Questions tagged [mathematical-physics]

DO NOT USE THIS TAG for elementary physical questions. This tag is intended for questions on modern mathematical methods used in quantum theory, general relativity, string theory, integrable system etc at an advanced undergraduate or graduate level.

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How are topological invariants obtained from TQFTs used in practice?

Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places: Atiyah, Topological quantum field theory Lurie, Topological Quantum Field Theory ...
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Wave equation: predicting geometric dispersion with group theory

Context The wave equation $$ \partial_{tt}\psi=v^2\nabla^2 \psi $$ describes waves that travel with frequency-independent speed $v$, ie. the waves are dispersionless. The character of solutions is ...
Sal's user avatar
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Was von Neumann's 1954 ICM address "Unsolved Problems in Mathematics" outdated?

I recently tried to "explain" the generalized probability theory aspect of quantum theory (as one common part of both quantum field theory and quantum mechanics), in the sense of motivations for the ...
Thomas Klimpel's user avatar
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Learning roadmap to Topological Quantum Field Theories from a mathematics perspective

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
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What are D-branes (in a topological field theory)?

In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
Dan Kneezel's user avatar
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Open questions in Topological K-Theory

I am interested in knowing about current research in the Topological K-Theory, especially its interactions with String Theory. About one and a half decade back, there were some papers by Physicists (e....
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Connection between distributional and renormalized solutions for Boltzmann equation

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem ...
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Multivariable Integral, How to compute it?

Q How to evaluate a multivariate integral with a Gaussian weight function? $$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm f}\left(x_{1},x_{2},\...
Sijo Joseph's user avatar
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Why are Wigner matrices the appropriate representation for rotations in physics?

Question. Why are the Wigner-D-matrices the appropriate representation for rotation of operators with respect to space-coordinates in physics? Background. A physics paper brought up this question in ...
Mikkel Rev's user avatar
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Motivation for quantum cohomology rings

I can't seem to find a good source for the motivation for defining the big quantum cohomology ring with its quantum product. Collecting the Gromov-Witten invariants in a generating function seems like ...
failedentertainment's user avatar
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Is this physical model exactly solvable?

There exists a popular model in the Physics of heavy quark bound systems, called the Cornell potential model, in which the inter-quark potential is modeled to vary with radial distance $r$ as $$V(r) ...
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What are the mathematical foundations of the renormalisation group?

Briefly, RG refers to mathematical tools that allows systematic investigation of the changes of a physical system as viewed at different distance scales. These methods are very important in quantum ...
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Could you explain the failure of the Hodge decomposition to exist for non-compact manifolds?

I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while). Specifically what I'm (pragmatically)...
user223345's user avatar
9 votes
1 answer
657 views

In what sense is quantum field theory mathematically incomplete?

Is the Yang-Mills existence and mass gap (Millenium Prize problem) essentially what is required? Or are there more problems in putting QFT on strong mathematical foundations? For example, the ...
user68441's user avatar
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Looking for a rigorous treatment of the functional derivative the way it's used by physicists.

A lot of theories in physics can be derived from a variational principle: some action functional S on a space of e.g. field configurations $\phi$ is given, and the equations of the theory follow from ...
Inzinity's user avatar
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What is the meaning and importance of the Hodge codiferential?

In differential geometry given a smooth manifold $M$ we can define the exterior derivative $d$ acting on $k$ forms giving back $k+1$ forms. It is a map $d : \Omega^k(M)\to \Omega^{k+1}(M)$ which is in ...
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Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ...
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7 votes
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Tautochrone and Brachistochrone problems are equivalent

It is a well known fact that the brachistochrone (the problem of finding the curve of quickest descent in a uniform gravitational field) and the tautochrone (the problem of finding a curve from which ...
failedentertainment's user avatar
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131 views

Invariance of the Lagrangian vs invariance of its integral

I'm having a little bit of trouble sorting out the right definitions. I have two sources to work from (but feel free to suggest others I should consult.) . It's in the context of trying to understand ...
rschwieb's user avatar
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Heisenberg's inequality

