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

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Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
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1answer
191 views

How to classify continuous/differentiable maps from $T^2$ to $U(N)$?

I read a physics paper, of which the main idea is based on the topological classification of maps from 3-torus to the space of $N\times N$ unitary matrices. To quote their equation (4), which gives a ...
3
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1answer
214 views

Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?

Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding ...
4
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2answers
148 views

Lagrangians with the same extremal curves

It is well know that (in the sense having the same image not parametrization) the extremal curves of the energy functional on a Riemanian manifold $(M,g)$: $E[\gamma(t)] = \int ...
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1answer
268 views

A question about complex polarization

Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar P ...
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84 views

How can the interersection number of $2$ $D6$ branes wrapping around a CY manifold be derived?

For two intersecting $D6$ branes $a$ and $b$ wrapped around a $6$ dimensional torus $T^6 = T^2 \times T^2 \times T^2$ specified by $$ \textrm{D6-brane a:}\, (l_1^a,l_2^a,l_3^a) $$ $$ ...
4
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1answer
162 views

Lieb: Stability of matter, problem with variational method

I am trying to recalculate 'Stability of Matter' Paper from Lieb. I have a problem at the last step of the proof at page 3, where Lieb calculates the minimizer. I will explain my ideas and my problem ...
4
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1answer
109 views

Closed form for 3j-symbol ratios

I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch–Gordan ...
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1answer
87 views

What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE?

What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? Is it possible to write set-theoretical solutions of Quantum ...
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0answers
198 views

Sum over a product of binomial coefficients related to a collision problem

I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation: ...
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0answers
66 views

range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants

The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
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1answer
205 views

Stokes-Einstein rotational diffusion and vector orientation time

The Stokes-Einstein rotational diffusion relation tells us that we can write down a rotational diffusion coefficient for a sphere as: $D_r \approx \frac{k_B T}{\zeta_f} \approx \frac{k_B T}{(8 \pi ...
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1answer
133 views

Does this Mean Value have a Name?

Question: does the following mean value have a name? $$v^*=\sqrt{\frac{\sum_{i=1}^{n}\alpha_iv_i^2}{\sum_{i=1}^{n}\alpha_i}}, \alpha_i\in\mathbb{R}^+$$ where the $v_i$ are individual speed limits ...
3
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1answer
234 views

Electromagnetic duality symmetry

This question arose while reading http://prd.aps.org/abstract/PRD/v13/i6/p1592_1 (Duality transformations of Abelian and non-Abelian gauge fields, by Stanley Deser and Claudio Teitelboim). It is well ...
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5answers
480 views

How to characterize Dirac's gamma matrices in differential geometry?

I want to understand what is the interpretation of Dirac gamma matrices in differential geometry. Basically, I am considering the Dirac matrices as 3-indexed tensors, which means a tensor with 1 ...
2
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1answer
175 views

How to obtain the Period matrix from the Igusa Invariants of a genus two curve?

I am looking for an algorithm to obtain the period matrix tau= (tau1 & tau12\ tau12 & tau2) from the Igusa invariants of a genus two curve. More precisely: I consider a family of genus two ...
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0answers
188 views

quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement. Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...
3
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0answers
113 views

Physical relevance of either fundamental identity generalizing Jacobi [closed]

There are two fundamental identities for n-ary generalizations of the Jacobi identity. One fundamental identity is right for Nambu mechanics and such, the other for L_\infty algebras as in CSFT. Which ...
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2answers
322 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 ...
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0answers
38 views

Gordon Lemma for Jacobi operators

The Gordon lemma is a useful tool in the theory of 1-D discrete time-independent Schrödinger operators that exploits local repetition in the potential to prove absence of point spectrum. Has anyone ...
10
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3answers
418 views

Resource for learning quantum mechanics from the viewpoint of representation theory

Quantum mechanics is deeply connected with representation theory. Therefore, I'm looking for a textbook or article which presents quantum mechanics in a representation theoretic manner. Could anyone ...
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0answers
203 views

Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help. The integral itself is, with $\tau$, $\lambda$, and ...
12
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4answers
812 views

What properties of knots lead Lord Kelvin to hypothesize that atoms were knots in the ether?

