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

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17
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2answers
774 views

How does hyperbolicity of space time affect our lives?

My main research has been in hyperbolic geometry and geometric group theory. I always thought that the only real "application" of my work was that the universe is a 3-manifold. But recently I found ...
1
vote
2answers
302 views

Translation of an article

I need to read this article "On the spectrum of an energy operator for atoms with fixed nuclei in subspaces corresponding to irriducible representations of permutation groups" authors:G.Zhislin, A. ...
2
votes
0answers
113 views

universal connection for SU(3)

In 1961, Narasimhan and Ramanan (Am J Math 83 563) showed that one could represent an arbitrary SU(3) connection as $ i t^c_{a b} A^c_\mu(x) = e^*_a \cdot e_{b, \mu}(x) $ in which the $ t^a $'s are ...
2
votes
0answers
101 views

Three body problem with two fermions and a different particle

I'm studying the three body problem with two fermions of unitary mass and another different particle. I need references of the HVZ theorem in this case. Is there someone who knows them?
1
vote
1answer
350 views

maxwell's equations and hodge theory

How is Hodge theory of harmonic forms related to maxwell's equations.Atiyah says that Hodge was directly motivated by considerations of maxwell's equations while commenting on donaldson.
1
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1answer
180 views

Gaussian measures on non-separable spaces

Let $X$ be a topological affine space which is neither separable nor metrizable. There are plenty of trivial Gaussian measures: each Dirac point-mass $\delta_x$ are the Gaussian measure with zero ...
1
vote
0answers
173 views

Is connected correlation/cumulant expansion additive?

Say X is a free field or a Gaussian random variable. Then I want to analyse the connected correlation, $<(X + a (X^2 - \langle X^2 \rangle))^n>_c$ I think that for $n \geq 4$ there are no ...
1
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0answers
128 views

Self Dual 2-Forms on Complexified Minkowski Space

I'm trying to get my head around integrability in twistor theory, but am struggling with interpreting the concept of self-duality on complexified spaces. One can complexify Minkowski space to ...
3
votes
1answer
221 views

Solvable models in quantum mechanics

Is there anyone who studied on the book "Solvable Models In Quantum Mechanics" by Albeverio? I don't succed in understanding the proof of page 116 about the eigenvalues of the Hamiltonian with point ...
1
vote
2answers
213 views

The limiting behavior of geometric random walk

I would like to know what the asymptotic limiting behavior is for the following random walk on $\mathbb Z^d$. By Donsker's invariance principle, I suspect that its behavior is diffusive, i.e., the ...
1
vote
1answer
361 views

Wedge Product of Lie Algebra Valued One-Form

I've been reading about the formal structure of gauge theories and am a little confused by the notation. Could someone clarify this for me? Suppose that $A$ is a Lie algebra valued 1-form ...
1
vote
3answers
341 views

Explicit model of BSU(2) in terms of singular complex

What is the explicit model of BSU(2) in terms of singular complex, up to 5 dimensions, so that one can compute $\pi_5(BSU(2))=\mathbb{Z}_2$ explicitly? This question is related to another question of ...
1
vote
1answer
296 views

$\pi$-cohomology class — a variant of cohomology class

Let $X$ be a topological space with a triangulation. The triangulation defines a chain complex in $X$. Let $\mu_d$ be a cochain and $M^d$ be a chain. We use $< \mu_d, M^d > \in M$ to denote ...
1
vote
3answers
183 views

Applied examples of (non)uniformly hyperbolic and/or ergodic systems

I try to give reference to completely applied examples of (non)uniformly hyperbolic and/or ergodic systems. With completely applied I don't mean an irrational rotation on the torus but from other ...
1
vote
0answers
153 views

multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ , is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $ for $\left | \sigma \right |\leqslant ...
5
votes
1answer
187 views

Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...
3
votes
0answers
95 views

Lower bounds of laplace transform of characteristic functions

Cross-posted on maths.stackexchange I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function ...
8
votes
1answer
547 views

What is quantum Brownian motion?

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in ...
0
votes
0answers
261 views

Moment of Inertia of a Polygon

I'm programming a game and I have to find the moment of inertia of a flat polygon (axis of rotation is perpendicular to the polygon, from now on I will call it the point of rotation). My idea was to ...
0
votes
1answer
274 views

From Brownian Motion to the Heat Equation

Consider a set of N balls that start at the origin. In a given unit of time, $\delta t$, the balls have a probability $p = 0.5$ of jumping a distance $\delta x$ to the right, and the same probability ...
0
votes
0answers
374 views

Solving a linear recurrence relation with variable coefficients.

I have the following recurrence relation: \begin{equation} A[n]=f_A[n-1] A[n-1] + f_B[n-1]B[n-1], \\ B[n]=g_A[n-1] A[n-1] + g_B[n-1]B[n-1], \end{equation} where ...
5
votes
0answers
149 views

Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
4
votes
2answers
779 views

Eigenvalues of random Hamiltonian matrices

A real $2n\times 2n$ Hamiltonian matrix has the general form $$H=\begin{pmatrix} A & B \cr C & -A^T \end{pmatrix} $$ where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are ...
7
votes
1answer
365 views

necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds

Is there any necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds $M$?
8
votes
2answers
394 views

Is the quantum algebra unique (up to isomorphism) in deformation quantization ?

