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

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1answer
245 views

Applications of set theory in physics

In the introduction of the paper "Links between physics and set theory", the following quote of Eris Chric is stated: ...
2
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1answer
100 views

Two equivalent descriptions of a physical system yielding a non-trivial mathematical formula

First I would like to admit that this question may not be entirely appropriate for this site, but I will give it a go none the less. One often hears stories about how string dualities lead to highly ...
0
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1answer
62 views

Mathematical statistical qm book-recommendation

I feel that there are quite a few good and rigorous books on the mathematical foundations of quantum mechanics, but I am currently looking for a book that covers mathematical statistical quantum ...
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2answers
233 views
+50

Certain inverse problem related to moments

Suppose $D\subset \mathbb C$ is a smoothly bounded domain and it contains the origin. Let $ds$ denote the arc length measure on $\partial D.$ I am interested in the following two inverse problems ...
9
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4answers
288 views

Obtain any 3-manifold from repeating surgeries on knots in $S^3$

In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) ...
4
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1answer
85 views

Isomorphism of various gauge groups under homotopy

Given a compact, simple Lie group $G$, and a compact, oriented three manifold $M$, we can consider the following smooth homotopy groups: $(C^\infty(M;G)/{\sim},\cdot )$: Here, $\sim$ is smooth ...
3
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1answer
66 views

Blow-Up for Semi-Linear Wave Equations

I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...
79
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25answers
15k views

A soft introduction to physics for mathematicians who don't know the first thing about physics

There have been similar questions on mathoverflow, but the answers always gave some advanced introduction to the mathematics of quantum field theory, or string theory and so forth. While those may be ...
10
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1answer
502 views

Associativity of Kontsevich's star product up to second order

In Deformation quantization of Poisson manifolds, Kontsevich gives the quantization formula $$f \star g = \sum_{n=0}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma,\alpha}(f,g).$$ He gives ...
4
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1answer
534 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: ...
4
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1answer
538 views

basic questions on quantum integrable systems

I have been learning about (classical) integrable systems lately, e.g. in the examples of a lax pair etc. I frequently run into the term 'quantum integrable system'. May I ask a few questions: What ...
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0answers
76 views

power laws emerging from the sandpile model

Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?
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0answers
203 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 ...
3
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0answers
218 views

Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...
1
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1answer
146 views

Yang-Mills Functional and Energy

I have a question about the meaning of Yang-Mills Functional. It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is: ...
13
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2answers
964 views

Symplectic formulation of statistical physics

Does there exists a symplectic formulation of statistical physics? I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated ...
5
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1answer
249 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 ...
62
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9answers
6k views

Theoretical physics: Why not just R^4?

You and I are having a conversation: "Okay," I say,"I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles." "Something like that" "...And ...
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3answers
233 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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0answers
37 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? ...
0
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1answer
186 views

How would I apply Wick's theorem to the time-ordered product of three fields?

I think I know how to apply Wick's theorem in order to expand the time-ordered product of quantum fields, but I just want to verify my understanding. Could someone perform it for the arbitrary ...
0
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1answer
180 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 ...
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2answers
614 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 $ ...
5
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1answer
129 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 ...
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3answers
568 views

Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In the paper "Exotic ...
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1answer
146 views

2D semilinear elliptic PDE

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. ...
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3answers
4k views

What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...
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0answers
129 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?
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2answers
350 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 ...
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0answers
108 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 ...
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0answers
128 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 ...
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0answers
70 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. ...
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1answer
143 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?
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2answers
882 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 ...
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1answer
220 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 ...
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0answers
106 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
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1answer
234 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 ...
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5answers
3k views

Good references for Rigged Hilbert spaces?

Every now and then I attempt to understand better quantum mechanics and quantum field theory, but for a variety of possible reasons, I find it very difficult to read any kind of physicist account, ...
3
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2answers
499 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: ...
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0answers
84 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
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2answers
208 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
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1answer
103 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}} + ...
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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 ...
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1answer
102 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, ...
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2answers
256 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 ...
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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}} = ...
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0answers
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Dropping three bodies

Consider the usual three-body problem with Newtonian $1/r^2$ force between masses. Let the three masses start off at rest, and not collinear. Then they will become collinear a finite time ...
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2answers
637 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 ...
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29answers
9k views

Where does a math person go to learn statistical mechanics?

The more math I read, the more I see concepts from statistical mechanics popping up -- all over the place in combinatorics and dynamical systems, but also in geometric situations. So naturally I've ...
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2answers
146 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 ...