Mathematical methods in quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.
5
votes
1answer
110 views
Foliations of Lorentzian manifolds by Spacelike Hypersurfaces
Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike ...
18
votes
2answers
589 views
Reflection of light from function graph
Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$ and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ ...
2
votes
0answers
122 views
How to write BRST-BV for dg-Lie?
The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc.
Where is there written a corresponding formula incorporating the differential of
a dg Lie algebra and module?
4
votes
0answers
87 views
PDE with distributional coefficient [migrated]
I'm a undergraduate major in physics and when learning mathematical aspects of quantum physics I run into for example this problem: for a 1-dim system with delta potential, $-\frac{d^2}{dx^2}\phi + ...
2
votes
2answers
341 views
space at the Planck scale [closed]
All models of space that I know from physics use real or complex manifolds. I was just wondering if it is still the case at the level of Planck scale. In string theory, physicists still use strings ...
1
vote
0answers
101 views
Spectral sequences and Batalin-Vilkovisky formalism
I have been studying the BRST quantization in quantum field theory recently and noticed that the subject is very much related to algebraic topology and cohomology. A quick google search led me to the ...
1
vote
0answers
122 views
Dirac operator in Generalized Geometry
I am wondering how the Dirac operator can be built in the context of Hichin's generalized geometry.
In particular, I have the following questions:
On a spin manifold, is the conventional spin ...
1
vote
0answers
66 views
Dixon's diagram for BRS cohomology
The article by J. A. Dixon titled Calculation of BRS cohomology with spectral sequences (Comm. Math. Phys. Volume 139, Number 3 (1991), pages 495-526) describes in words a diagram that is not printed. ...
1
vote
0answers
35 views
modern exposition of exact ground state of classical XY model or Ising model
What is the state of art technique in solving exact ground state of Heisenberg model, meaning minimization of the H terms (hamiltonian) provided infinite spin space?
...
1
vote
1answer
211 views
How to construct (another) Landau-Ginzburg model for a compete intersection Calabi-Yau?
For Calabi-Yau variety $X$ which is a complete intersection
$$
f_1=f_2=\ldots=f_r=0
$$
in ${\mathbb P }^n$ (hence $\mathrm{dim}\,X=n-r$) it is possible to construct a Landau-Ginsburg model in the ...
5
votes
2answers
852 views
Physicist trying to understand modern mathematics
I'm a physicist trying to gain a deep understanding of mathematics that is required for my work.I intend to specialize in string theory which is a very math intensive branch of theoretical physics ...
0
votes
0answers
99 views
Perturbation of Laplacian via Kato-Rellich theorem
Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian
$$-\Delta+V(x)$$
is self-adjoint on $H^2(\mathbb{R}^3)$.
My idea is to use Kato-Rellich theorem; ...
3
votes
1answer
225 views
book about string theory a la Von Neumann [duplicate]
Can we summarize string theory (in its actual state) in some principles and fundamental equations like electromagnetism, general relativity, quantum mechanics and classical mechanics ?
I am looking ...
4
votes
1answer
141 views
reference for higher spin - not gravitational nor stringy
Other than the papers of Berends, Burgers and van Dam, are there any papers that study the general case of deforming a free field theory with higher spin fields to be interactive?
0
votes
0answers
81 views
What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?
I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction of instantons. At the heart of this is the complex
$$
V ...
7
votes
2answers
189 views
How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?
Mathematical physicists in solid state physics and topological insulators talk a lot about Walker-Wang models, which are a family of Hamiltonians defined on a 3d lattice. Unfortunately, the original ...
2
votes
1answer
100 views
Proper domain for operators
in this paper on arxiv in equation 27, two operators
$$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$
and $$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + ...
1
vote
1answer
98 views
Commutators of Schur polynomials of Lie algebra elements
Question:
Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...
1
vote
2answers
252 views
Witten index non-trivial in the context of Quantum Mechanics?
Let $H$ be a self-adjoint Hamiltonian and $H$ admits a decomposition into closed operators $D,D^*$, such that we have $H = D^*D$.
I will now consider the one-dimensional case on a compact set:
So ...
14
votes
2answers
609 views
Exact Definition of Dirac Operator
Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator ...
12
votes
3answers
1k views
On $e^{\pi\sqrt{4\cdot163}}$ and unusual connections
We are familiar with the expansion of the j-function,
$$j(\tau) = \tfrac{1}{q}+744+ 196884{q} + 21493760{q}^2 + \dots\tag1$$
and maybe with the approximation,
$$e^{\pi\sqrt{652}} = ...
0
votes
1answer
178 views
Some quantities which definitions are (somehow) similar to the classical Divergence
Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...
31
votes
7answers
3k views
Companion to theoretical physics for working mathematicians
In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to ...
1
vote
0answers
83 views
How can I express the following integral as special function?
How can I express the following integral as special function? where the integral is from 0 to infinity
$ f_{int}(x)=\int^{\infty}_{0}dy\frac{e^{(-x+y)}}{(1+e^{(-x+y)^{2}})\sqrt{y}} $
5
votes
1answer
169 views
Survey paper on isoperimetry
I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some ...
1
vote
0answers
76 views
Floquet-Bloch solutions of the quasiperiodic Schrödinger equation
I am concerned about the Schrödinger equation
$-x''(t)+q(t)x(t)=Ex(t).$
Here, the potential $q$ is real and quasiperiodic with frequency vector $\omega$. That is, we let $T^d$ be the $d$-dimensional ...
