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9
votes
3answers
488 views

Applications of Jordan algebras

Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product ...
0
votes
0answers
22 views

Is there a “natural” way to interpolate between a set of bound state wave functions?

Consider for example the Coulomb potential, $-Z/r$, for which there exist a set of bound states with energy $\epsilon_n := {-Z^2 \over 2 n^2}$ (in Hartree). If I want the "wavefunctions" for some ...
7
votes
2answers
202 views

What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$?

Physicists routinely wrote all 3 Pauli spin matrices as a vector. $$ \sigma_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right) \hspace{0.25in} \sigma_2 = \left( ...
7
votes
2answers
372 views

States in C*-algebras and their origin in physics?

in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$ Now, of course there is also in classical physics and quantum ...
1
vote
0answers
57 views

Partial trace with spectral measure

I'm a physicist who needs mathematical advice: Let $A= \sum_{i=1}^{\infty} a_i P_{\phi_i}$ be a self-adjoint operator with projectors $P_{\phi_i}$ on the orthonormal eigenbasis $(\phi_i).$ Let $$B= ...
18
votes
4answers
1k views

Why are quantum groups so called?

I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of ...
9
votes
0answers
155 views

Is this generalization of eigenvalue and eigenvector studied?

While thinking about what it means for observables to be simultaneously measurable in quantum mechanics I came up with the following concepts, which I will call "linearly indexed" versions of standard ...
5
votes
0answers
93 views

Gleason’s Theorem for non-separable Hilbert spaces: On a theorem of Solovay

Gleason’s theorem plays an important role in the foundations of quantum mechanics. On the positive side it demonstrates how the probabilistic structure of quantum theory follows from its logical ...
11
votes
0answers
162 views

Squeezing physics out of formal deformation quantizations

I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested ...
4
votes
1answer
173 views

Equation of motion for the Lagrangian $\mathcal{L} = \text{Tr}(\partial^\mu G \partial_\mu G^{-1})$, $G$ is unitary $N \times N$ matrix? [closed]

What is the equation of motion for the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G \partial_\mu G^{-1})$$where $G$ is an $N \times N$ unitary matrix? Could anyone supply a reference to its ...
4
votes
1answer
118 views

Generating Functional for the Dirac Field, equivalence of expressions

As with the Klein-Gordon field, we can alternatively derive the Feynman rules with the free Dirac theory by means of a generating functional. In analogy with the scalar field theory where $Z[J]$ is ...
3
votes
0answers
81 views

Branches of 3j symbols

Question Is there a quick way to identify the branches in a 3J symbol? Context I need to compute Wigner 3J symbols/Clebsch–Gordan coefficients, $$ \begin{pmatrix} \ell_1 &\ell_2 &\ell_3\\ ...
5
votes
0answers
101 views

Relativistic correction to the Hydrogen atom spectrum, reference? [closed]

Can anyone provide me a reference to the relativistic correction to the Hydrogen atom spectrum worked out? I've tried googling, but I haven't found anything (perhaps I'm just bad at googling). Thanks! ...
2
votes
1answer
66 views

Supercommutator of exterior multiplication operators and their adjoints

Let $\mathfrak{h}$ be a complex Hilbert space and consider Grassmann algebra $\mathcal{F}=\bigwedge\mathfrak{h}$ with its induced inner product. For $\omega\in\mathcal{F}$ we also consider the ...
2
votes
0answers
91 views

Complex scalar field, computation of propagators, four point function [closed]

This is a followup to my previous question here. In quantum field theory, the Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^*\phi.$$Can ...
4
votes
1answer
154 views

Sign of 3j symbol (in view of interpolation)

Question Is there a closed formula for the sign of a 3j symbol? Context I need to compute Wigner 3J symbols/Clebsch–Gordan coefficients, $$ \begin{pmatrix} \ell_1 &\ell_2 &\ell_3\\ ...
6
votes
1answer
241 views

Complex scalar field, generating functional?

The Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^* \phi.$$Can anyone work out or provide me a reference to the computation of the ...
6
votes
1answer
193 views

Gauge field quantization, electromagnetism

Classical electromagnetism (with no sources) follows from the actions$$S = \int d^4x\left(-{1\over4}F_{\mu\nu}F^{\mu\nu}\right),\text{ where }F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$The ...
4
votes
1answer
139 views

Centralizer of hermitian matrices with zero trace

In Quantum Physics one often has to deal with commutators. Here I want to denote by $H_0$ the set of all hermitian matrices with trace equal to zero! One can easily relate it to ...
2
votes
0answers
113 views

From symplectic manifold to Hilbert spaces [closed]

What could be a mathematical model of such physical wish? I'm looking for something sending a symplectic manifold $(M,\omega)$ to a Hilbert space $H_{M}$ with the following properties: 1- We should ...
2
votes
2answers
139 views

Appropriate Recursion relations for Wigner 3j Symbols

I am attempting to code the Cosmic Microwave Lensed Temperature and Polarisation power spectra from first principles and have been told to code the relevant Wigner 3j symbols using recursion rather ...
6
votes
3answers
227 views

Quantum Mechanics and bilinear optimal control theory

I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators. So my question is something like this: Let $i \partial_t \psi(x,t) = ...
1
vote
1answer
112 views

