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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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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, describing the bottom of a well of ...
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1answer
27 views

Prove that the electric field produce by a punctual charge is isotropic and radial

I would like to prove mathematically that the electric field produced by a punctual charge is isotropic and radial, i.e. $$\vec{E}(r,\phi,\theta)=E(r)\vec{e}_r\tag{1}$$ I think that this statement ...
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2answers
117 views

Curl and circulation of a vector field that is ill-defined at the origin: any interesting physical effects?

In the cylindrical polar $(\rho,\phi,z)$ coordinate, suppose the velocity field in a liquid is given by $$\vec{v}=\frac{K}{\rho}\hat{e}_{\phi}, \qquad K=\text{constant}.\tag{1}$$ It can be easily ...
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2answers
49 views

Gauss divergence theorem (GDT) in physics

Some of the statements for $GDT$ which I find in modern textbooks (both electromagnetism and multivariable calculus textbooks) are: (1) Calculus: Several variables Adams Let $D$ be a regular, ...
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1answer
40 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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0answers
20 views

Doubt in application of $GDT$ in electrostatics

Consider a volume charge distribution with continuous density $\rho({\bf r'})$. The electric field at ${\bf r}$ is: $${\bf E}({\bf r})=k\int_V \frac{\rho({\bf r'})}{R^2}\hat{\bf R}\, \mathrm dV$$ ...
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0answers
24 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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0answers
52 views

Positive frequency definition in general spacetime for general fields

In Quantum Field Theory the positive frequency solutions to the classical field equations are quite important since they are the basis of the definition of particles. In Minkowski spacetime we have a ...
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1answer
18 views

What is meant by “collective behavior” in the definition of plasma?

"Plasmas are many-body systems, with enough mobile charged particles to cause some collective behavior ." [M.S. Murillo and J.C.Weisheit Physics Reports 302, 1-65 (1998)]. In the above definition ...
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0answers
39 views

Path integrals and fourier series

I am currently reading the Feynman and Hibbs about Quantum mechanics and path integrals and I found something pretty confusing ( for me ) at page 72. At this page, they are replacing an integration on ...
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1answer
99 views

Normal ordered products of operators and inverses

I have been reading this paper ("Operator ordering in quantum optics theory and the development of Dirac’s symbolic method" by Hong-yi Fan), and on page 3 (right-hand column) the author writes that $:...
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64 views

Intuitive/Physical reason why fields are distributions

I read in Urs Schreiber's notes on mathematical QFT that the infinities in the standard approach to QFT appear because the product between operator-valued field distributions is not always well ...
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1answer
62 views

Intuition between this construction of the sympletic form for classical fields

In this paper, Wald presents a quite general construction of a sympletic form for classical fields. If I understood (which I might have not, and in that case corrections are highly appreciated), the ...
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1answer
59 views

Are the horizon generators radial null geodesics also?

What I am going to ask is probably a result of unrigorous treatment of the submanifold in question. Radial Null Geodesics of Schwarzschild So start with Schwarzschild spacetime. The metric tensor is ...
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1answer
60 views

Gödel undecidability in physics [duplicate]

According to Gödel's Incompleteness theorems, there exist problems in any sufficiently powerful, consistent system of arithmetic that are undecidable form the axioms of said system. *What known ...
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0answers
51 views

Occurances of integrals of the form $Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx$ (and perturbation techniques) [closed]

I am writing a review on perturbation techniques (actually hyperasymptotic techniques) for integrals of the form $$Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx,$$ where the interest is in the ...
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0answers
67 views

Euler-Maclaurin formula for path integral

Is there a corresponding Euler-Maclaurin formula for path integral when we divide the path integral into discrete lattice? What is the error correction when we divide the space into lattice of length ...
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0answers
55 views

Why is the singularity not taken into account?

In this article "Reflections on Maxwell’s Treatise", Section 4.2, it says: He replaces $\mathbf{m}$ with a volume element of magnetization $\mathbf{M}\ dV$ , integrates over $V$ , and lets the same ...
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1answer
105 views

What classifies gaugings?

Gauging a global symmetry $G$ introduces several free parameters to the theory. For example, In $d=4$, gauging a simple and simply-connected Lie group introduces a coupling constant and a theta term, ...
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0answers
27 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 ...
5
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1answer
366 views

General derivative of the exponential operator w.r.t. a parameter

I am interested in the calculation of the general $N$th derivative w.r.t. a parameter $\lambda$ of a quantum mechanical exponential operator with the following structure: \begin{equation*} \frac{\...
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0answers
35 views

Identity involving Majorana spinors and Pauli matrices

How to prove that: $$(\sigma^\mu \bar{\xi}_2)_\alpha \partial_\mu (\xi_1 \psi)=-(\sigma^\mu\bar{\xi}_2)_\beta \xi_{1\alpha}\partial_\mu\psi^\beta-(\xi_1\sigma^\mu\bar{\xi}_2)\partial_\mu \psi_\alpha\...
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0answers
39 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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1answer
31 views

Is the disposition of $1$s and $0$s when writing orthogonal ket vectors purely conventional?