The Heisenberg's inequality in $\Bbb{R}$ reads $$\|f\|_{L^2}^4\leq \int_{\Bbb{R}}x^2f(x)^2dx\int_{\Bbb{R}}\xi^2\hat{f}(\xi)^2d\xi$$ where by $\hat{f}$ we refer to the Fourier transform of $f$. The ...
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Defining a trace-class operator with a Bochner integral

For complex numbers $z$ consider the system of $L^2(\mathbb{R})$-vectors with norm equal to $1$ \begin{equation} \psi_z=e^{-\frac{1}{2}|z|^2}\sum_{n=0}^{\infty}\frac{z^n}{\sqrt{n!}} |n\rangle, \end{...
Adomas Baliuka's user avatar
7 votes
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320 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
0xbadf00d's user avatar
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A conjecture on Schatten 1-norm

I have a conjecture on Schatten 1-norm. Before presenting the conjecture, let us first specify the notions used here. A matrix $A$ is said to be a density operator if $A$ is positive semidefinite ...
D. J.  Zhang's user avatar
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Implementation of a simulation of an incompressible Newtonian fluid with uniform density

Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I want to simulate an incompressible Newtonian fluid with uniform density $\rho$ and viscosity $\nu$. The evolution ...
0xbadf00d's user avatar
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7 votes
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Cauchy representation and branch point order

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. The main application is for dispersion ...
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Percolation and number of phases in the 2D Ising model.

Update. As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive After a long time I came back to try to understand an article on the Ising model. ...
Elias Costa's user avatar
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7 votes
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256 views

Reference for Dimensional Regularization

I am studying a bit of theoretical physics (QFT and string theory), and I obviously stumbled upon dimensional regularization. I have been told that this technique has in fact a solid mathematical ...
Daniel Robert-Nicoud's user avatar
6 votes
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200 views

Separation of variables of 2nd order PDE yields 1st order ODEs

Consider the telegraph or Klein-Gordon equation on a rectangle*, $$ \begin{align} \left(\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}\right)\psi(x,y)=\gamma^2\psi(x,y), \end{align} $$...
DanielKatzner's user avatar
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257 views

Solving a pair of dual integral equations

I have the equations $$ \int\limits_0^\infty dk \ A(k)k \sinh(ka)\cos(kx)=0 \ \ ; \ \ 1<|x|<\infty \tag{1} $$ $$ \int\limits_0^\infty dk \ A(k) \cosh(ka) \cos(kx)=1 \ \ ; \ \ |x|<1 \tag{2} ...
Sal's user avatar
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6 votes
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Is $\frac{d^2}{dx^2} + \frac{1}{4}x^2$ Self-Adjoint? Is this argument for it being Self-Adjoint correct?

I'm a physics student getting a bit out of my depth in maths, and I'd really like someone to confirm that my reasoning below is correct. We're working in the Hilbert space $\mathcal{H}=L^2(\mathbb{R},...
J_B_Phys's user avatar
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270 views

Witt algebra as infinitesimal symmetries of the diffeomorphism group of the circle?

Let $\mathcal G$ be the diffeomorphism group of the unit circle in the complex plane. Since the unit circle is a compact smooth manifold, $\mathcal G$ can be given the structure of a Lie group such ...
pre-kidney's user avatar
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6 votes
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115 views

Use of the word "topology" in physics

I don't know so much about topological order or topological phase transition or topological state of matter or other similar expressions from physics and mathematical-physics, but I can't see a direct ...
HaroldF's user avatar
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6 votes
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157 views

The Virasoro-Bott group and the KdV equations

The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group. For the famous $KdV$ equations these equations are given on the Virasoro-Bott ...
R Mary's user avatar
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6 votes
0 answers
553 views

Approximating a discrete measure with a continuous one

In physics it is common to approximate distributions of point masses or charges with continuous distributions. To do this, one typically defines a density function by moving throughout the space a ...
Massimo Ortolano's user avatar
6 votes
0 answers
439 views

Integration by parts on manifold with a boundary

Suppose $C$ is a 3-form, and $G$ is a 4-form defined by $G = dC$. Also, $M_{11}$ is an 11-dimensional manifold (without a boundary), $W_{6}$ is a 6-dimensional submanifold of $M_{11}$ and $D_{\epsilon}...
leastaction's user avatar
6 votes
0 answers
383 views

