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

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148 views

Does the notion of a “coherent state” exist in TQFTs? (ETQFTs?)

In the quantum harmonic oscillator, there exists a family of states called coherent states which form an overcomplete set of states. They are regarded as "the states most resembling classical states", ...
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0answers
17 views

Helmholtz equation Poynting vector integral

The Maxwell's equation for harmonic time dependent field in vacuum is \begin{align} \nabla \times B + i\omega E &= 0\\ \nabla \times E - i\omega B &= 0 \\ \nabla \cdot B &= 0 \\ \nabla ...
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0answers
76 views

target category of extended field theory

An A-S TFT is a functor from $\text{Bord}_{<n−1,n>}(\mathcal{F})$ to $\text{Vect}$ where $\mathcal{F}$ denotes a set of background fields, eg a spin structure. An extended theory is a functor ...
2
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1answer
338 views

a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?
6
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1answer
297 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 ...
1
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1answer
67 views

Minimum of Random Energy Model (REM) with logarithmically correlated potential

In the paper [FB] (ArXiv, J. Phys. A), the authors analyse a particular Random Energy Model (REM) with logarithmically correlated potential and conjecture in Eq. (2) that the distribution function of ...
4
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1answer
173 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 $(M,g)$ ...
4
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1answer
486 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: ...
7
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1answer
413 views

Learning roadmap to TQFT from a mathematics perspective

I had asked a question on math.stackexchange but did not receive any answers. I hope that this question is appropriate for this website as it is about an advanced subject. Hence I am posting it below. ...
6
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3answers
560 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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7answers
5k views

Why does bosonic string theory require 26 spacetime dimensions?

I do not think it is possible really believe or experimentally check (now), but all modern physical doctrines suggest that out world is NOT 4-dimensional, but higher. The least sophisticated ...
7
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2answers
379 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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1answer
117 views

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 ...
6
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1answer
601 views

Cartan 3-form on a Lie group G

Does anyone have a reference to learn more about the Cartan $3$-form on a group manifold $G$? I have read that the WZW term is nothing more than the integral of the pullback of the Cartan $3$-form via ...
32
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6answers
3k views

Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one? I've found some examples: 1) In MO-Q111339 ...
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1answer
52 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 ...
8
votes
1answer
618 views

Matrix integral identity

1) How to prove that $N\times N$ matrix integral over complex matrices $Z$ $$ \int d Z d Z^\dagger e^{-Tr Z Z^\dagger} \frac{x_1\det e^Z -x_2 \det ...
10
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2answers
422 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 ...
5
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1answer
397 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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3answers
139 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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2answers
236 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 ...
11
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2answers
524 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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3answers
186 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 ...
2
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1answer
205 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. ...
5
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3answers
95 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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0answers
91 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 ...
2
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2answers
285 views

Large vs Small Gauge transformations and Physical theories

I can't decipher the difference between large and small gauge transformations especially in its applications in physics.If perhaps one can engineer a simple physical theory that has such a ...
5
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2answers
949 views

What is the trace in the Chern-Simons action?

Warning: This is a very stupid question regarding a basic misunderstanding that I have. I realize that the question is very elementary, but I guess asking stupid questions is better than remaining ...
2
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0answers
45 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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0answers
74 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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0answers
99 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 ...
13
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2answers
643 views

Geometric Quantization

I'm curious about geometric quantization. Of course, I know the procedure: Start with a classical phase space $T^{*}X$, $X$ is the configuration space, then do prequantization by creating a ...
2
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0answers
116 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 ...
8
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3answers
2k views

What's the current state of Yang Mills Mass Gap question?

What's the current state of Yang Mills Mass Gap question, is there any place that does this problem? Especially I want to know if there is any progress (out of that mentioned in the introduction ...
6
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5answers
676 views

Symplectic structures from Lagrangians?

In Witten's paper 'Quantization of Chern-Simons Gauge Theory with Complex Gauge Group,' he makes the statement (p. 35) that the symplectic form on the moduli space of flat $G_\mathbb{C}$ connections ...
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7answers
1k views

A good primer for geometric quantization.

Hello everyone: I'm searching for a good primer on geometric quantization. I found the following: Mathematical foundations of geometric quantization (A. Echeverria-Enriquez, et al.) Symplectic ...
25
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4answers
4k views

What is Chern-Simons theory?

What is Chern-Simons theory? I have read the wikipedia entry, but it's pretty physics-y and I wasn't really able to get any sense for what Chern-Simons theory really is in terms of mathematics. ...
8
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2answers
570 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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0answers
53 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 ...
9
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1answer
389 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 ...
3
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0answers
301 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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4answers
3k views

Role for generalized geometries in string theory

What role do generalized geometries (in terms of Dirac structures, for instance, symplectic, Poisson, complex, and generalized complex structures in the sense of Hitchin, Cavalcanti, and Gualtieri) ...
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0answers
203 views

How can we formulate maximal time $T$ in Hyperbolic Kahler Ricci flow

In general, the exact maximal time $T$ of a Riemannian Ricci flow may not be easy to find. However, fortunately, for Kähler-Ricci flows, the maximal time of existence $T$ is explicitly determined by ...
4
votes
3answers
155 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 ...
5
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1answer
175 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 ...
4
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1answer
157 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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1answer
195 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 ...
5
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2answers
740 views

Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$

I would like to know what are the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$, and/or how to calculate them. It can be shown that $H^d[U(1), Z]$ is $Z$ for ...
4
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1answer
72 views

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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1answer
66 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 ...