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Questions tagged [mathematical-physics]

DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com. Mathematical physics is the mathematically rigorous study of the foundations of physics, and the application of advanced mathematical methods to problems in physics. Examples include partial differential equations (PDEs), variational calculus, functional analysis, and potential theory.

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87 views

Liouville theorem and the ergodic assumption

I am following a course on statistical mechanics. My instructor presented us the following Liouville theorem in two (claimed) equivalent ways: Differential statement: The probability distribution $\...
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1answer
37 views

Klein-Gordon equation propagators: intersection with the support of the source

Let $(M,g)$ be a globally hyperbolic. Let $P = \Box - m^2$ be the Klein-Gordon differential operator. Following Fewster's notes, we may define the retarded/advanced propagators $$E^\pm : C^\infty_0(M)\...
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46 views

When do the solutions of combinatorial Dyson-Schwinger equations generate a Hopf subalgebra?

Say I have a set of combinatorial Dyson-Schwinger equations of the form $$\begin{align} X_1 &= \mathbb{1} + \alpha B_+^a (f_1(X_1,...X_N)) \\ & ... \tag{1} \\ X_N &= \mathbb{1} + \alpha ...
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0answers
70 views

Wave function as a section of a complex line bundle to do QM in polar coordinates

If you want to change the coordinates of a Wave function $\Psi$ in 2D QM from cartesian to polar coordinates in a naive way one encounters a problem, namely the (naively defined) radial momentum ...
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46 views

How shall we show the surface integral approaches a limit (or does not blow up) at a field point near $S'$

Consider the electric field due to volume charge distribution in volume $V'$: $\mathbf{E}=\displaystyle \int_{V'} \rho' \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV'$ The integrand ...
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1answer
50 views

Discretization of path integral and linear interpolations

Consider the evaluation by discretization of the path integral $$\int e^{iS[x(t)]}\mathfrak{D}x(t),\quad S[x(t)]=\int_{t}^{t'}\left[\frac{m}{2}\dot{x}(\tau)^2-V(x(\tau))\right]d\tau.$$ One ...
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1answer
58 views

Physical interpretation of Dirichlet energy for a membrane

In the following model of a membrane with a mass particle in it, why does the integral represents the elastic energy of the system? Let $\Omega$ be an open connected region (the membrane) in $\Re^2$,$...
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113 views

Transient solution system of differential equations obtained from master equation

I have to solve the following equation (or at least obtain an approximate estimate) for the diagonal terms of the density matrix. We consider that the initial state is a coherent state $\rho_{n,n}(0)=...
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1answer
77 views

Domain of the infinite square well hamiltonian

I am reading the book by Gitman et al. 'self-adjoint extensions in quantum mechanics'. In the book, they give a precise definition of the domain of the hamiltonian of an infinite square well. For ...
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0answers
62 views

Is there a useful relationship between connection on space coordinates and material derivative?

I am referring to an important part of the question Relationship between Connection and Material Derivative. Here is a paste and cut of the relevant part. That is the directional derivative along $...
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50 views

Can real numbers dimensions exist?

This title may not explain my question right but I could not think of any better short explanation. My question is, if there is a possibility of a structure (or space) with the dimension $Dim = 3,5$, $...
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0answers
40 views

Size of quantum corrections at infinity

Suppose we have a one dimensional field theory for the field $\phi(r)\;r\in[0,\infty]$ and that the solution for the background (Euler Lagrange equations) give a function $\phi_0$ that goes to a ...
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0answers
26 views

Perturbative series in physics: why are coeffcieints of Gevrey-1 type (i.e. bounded by $\alpha C^n(n!)^1$

I have only been able to find this explicitly mentioned in this paper on resurgence techniques in physics. And have chased up the hints it gives, but they are not very explanatory. Essentially, the ...
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0answers
64 views

Does uncertainty principle truly represent the “lower bound” of the information we can obtained from a pair of noncommunicable operator?