Consider a Poisson algebra A (i.e. commutative algebra with Poisson bracket). Let $\hat A$ be a deformation quantization of the algebra A. We know that construction of deformation quantization and ...
0
votes
1answer
329 views

The real and imaginary parts of the Incomplete Gamma function for second argument being purely imaginary

Dear all, I am looking for explicit (at least more explicit than the original expression) for 1) Re$(\Gamma(a, i\omega))$ as well as 2) Im$(\Gamma(a, i\omega)),$ where i Re and Im denote the real ...
6
votes
7answers
589 views

Quantization of a classical system (e.g. the case of a billard)

There has been already several questions asking for an introduction to quantum mechanics for a mathematician, but this ons is slightly different, and more restrictive. I know (some) quantum mechanics, ...
3
votes
1answer
140 views

Second quantization of partial isometry

If we have a unitary map from Hilbert space $H$ to $H$, we get a unitary map from $e^{H}$ to $e^{H}$, where $e^{H}$ is the symmetric Fock space of $H$. But if we replace the unitary with partial ...
0
votes
0answers
104 views

What is the integer form of a projector into the intersection of the ranges of two integer projection matrices?

Consider two square integer matrices $X$ and $Y$ of the same dimension with the following properties: $X^2=rX$, and $Y^2=sY$ for integers $r$ and $s$. The $\gcd$ of the entries of $X$ is 1 and the ...
2
votes
2answers
414 views

smallest simplest $E_8$ -module

What is the smallest simplest(non-trivial) $E_8$ -module ?
10
votes
1answer
777 views

decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$

Do anybody know , why can we write the following decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$ 1) $E_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus ...
6
votes
1answer
285 views

Relation between TQFT and Wilson lines, boundary conditions, surface defects etc

I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered ...
2
votes
1answer
406 views

why the group $GL(6,V)$ has an open orbit?

N.Hitchen in his paper about geometry of three forms wrote that "for a Real vector space $V$ of dimension six, the group $GL(6,V)$ has an open orbit and he referenced it to a thesis which was written ...
11
votes
2answers
734 views

Hopf Algebra for a physicist

Hello, for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I ...
4
votes
3answers
581 views

Meaning of a phrase from “The algebra of grand unified theories”.

Motivated by an answer to this mathoverflow question I've been making an effort to understand Baez and Huerta's article "The algebra of grand unified theories". As far as I can tell, mathematically, ...
10
votes
0answers
304 views

Which limit to take as a key applied math decision

The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...
11
votes
2answers
506 views

What do correlation functions compute in CFT?

I would like to understand what correlation functions compute in Conformal Field Theory in mathematics. Let me begin with basic definitions. We define a free boson field $\phi(z)$ as a formal power ...
5
votes
2answers
388 views

Permuting Racked Pool Balls with a Single Break

Given reasonable physical assumptions (on friction, collisions, etc.), would it be possible to "break" in a pool game such that when all the balls come to rest, the only difference is that the racked ...
4
votes
1answer
191 views

Rotations, Harmonic Oscillators, Gaussians, Ladders

I am trying to understand better the quantization of the Harmonic Oscillator. Here are three ways of thinking about the Harmonic Oscillator. Eigenfunctions of the differential operator: $H = ...
6
votes
2answers
398 views

When is a space of measures a measurable space?

Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real ...
0
votes
0answers
77 views

Divisors, factorisations of matrix valued functions, and the Lorentz group

How to construct a complex projective variety with several classes of non-intersecting divisors? How to keep the answer concrete and simple, so that explicit calculations can be done? And the problem ...
2
votes
2answers
399 views

An interesting equation of some practical interest.

I have encountered a problem in my elec. eng research that I find rather challenging, being a bear of very little brain. The question has been considered under a slightly different aspect here: ...
8
votes
3answers
465 views

Space of sections of a fibre bundle with non-compact base space

Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$. For compact $M$ it is well known (Hamilton 1982, Part II Corollary ...
31
votes
7answers
3k views

The Unreasonable Effectiveness of Physics in Mathematics. Why ? What/how to catch?

Starting from 80-ies the ideas either coming from physics, or by physicists themselves (e.g. Witten) are shaping many directions in mathematics. It is tempting to paraphrase E. Wigner, saying about ...
4
votes
0answers
414 views

Open question: non-commutative site following Grothendieck, Quillen, Connes and Crane for quantum gravity.

This is an open question and it's to find out who is interested in this kind of thing, who can benefit from thinking about this. It is very brief but hopefully will only be unclear to people who are ...
19
votes
6answers
2k views

Mathematician trying to learn string theory

I'm a mathematician. I want to be able to read recent ArXiv postings on high energy physics theory (String theory) (and perhaps be able to do research). I want to understand compactifications, ...
0
votes
1answer
88 views

orthotropic materials solution of boundary value problems

What are the methods or approaches for the analytical solutions of boundary value problems in the theory of elasticity for orthotropic materials?
10
votes
1answer
330 views

What are the simple Lie superalgebras of type E?

Background Simple finite dimensional Lie superalgebras over $\Bbb C$ have been classified. There are the Cartan type superalgebras (algebras of purely odd vector fields), two strange families P(n) ...
1
vote
1answer
234 views

Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal isotropics of $V\bigotimes \mathbb{C}$ ?

We say $L< (V\oplus V^{*})\bigotimes \mathbb{C}$ is isotropic when $< X,Y>=0$ for all $X,Y\in L$ Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal ...
0
votes
0answers
107 views

Is there “harmonic potential” for classical bosonic string?

I am sorry that I did not sound understandable in my pervious questions. There is a stringy version i.e. categorical version of the classical (non-quantum) physics situation in which a test massive ...