangian in a book of Sternberg to the “usual” construction

I have a question regarding the following cited text from [1]: Let $F$ be a representation of the structure group $G$ of the principal bundle $P_G\to M$ (a (semi-)Riemannian manifold), and let ...
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115 views

Extension of a bounded operator on manifold

I have a problem, which is quite urgent, as I have only today discovered an error in a proof i had in a thesis which is to be handed in tomorrow. The problem, if stated in as full generality as ...
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534 views

References for a physicist migrating to stochastic processes

I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...
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167 views

System of quadratic complex equations

I want to solve this system of N non-linear equations without using a numerical method: $x_{k}^{2}= \alpha_{k }+ \sum\limits_{m=1}^{N} (\beta_{km} x_{m} + \psi_{km} x_{m}^{*})$ With $\left| ...
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138 views

The mathematics in understand anyons [closed]

I've been about particles called anyons which exist within a two dimensional framework. I've also found out that these particles can have an angular momentum equal to any real number. Normally, in ...
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174 views

quantum states and observables for the non-commutative torus

The noncommutative torus $A_\theta$ is a $C^*$-algebra corresponding to an irrational foliation of the torus $\mathbb{T}^2$ by lines of slope $\theta \notin \mathbb{Q}$. As far as I am reading it is ...
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220 views

How to calculate the first and second homotopy groups of the following space constructed from $U(4)$

In solving a physics problem, I came across a weird topological space constructed from $U(4)$, the group of $4\times4$ unitary matrices. I want to know the first two homotopy groups of it. Here is how ...
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176 views

Orbital integral by using symplectic quotient

Let $(M,\omega)$ be a compact symplectic Hamiltonian $G$-manifold and $\Gamma_{\text hol}(M,L)$ be the space of holomorphic sections of the line bundle $L\to M$ I am looking for a proof for ...
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364 views

4d Constructive Quantum Field Theory

As a follow up to my previous question (How does Constructive Quantum Field Theory work?), I was wondering what difficulties physicists have had constructing 4d axiomatic qfts. Why has CQFT's success ...
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291 views

How does Constructive Quantum Field Theory work?

Please correct me if I'm wrong, but it seems to me that two and three dimensional axiomatic quantum field theory were constructed as follow: the wightman axioms were formulated in euclidean space via ...
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310 views

A question about Quantized closed Kaehler manifolds

Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding $$\phi: (M,\omega)\to  (\mathbb CP^N, ...
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314 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 ...
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303 views

Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation

Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$? Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...
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251 views

The dogma of the natural numbers in physics

As is well known "God made the natural numbers; all else is the work of man" (Leopold Kronecker). However, "what would correspond more to the spirit of physics would be a mathematical theory of the ...
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128 views

Is the structure constant additive on connected components?

Let $M$ be a Riemann surface and $\mu$ a metric on it, which could be non-compact. Moreover let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and $\mathrm{det}^*(\Delta_{\mu,\,M})$ its ...
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623 views

Mathematical equivalent to ladder operators?

A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...
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192 views

What are some good sanity checks for simulating BNLS?

After doing some googling, I failed to find any explicit solution for the Biharmonic Nonlinear Schrodinger Equation, which states: $$ i\psi (x,t) _t - \Delta ^2 \psi (x,t) + |\psi (x,t) | ^{2 \sigma} ...
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197 views

A question on existence of $Spin^c$-structure $P\to M$

Let $(M,\omega)$ be a compact symplectic manifold and the cohomology class $$[\omega]+\frac{1} {2}c_1(\wedge_{\mathbb C}^{0,n}(TM, J))\in H_{dR}^2(M)$$ is integral, for some almost complex structure ...
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138 views

All symplectic manifolds have $Mp^c$-structures?

Let $(M,\omega)$ be a symplectic manifold, and $Mp^c(n)=Mp(n)\times_{\mathbb Z_2}U(1)$ which $Mp(n)$ here is Metaplectic group which is the double cover of symplectic group. I am looking for a ...
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Characterize spin cobordism invariants in dimer models

The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...
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235 views

State of the art of BPS and Donaldson-Thomas invariants for toric Calabi-Yau threefolds

I am trying to understand what has been done with regards to computing BPS invariants and Donaldson-Thomas type invariants of Calabi-Yau threefolds. To make the question more focused, let's say that I ...
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83 views

Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diffusing Particles. The picture below shows the idea how permutations and inversion numbers reflect ...
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371 views

$\{\phi:\int \phi d\mu=0\}$ for a fixed shift invariant $\mu$

Given a shift invariant probability measure $\mu$ on a mixing subshift of finite type. What are the Lipschitz functions with zero integral with respect to the measure $\mu?$ Clearly any ...