Background I: Suppose the commonly used non commuting operator $\hat p$ and $\hat x$. The uncertainty principle told us that $\sigma_p\sigma_x\geq \frac{\hbar}{2}$. In standard quantum mechanic ...
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47 views

Finding approximate eigenfunctions solutions with small eigenvalues

This question is about an appendix to chapter 7 of Aspects of Symmetry Erice lectures by Sidney Coleman. We have a SE for a 1-dimensional simple harmonic oscillator with $\omega = 1$, describing the ...
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1answer
62 views

Equivalence Picard-Lefschetz path integrals and “Feynman's” path integrals

I have just seen the Picard lefschetz method applied to path integrals in order to make these more convergent. I understand how we could modify the contour of integration for a real integral but what ...
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28 views

Arbitrary function on the Faddev-Kulish dressing

On this paper the authors review the Faddev-Kulish dressing in QED which is a solution to the IR divergence problem. Given one electron momentum $\mathbf{p}$, They define the soft factor by $$F_\ell(...
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35 views

Spectral representation of a BCS gap function

I am playing with the spectral representation of a BCS gap function and I have trouble verifying causality properties. I find a divergence and I don't know what is the problem. Assume the gap ...
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0answers
59 views

Looking for lecture videos that follow Arnold's Mathematical Methods of Classical Mechanics

I'm an undergrad and I'm looking for lecture videos (on youtube and such) that follow this textbook. My course roughly follows it, but glosses over some mathematical details that I feel would be ...
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27 views

conventional matrix notation for distance interval

Why matrix notation for distance interval is represented by this? $$g_{\mu \nu}\Delta X^{\mu}\Delta X^{\nu}=(\Delta X)^Tg (\Delta X)=\Delta X^{\mu}\Delta X^{\nu}g_{\mu \nu}$$ Could you explain ...
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0answers
48 views

self-adjoint extension of the momentum operator in an infinitely deep potential

Theta parameter arises when calculating self adjoint extensions of the momentum operator of a particle in an infinnitely deep potential, what does this means physically?
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0answers
35 views

What is the magic behind Sector Decomposition?

I have a question regarding Sector Decomposition, which is briefly introduced in this paper arXiv: 0803.4177. So far I played around with a toy example and applied the Sector Decomposition method to ...
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0answers
50 views

Have fractional order differential models been explored as an alternative to standard gravitational field theory?

Since Einstein introduced his field equations and general theory of relativity, experimental evidence, at least on the cosmic scale has repeatedly supported the theory. Nevertheless, many seeking to ...
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0answers
113 views

Wave Equations from Decoupling Maxwell's Equations in Bianisotropic Media

For several days now, I have been trying to decouple Maxwell's equations in bianisotropic media so that I end up with a form that involves only one variable (of E, D, B, H), i.e. a so-called 'wave ...
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0answers
99 views

Electric flux over a closed surface when point charge lies on the surface

What will be the electric flux over a closed surface when point charge lies on the surface, that is neither inside nor outside? I ask this question because electric field at that point will be ...
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0answers
58 views

How do I proceed with the following coulomb integral?

I am trying to solve the $H_{2}^{+}$ ion problem using Fourier transform approach. The Hamiltonian that I am trying to solve is as follows, $$H=-\frac{\hbar^{2}}{2m_{e}}\nabla^{2}_{e}-\frac{e^{2}}{4\...
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0answers
33 views

Is there a way to describe gravitational waves and time-dependent gravitation without tensors?

I have been reading about gravitational waves, and they fascinate me. However I struggle to follow the mathematics behind it, because they are described using tensors, index gymnastics, et cetera. Is ...
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0answers
49 views

why is a Lagrangian submanifold a semi-classical state and not a classical state?

I read that the Lagrangian submanifold can be regarded as a semi-classical state when classical mechanics is formulated using symplectic geometry. Does anyone know why it would be a semi-classical ...
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1answer
86 views

Derivation of $j$ being a 4-vector in Landau-Lifschitz: Formulation with rigorous mathematical treatment?

Here on Stack exchange, there appeared the question on how to derive the 4 current actually being a Lorentz-tensor. One of the answers (How do we prove that the 4-current $j^\mu$ transforms like $x^\...
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0answers
57 views

Which geometry does not allow the existence of matter?

I have seen these lectures by Fredric Schuller that discuss the obstruction theory and the role of global geometric properties in admitting a spin structure. See the video at 01:27:52 https://youtu....
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1answer
66 views

Is the string-net model Hermitian?

In Kitaev and Kong's paper, they define the Hermitian inner product on morphism spaces in Eq. (11). My question is that: Given that F symbols satisfy the pentagon identity, does that the string-net ...
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0answers
33 views

periodic orbits in Gutzwiller's trace formula

It is said that in the Gutzwiller trace formula, one sums over the periodic orbits. I do not know how to derive the formula, but a simple question arises for me. That is, for some classical ...
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0answers
61 views

How to experimentally construct an imaginary(complex) potential, and what does it mean?