5
votes
1answer
230 views
Domains of raising and lowering operators in QM
Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and ...
2
votes
1answer
120 views
Well-posedness of heat equation with distributional right hand side
The question is about well-posedness of heat equation
$$
\frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T],
$$
subjected to ...
4
votes
1answer
216 views
Uhlenbeck's theorem novelty
This link provides a short introduction to the contributions of Uhlenbeck about regular gauge fixing. However, I feel quite puzzled about it and I do not understand the real novelty apported by this ...
6
votes
2answers
140 views
Is there a maximum principle for stress in continuum mechanics?
I'm working with the equilibrium equations in linear elasticity, which I have not worked with in the past. My engineering colleagues seem to "know" that the maximum Von Mises stress occurs on the ...
2
votes
0answers
83 views
Do position and momentum measurements determine a wave function?
Suppose we have a function $f\in L^2(\mathbb R^n)$ and we know the functions $x\mapsto|f(x)|$ and $p\mapsto|\hat f(p)|$, where $\hat f$ is the Fourier transform of $f$.
Can we reconstruct the function ...
2
votes
2answers
282 views
Geometrical interpretation of a Schrödinger operator
Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function
$$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$
such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some ...
2
votes
1answer
107 views
Quaternion Wishart matrices of half-integer dimension?
For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution
...
1
vote
0answers
140 views
Electrodynamics modelled by U(1) gauge theory [closed]
As the article 'Electrodynamics in general spacetime' greatly explains, the U(1)-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to ...
2
votes
0answers
82 views
“Semiclassical approximation” in random matrix theory
I am reading Planar Diagrams by Brezin, Parisi, Itzykson and Zuber. If you specialize the discussion in section 5 there we seem to derive the eigenvalue distribution of $N \times N$ random Hermitian ...
1
vote
0answers
92 views
translation invariance of the Laughlin wave function
This is a translation into math of the following question, posted on PhysicsOverflow.
Let $H:=L^2(\mathbb C)$.
For every $N$, let $\psi_N\in\Lambda^N H\cong (L^2(\mathbb C^N))^{S_N}$ be the function
...
0
votes
0answers
32 views
Probability of close approach for multivariate normal variables
The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...
6
votes
0answers
202 views
Can one classify irreducible unitary representations of the Weyl algebra?
I saw in this MO post: Is there a machinery describing all the irreducible representations ? that classifying irreducible representations of the Weyl algebra is essentially intractable. My question is ...
8
votes
2answers
311 views
What is the BRST-anti-BRST formalism?
What is the BRST-anti-BRST formalism?
Is the Sp(2) doublet the ghost, antighost pair?
Introductory accounts of this subject seem to be hard to find. I would appreciate a reference for someone who ...
5
votes
1answer
400 views
Harmonic Crystal using Random Walk
Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here.
Basically, it is a ...
16
votes
3answers
921 views
How mirror of quintic was originally found?
In the 90-91 pager
"A PAIR OF CALABI-YAU MANIFOLDS AS AN EXACTLY SOLUBLE SUPERCONFORMAL THEORY",
Candelas, De La Ossal, Green, and Parkes, brought up a family of Calabi-Yau 3-folds, canonically ...
1
vote
0answers
54 views
Bogomol’nyi’s Formula for the Critical Action
I'm studying Aigner's paper 'Existence of the Ginzburg-Landau Vortex Number' (2001) and I have some difficulties to prove the equality (3.1) , which is
...
3
votes
0answers
83 views
A technical question in Feix's construction of hyperkahler metric on cotangent bundles
I am now reading Feix's paper Hyperkahler metrics on cotangent bundles
and I have a technical question to ask.
In his paper, for an analytic Kähler manifold $(X,J,\omega)$, Feix considered its ...
2
votes
0answers
82 views
Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom
Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation
$$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$
can ...
4
votes
0answers
88 views
Ground State Degeneracy of 2+1D U(1) Chern Simons Theory?
I am a physics graduate student trying to understand more mathematical aspects of gauge theories.
How can I understand ground state degeneracy of a simple Chern Simons Theory: 2+1D U(1) $S= \int_M ...
1
vote
2answers
131 views
Deformation quantization of a closed Riemann surface with genus >1
Quantization of of an elliptic curve can be done in different ways.
In C^*-algebraic version,
one can start with the C^*-algebra ...
1
vote
0answers
83 views
The normalization axiom of a quantization
Guillemin, Ginzburg and Karshon explain a quantization in their book [Chap 6,MR1929136] as follows.
The quantization is a process which associates to a symplectic manifold $M$ a Hilbert space ...
10
votes
0answers
237 views
“extended TQFT” versus “TQFT with defects”
There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a ...
7
votes
0answers
142 views
What's the appropriate notion of a Unitary representation of a Lie algebra?
Here Lie algebras/groups are real. The most straightforward definition might be:
Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...
0
votes
0answers
42 views
Self-adjointness of the components of the magnetic derivative
On $L^{2}(\mathbb{R}^{n})$ define the operator $\Pi_{j} u := (-i\partial/\partial x_{j} - A_{j})u$, where $A_{j} \in L^{2}_{loc}(\mathbb{R}^{n})$ represents the $j$-th component of the magnetic ...