I've often heard that Lord Kelvin was one of the first people to study knot theory, as he hypothesized that atoms were knots in the ether. I assume that he had some compelling evidence for this fact. ...
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3answers
424 views

References request: constructive quantum field theory

I am taking a course this semester on QFT, which deals much with constructive quantum field theory. Some of its topics so far involve relationships between non-Gaussian probability measures,Feynman ...
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0answers
283 views

Curvature as infinitesimal holonomy

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy ...
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2answers
404 views

Kummer's quartic surface and the Dirac operator

The picture shows 1936 photograph by Man Ray of a Kummer surface plaster model in the collection of the Institute Henri Poincare (from Mathematics, Art and Science of the Pseudosphere by Kenneth ...
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0answers
111 views

When the leaves of a distribution are compact

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
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0answers
247 views

What is the characteristic functional for Brownian motion on a sphere?

I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to ...
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1answer
898 views

Mathematical study of Mpemba effect?

It has been known since the days of Aristotle and Descartes that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...
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0answers
142 views

Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...
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0answers
139 views

RG flow and Ricci flow

It looks like the Laplace operator in the nonlinear sigma model (say the Polyakov action) is different from the Laplace-Beltrami operator, how can one get the Ricci flow as a low order approximation ...
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1answer
159 views

An example for the case tht if the leaves of polarization be non-compact then the polarized sections are not square integrable

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
2
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1answer
249 views

Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$

Let $\lambda_1, \dots, \lambda_n$ be the eigenvalues of a random Unitary matrix. I am interested in the expected value: $$\mathbb{E}_{X \in U(N)}\left[ \prod_{i=1}^n \frac{1}{(1 - ...
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1answer
275 views

The space of holomorphic sections are finite dimensional?

I start my question with a definition and some motivation. Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex ...
2
votes
1answer
83 views

An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive

I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar ...
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1answer
142 views

$S^n$ admit a real polarization $D\subset TS^n$?

When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$
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0answers
200 views

Elementary proof of lack of phase transition in Ising models with external fields

I have a question about the phase transitions in the Ising model in the presence of a (constant) external magnetic field. I will state the question on $\mathbb Z^2$ for simplicity. A definition of the ...
7
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2answers
261 views

What is the definition of picture changing operation?

What is the definition of picture changing operation? What is a standard reference where it is defined - not just used?
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144 views

Energy barriers between Hadamard matrices

Hadamard matrices may be characterized as $n\times n$ real orthogonal matrices $U$ that achieve the lowest possible "energy" as defined by the (scaled and shifted) entry-wise 1-norm: $$ E(U)=n^2 ...
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0answers
96 views

Master Equation to Fokker-Planck for a Jump-Diffusion

Does anyone know if there is a derivation of the Master Equation approximation by a Kolmogorov backward equation (Fokker-Planck) to a jump-diffusion with a compensated Poissonian integral? If not, can ...
2
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0answers
250 views

How to perform this matrix integral?

Edit: some backgrouds added. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...
0
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1answer
220 views

On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
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2answers
195 views

Positivity of the Coulomb energy in two dimensions

In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{|\cdot|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...
2
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1answer
196 views

Probability measures on $L^p$

Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
3
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1answer
179 views

Question about the Aganagic-Vafa A-brane

According to Aganagic-Vafa (hep-th/0012041) and Fang-Liu (arXiv:1103.0693), for a semi-projective toric Calabi-Yau 3-manifold $X$, the Aganagic-Vafa A-brane $L_{AV}\subset X$ is defined by the ...
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1answer
217 views

Understanding the intermediate field method for the $\phi^4$ interaction

In Rivasseau's and Wang's How to Resum Feynman Graphs, on page 11 they illustrate the intermediate field method for the $\phi^4$ interaction and represent Feynman graphs as ribbon graphs. I had to ...
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2answers
502 views

Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
3
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0answers
104 views

Where is there a treatment of double field theory other than in local coordinates?

The n-lab seems to lack a treatment of double field theory. Where is there a treatment other than in local coordinates? Or at least one which identifies the coordinates as local coordinates for a ...
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1answer
216 views

direct proof that schrodinger's equation kernel corresponds to delta-function initial value [closed]

I want to show directly, that the kernel for the n-dimensional free linear schrodinger equation, if taken to time t=0, is dirac's $\delta $ function. I can show that the integral is constant, but it ...
4
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1answer
362 views

Quantization of symplectic vector space and choice of lagrangian subspaces

My question is related to Geometric Quantization. I don't undrestand the philosophy of following assertion If $(V,\omega)$ be a symplectic vector space then the quantizations of $V$ ...