Some useful reference may be from: Finding Stagnation Points from the complex potential Imaginary potential and stationary wavefunction What is the physical interpretation of imaginary terms in the ...
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0answers
38 views

Frenkel-Kontorova model

The Hamiltonian of the Frenkel-Kontorova model is given by \begin{eqnarray} H&=&\sqrt{2}\hbar\sum_{i=1}^{N}\{(\hat{n}_{i}+\frac{1}{2})-\frac{K}{\sqrt{2}\hbar}\cos(\frac{1}{\sqrt{2}}\frac{\...
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0answers
61 views

Scattering amplitude of strong interactions, Euler's Beta function and strings

On reading the answer by Ron Maimon to the "Gabriele Veneziano, strong nuclear force and beta-function" I became puzzled: Why the fact of the Beta function implication leads to the idea of a string ...
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0answers
51 views

Quantum mechanics demanding continuous vector space?

I recently came across Solèr's theorem which seems to state that for quantum mechanics to make sense, we have to use a continuous vector space. But how can that possibly be? I believe we can, in ...
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0answers
133 views

Derivation of Relativity of Simultaneity?

Suppose you have a train moving forward relative to an inertial observer at velocity $v$. Suppose you have a clock 1 at the front of the train and a clock 2 at the back, and, in the frame of the train,...
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0answers
31 views

Constraints on creation operators operating on the electron field in order to produce a particle

My background covers two years of QM, and I am now starting into QFT using Zee and Srednicki. I am familiar with the math behind wave packets and their rapid dispersion, as one of the reasons they ...
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0answers
108 views

What is the physical intuition for the definition of elliptic PDE's?

Consider the Laplace equation $$\Delta u=0.$$ I am aware of the relation of this equation to electrostatics, and that one can "guess" many of its mathematical properties from this physical framework. ...
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0answers
272 views

What's the difference between a Green's function and a fundamental solution?

The Wikipedia article on fundamental solution says In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the ...
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0answers
36 views

How does one calculate the evolution of a species in a low energy plasma over time?

I am currently working on a global modeling program for low temperature plasma and I have hit a wall. My goal is to graph the evolution of a species as a function of time using an approximation of the ...
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0answers
138 views

Hadamard expansion of interacting Klein Gordon 2-point function

Context: There is an algorithm due to Hadamard, I believe, for constructing local bi-distributional solutions to elliptic and hyperbolic equations for the purpose of proving existence and uniqueness ...
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1answer
54 views

What is the physics behind the energy transferred in a 'rotating collision'?

https://www.youtube.com/watch?v=eLXHLRa37_g In this video, we can see that rotating the paper somehow makes it less susceptible to being damaged. I can tell that the paper has some angular momentum. ...
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0answers
40 views

Boundary conditions for a complicated diffusion equation problem

this is my first post/question, so any constructive criticism on how to ask questions better is also welcome I have this problem: we have an infinite(in z axis) cylindrical tube, from $R1$ to $R2$ ...
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0answers
215 views

Does string theory solves the rigor problems of QFT?

String theory is often said to be one possible way to a theory of everything. In that setting it must obviously not just encompass gravity, but everything described by quantum field theory also. In ...
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0answers
59 views

Algebraic curves from (quiver) gauge theory

I'm interested in algebraic curves one can associate to gauge or string theories. Examples involve Seiberg-Witten curves or family of A-polynomials which define holomorphic Lagrangian submanifolds for ...
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0answers
148 views

Intuition about this derivation on QFT

I've found on nLab this post on Wightman axioms which in particular contains a nice example about the quantization of the Klein-Gordon Field. This is a remarkably clean approach from the point of view ...
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0answers
112 views

Interpretation of Dynkin diagrams

I am having trouble in understanding the physics represented by dynkin diagrams. Say I have the following diagram: What is the difference between the square nodes and circular nodes? What does the ...
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2answers
146 views

Demonstrate sinusoidal steady state in linear circuit

Suppose a linear passive circuit is hooked up to an AC generator which outputs a sinusoidal voltage of angular frequency $\omega$. How can we demonstrate rigorously that after a sufficient amount of ...
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0answers
53 views

3d spatial string-net model and its plaquette term

Can someone explain the following sentence, saying the difference betweeen the 2d string-net plaquette intersections and 3d string-net plaquette intersections? Thus, for 3d sting-net model, if we ...