Solving Schroedinger Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system [closed]

Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$ $r_1$ is the distance between the proton $1$ and the electron. $r_2$ is the distance between the proton $2$ and the electron. $R$ is the ...
2
votes
0answers
37 views

Error bounds for eigenvalue expansion of the Mathieu equation

The Mathieu equation is an important eigenvalue problem in Mathematical Physics that is completely understood in its properties, although there is no "direct way" of expressing eigenvalues and ...
0
votes
2answers
274 views

Existence and uniqueness for two-dimensional time-dependent Schrödinger equation

I currently have to deal with time-dependent Schrödinger equations in two variables on bounded domains and wanted to find out about uniqueness and existence of solutions. Unfortunately, I am a ...
5
votes
3answers
652 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0. $$ Then, to find ''bound states'', you solve on the right and find the ...
7
votes
1answer
110 views

Infinitesimal variation of spectrum of Schrödinger operator with changing domain

Suppose we have a Schrödinger operator $$-\frac{d^2}{dx^2}+V(x)$$ defined on $[a,b]$ with Dirichlet boundary conditions. I am interested in whether there are any results for the variation of the ...
1
vote
1answer
1k views

Does this linear algebra construction based on a graph have a name, and where has it been studied?

In the paper Kochen-Specker set with seven contexts by Lisonek, Badziag, Portillo and Cabello, the following construction is used : Question : Have such constructions been used elsewhere, and if so ...
1
vote
1answer
83 views

Self adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator H acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
1
vote
2answers
209 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 ...
4
votes
1answer
499 views

What is the current state of generalizations Noether's theorem?

The well-known Noether's theorem is a vital tool in classical physics. But it assumes some hypothesis, many of which could be removed by a detailed look. So my question is: In what directions has ...
2
votes
2answers
220 views

Quantum Field theory - integral notation

I have a problem with understanding how the resolution of the identity of an operator is presented in some literature for physicists. I'm a student of mathematics, and I understand the notion of a ...
3
votes
3answers
375 views

Reference request: a guide through quantum probability

Could you point out a comprehensive reference book (or more than one, if it is the case) on Quantum Probability that introduces the subject and then gradually builds up to the edges of contemporary ...
1
vote
2answers
271 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 ...
2
votes
0answers
97 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
119 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 ...
4
votes
0answers
32 views

Second-order term of the Fedosov quantised product

In Fedosov's version of quantisation of functions on a symplectic manifold, the product is given in terms of a symplectic connection. I have looked through Fedosov's book in deformation quantisation, ...
3
votes
1answer
124 views

Average entropy of quantum system in bipartite pure state for finite temperature

[I got halfway through writing this when I found the paper that answers the question in (essentially) the affirmative. I'll post it anyways in case anyone is interested.] Background: If a random ...
1
vote
1answer
48 views

When does an orthomodular projection lattice have a non-trivial centre?

When does an orthomodular lattice $L$ of projections onto a given Hilbert space have a non-trivial centre $Z(L)$ and what can we generally say about the cardinality of $Z(L)$?
6
votes
0answers
145 views

From 3-framings on $\Sigma$ to $\mathrm{Spin}^c$-structures on $\mathrm{Loc}_G(\Sigma)$?

Here is my question, below that some motivation: For $G$ a compact abelian Lie group and $\Sigma$ a surface, with $M_G = \mathrm{Loc}_G(\Sigma)$ denoting the space of flat $G$-connections on $\Sigma$ ...
4
votes
0answers
98 views

$q$-deformations of fundamental equation of information and entropies

Classical information theory: fundamental equation of information In classical information theory, the information $I(A)$ of an event $A$ (any element of the $\sigma$-algebra $\mathcal F$ of a ...
3
votes
2answers
105 views

Proof the solution of von Neumann equation will fluctuate forever if Hamiltonian and initial density matrix does not commute

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$ If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will fluctuate ...
1
vote
1answer
237 views

Perturbation theory of eigenvalues - Effects of degeneracy/ multiplicity

In Quantum mechanics Schrödinger's perturbation theory is very important (see Wikipedia) which deals with perturbation of the discrete spectrum of a self-adjoint operator. Where can I find a ...
5
votes
0answers
342 views

Why is geometric quantization (esp. Berezin-Toeplitz quantization) interesting for a symplectic geometer/topologist?

I know that many symplectic geometers are interested in quantization as well. From what I understood, quantization isn't expected to be used as a tool to answer symplectic questions (as in ...
5
votes
3answers
136 views

graded generalization of the Moyal–Weyl product

Has anyone written about the graded generalization of the Moyal–Weyl product/star product, that is, where the original algebra is already graded? Is it just a matter of signs?
6
votes
3answers
805 views

Some explanation about Dynin's formalism

I have seen this claim on the Wikipedia page for the Yang-Mills Millenium problem by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I ...
1
vote
2answers
307 views

How to evaluate the wiener measure of sets?

I would like to understand how the Wiener measure of some simple sets can be evaluated. I will sketch the construction of Wiener measure I have in mind: We denote the one point compactification of ...
19
votes
0answers
739 views

conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
2
votes
0answers
183 views

The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$: ...
8
votes
2answers
769 views

Mathematical equivalent to ladder operators?

A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...