Hawking's and Ellis' derivation of the form of Einstein's field equations

On pages 72-73 of the book "The large scale structure of space-time" Hawking and Ellis show while determining the form of the field equations of general relativity that there is a relation of the form ...
Stephan Fackler's user avatar
6 votes
0 answers
128 views

Green's function the way George Green defined it

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x-x')$ function as their source on the RHS. But ...
user35952's user avatar
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Deconvolution of distribution of diffraction reflexes

I'm a chemist stuck in a mathematical problem. Please bear with me as I'm trying to express myself in Math language. Let me explain in short terms the experimental method I'm using: X-ray diffraction....
Christian's user avatar
6 votes
0 answers
437 views

Is there a relation between Super Riemannian manifolds and Kähler manifolds?

(This question has a physics motivation). Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kähler manifolds, or at ...
Trimok's user avatar
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6 votes
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696 views

Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
user6818's user avatar
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5 votes
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What is a distinct feature of an ambiguous result

This question comes from my experience in radar signal processing. As I am going more deep into the theory of sampling, statistical signal processing and estimation theory in general, I have a very ...
CfourPiO's user avatar
5 votes
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108 views

Representation of the C*-algebra of a Kähler manifold

This question is inspired from mathematics of quantum mechanics. In quantum mechanics, we start with a Kähler manifold $\mathcal{M}$ (which is $\mathbb{CP}^n$), with the symplectic form $\omega$ and ...
johnnydines's user avatar
5 votes
0 answers
60 views

Do these conformal Killing vectors have a name? If not, what should we call them?

I have been investigating conformal Killing vectors on pseudo-Riemannian manifolds, that is, vectors which obey $$ \mathcal{L}_X g = \lambda g$$ where $g$ is the metric and $\lambda$ is some function. ...
Klein Four's user avatar
5 votes
0 answers
81 views

Is every bounded linear functional on a subspace of a Hilbert space given by a function?

This question comes from the Limiting Absorption Principle (LAP). I want to obtain the most general statement possible, so I proceed as follows. Let $M$ be a topological space, $(H, \langle \cdot, \...
Colette's user avatar
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5 votes
0 answers
130 views

Understanding Quantum Measurement in infinite dimensional systems

I have been relearning quantum mechanics recently since I realized I got stuck in some misunderstanding of fundamentals when trying to solve certain problems. My question is actually simply the ...
Mirod the best's user avatar
5 votes
0 answers
213 views

Deriving the equation for radial wave function

I'm trying to solve Schrodinger's equation of an exciton using the separation of variables method: $\psi = RY$. Here's the equation I've already derived: $$ \frac{2\mu r^2}{\hbar^2}(E+\frac{e^2}{\...
IGY's user avatar
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5 votes
0 answers
73 views

Non-smooth aspects of classical electrodynamics

There are many books which treat classical electrodynamics on smooth manifolds using the language of differential forms and connections. I think this is a wonderful approach, especially fruitful when ...
Paweł Czyż's user avatar
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5 votes
0 answers
102 views

RAGE Theorem and Fermi's Golden Rule

I'm currently trying to understand Fermi's golden rule more rigorously. The basic idea is that if $H$ is a self-adjoint operator and $|i\rangle, |f\rangle$ are states in the Hilbert space, then $|\...
Andrew Yuan's user avatar
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5 votes
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126 views

Show that surface integral of absolute vorticity is constant over time

In a rotating frame the (unforced, incompressible) Euler equation is $$ \frac{∂\vec{u}}{∂t}+\vec{u}\cdot\nabla\vec{u}=-\nabla\left(\frac{p}{\rho_0}\right)-2\vec{\Omega}\times\vec{u}+\nabla\left(\frac{...
Azamat Bagatov's user avatar
5 votes
0 answers
103 views

Not following this formula for coincidence rate among 'simultaneous' Gaussian distributions?

I'm reading this paper, which describes (among other things) the "triggering" system for a peice instrumentation used in the detection of subatomic particles in an astroparticle physics ...
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