Mathematical methods in quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.
1
vote
1answer
51 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 ...
2
votes
0answers
80 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
votes
2answers
110 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 ...
-7
votes
0answers
42 views
Math 11 grade help please [on hold]
All isotopes of technetium are radioactive, but they have widely varying half-lives. If an 256 g sample of technetim-99 decays to 16 g of technetium-99 in 32,000 years, what is its half-life ?
A 400 ...
-5
votes
1answer
63 views
Applications of logs and exponential: Compound interst formulas A=P(1+r/n)^nt A=Pe^rt [on hold]
Five years ago I places some money in a five-year certificate of deposit earning 4.8% compounded quarterly. It is nos worrh $3,650.25 and it has to be renewed at a new rate. How much money did I ...
1
vote
1answer
209 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 ...
1
vote
0answers
74 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
$$
\textrm{D6-brane a:}\, (l_1^a,l_2^a,l_3^a)
$$
$$
...
4
votes
1answer
125 views
Lieb: Stability of matter, problem with variational method
I am trying to recalculate 'Stability of Matter' Paper from Lieb.
I have a problem at the last step of the proof at page 3, where Lieb calculates the minimizer. I will explain my ideas and my problem ...
3
votes
0answers
44 views
Closed form for 3j-symbol ratios
I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch–Gordan ...
1
vote
0answers
54 views
What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE?
What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? Is it possible to write set-theoretical solutions of Quantum ...
0
votes
0answers
186 views
Sum over a product of binomial coefficients related to a collision problem
I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation:
...
1
vote
0answers
55 views
range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants
The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
1
vote
1answer
67 views
Stokes-Einstein rotational diffusion and vector orientation time
The Stokes-Einstein rotational diffusion relation tells us that we can write down a rotational diffusion coefficient for a sphere as:
$D_r \approx \frac{k_B T}{\zeta_f} \approx \frac{k_B T}{(8 \pi ...
1
vote
1answer
122 views
Does this Mean Value have a Name?
Question:
does the following mean value have a name?
$$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
votes
1answer
204 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 ...
4
votes
5answers
389 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
votes
1answer
132 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 ...
1
vote
0answers
166 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
votes
0answers
95 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 ...
1
vote
1answer
176 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 ...
0
votes
0answers
28 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
votes
3answers
377 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 ...
0
votes
0answers
105 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
votes
4answers
720 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.
...
8
votes
3answers
331 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 ...
5
votes
0answers
190 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 ...
6
votes
1answer
248 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 ...
0
votes
0answers
103 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
...
1
vote
0answers
147 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 ...
4
votes
1answer
787 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 ...
0
votes
0answers
136 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 ...
0
votes
0answers
100 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 ...
0
votes
1answer
151 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
votes
1answer
223 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 - ...
1
vote
1answer
263 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
75 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 ...
-1
votes
1answer
137 views
$S^n$ admit a real polarization $D\subset TS^n$?
When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$
3
votes
0answers
152 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
votes
2answers
240 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?
5
votes
0answers
130 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:
$$
E(U)=n^2 ...
0
votes
0answers
76 views
Master Equation to Fokker-Planck for a Jump-Diffusion
Does anyone know if there is a derivation of the Master Equation approximation by a Kolmogorov backward equation (Fokker-Planck) to a jump-diffusion with a compensated Poissonian integral? If not, can ...
2
votes
0answers
120 views
How to perform this matrix integral?
Edit: some backgrouds added.
In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral
$$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...
0
votes
1answer
195 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 ...
1
vote
2answers
187 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
votes
1answer
183 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 ...
2
votes
1answer
140 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 ...
6
votes
1answer
189 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 ...
1
vote
2answers
465 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
votes
0answers
95 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 ...
0
votes
1answer
134 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 ...