If I want to define the basis in the form of $4$-vectors, how do I proceed to make sure they are orthonormal with one $1$ and three $0$ in each vector? Is it just by convention? Does it matter if I ...
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1answer
92 views

Gauge-invariance of Lagrangians

I am rereading David Bleecker's Gauge Theory and Variational Principles, and I have realized I don't understand something. The offending part is in 3.3 (page 50-52), however I am reproducing the ...
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1answer
60 views

Physical meaning of theorem

This is the image of theorem from V.I Arnold's Mathematical method of mechanics. I understood the example given in text. But I want to know what is physical meaning of example? Can anybody help?
2
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1answer
51 views

Taking a trace using a continuous spectrum of eigenstates

This may be a simple question, but I have not been able to find an adequate discussion in any source that quite answers it. In many cases in quantum mechanics, traces are evaluated using the ...
3
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1answer
130 views

Why there's a Lorentz inner product in the unitary representations of the translation group?

Consider Minkowski spacetime. Its translation group is just the additive group $\mathbb{R}^4$. This is an abelian locally compact group. Next, consider one unitary representation $T : \mathbb{R}^4\to ...
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1answer
27 views

Meaning and Origin of an Expression which Involves Virtual Displacement

As an additional point of confusion related to the answer given here: Confusion with Virtual Displacement I have encountered the following expression in my study of virtual displacements. $$\delta{...
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2answers
97 views

Confusion with Virtual Displacement

I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs ...
2
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1answer
98 views

How to make sense of $\mathcal{I}^-$ as a Cauchy surface rigorously?

In some references, like Hawking's derivation of black hole radiation, one considers that $\mathcal{I}^-$ is a Cauchy surface. One recent reference with such a claim is the paper "Soft Hair on Black ...
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0answers
67 views

Pure math courses for physicists: Topology [closed]

I'm in my bachelor in physics. In a couple of weeks I start my last year, and I'm interested in taking some pure math courses. As you see, I like the theoretical point of view, but I don't know if the ...
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1answer
42 views

Negative unity matrix not hermitian? (stabilizer formalism)

I read the section in the attached picture about the stabilizer formalism and was wondering about the last sentence in the pic. It says that all operators of the stabilizer group are hermitian, ...
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0answers
25 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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1answer
53 views

What is the meaning of “representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space”?

What is the meaning of "representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space"?
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1answer
58 views

Post-measurement density matrix derivation

This is something standard, by I'm trying to redo this with spectral theory. Suppose we start with the usual postulates of quantum mechanics: States are unit rays on a separable Hilbert space. In ...
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1answer
47 views

Position of a particle sliding down an arbitrary curve as a function of time

Given a curve in a frictionless environment with parameterization $\displaystyle \mathbf{r}(\theta)=x(\theta)\hat{\mathbf{i}}+y(\theta)\hat{\mathbf{j}}$ for $\theta\in[0,\theta_f]$, how can I find the ...
2
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1answer
74 views

Utility of the time-ordered exponential

Is the time-ordered exponential $$ \mathcal{T}\exp\left\{-i\int_{t_0}^tdt'V(t')\right\}\tag{1} $$ just a mnemonic device for the series $$ \begin{aligned} 1 + (-i)\int_{t_0}^tdt_1 \, V(t_1) +{} &...
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1answer
187 views

Proof $\exp(-\beta H)$ trace-class operator

Let $H=\frac{p^2}{2}+\frac{x^2}{2}\, : D(H) \to L^2(\mathbb{R})$ be the Hamiltonian of the harmonic oscillator with $m=\hbar=\omega=1$. Prove that $\exp(-\beta H)$ is a trace-class operator if $\beta&...
2
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1answer
72 views

Trace over configuration basis

Let us take a many-body quantum system, whose phases in the configuration basis are labeled by $\mathbf {\hat q}=(q_1,\cdots, q_N)$ and momenta $\mathbf {\hat p}=\left(-i\frac{\partial}{\partial \hat ...
7
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1answer
86 views

Distinct choice of partition in the Path Integral

Practically all books in Quantum Mechanics and Quantum Field Theory define the non-relativistic path integral by taking one interval $[a,b]$ and breaking it up into $N$ subintervals of equal length. ...
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0answers
52 views

Pictures of Different Coordinate Systems in General Relativity

In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to ...
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1answer
69 views

2D global conformal transformations and the $z= \frac1w$ argument

For instance in Blumenhagen's CFT, there is a standard argument which determines that globally defined conformal transformations on the Riemann sphere where $$l_n = -z^{n+1} \partial_z$$ is an ...
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3answers
130 views

Domains of $H$ and $U(t) = \exp(-iH t )$

I am not so familiar with functional analysis. But in my impression, the Hamiltonian $H$ is often not defined everywhere on the Hilbert space. On the other hand, the time evolution operator $U(t)$, ...
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1answer
65 views

Sound wave equation: Neumann boundary conditions

In this paper it's described the solution of the damped wave equation in cylindrical coordinates $$ \nabla^2\left(c^2\rho_1+\nu\frac{\partial\rho_1}{\partial t}\right)-\frac{\partial^2\rho_1}{\...
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0answers
57 views

Motivating the Unintuitive Properties of Spinors

In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
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0answers
31 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
32 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 ...
6
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2answers
191 views

Is this actually the rigorous definition of the path integral in Quantum Mechanics?

Let a quantum system with a single degree of freedom be given. We want to define the path integral so that we get the representation for the propagator as $$\langle q' |e^{-iHT}|q\rangle=\int_{x(a)=...
4
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1answer
84 views

Physical significance of no self-adjoint momentum operator on half line?

I am watching a quantum mechanics lecture by professor Schuller. He mentioned that there does not exist any self-adjoint momentum operator defined on the half line. What is the physical significance ...