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4
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1answer
920 views

Rayleigh-Lamb dispersion curves

In an infinite plane elastic plate of thickness $d$, it is shown that the modes of oscillation corresponding to a fixed time-frequency $\omega$ have wave-numbers given by solutions of the ...
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0answers
453 views

errata for Morse & Feshbach - Methods of Theoretical Physics [closed]

Anyone knows where I can find an errata (or any related material, such as solution sheets, etc) for this book? Thanks. Note: This is not a physics question, but this book is so popular among ...
2
votes
2answers
734 views

Helmholtz decomposition in the plane

Prove or disprove the following proposition: For any smooth plane vector field $\mathbf{H}=\left(H_x,H_y\right)$, there exist scalar potentials $\phi$, $\psi$ such that $H_x=\frac{\partial \phi ...
4
votes
1answer
2k views

Uniqueness of Helmholtz decomposition?

Helmholtz theorem states that given a smooth vector field $\pmb{H}$, there are a scalar field $\phi$ and a vector field $\pmb{G}$ such that $$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$ ...
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votes
4answers
4k views

Trace of a commutator is zero - but what about the commutator of $x$ and $p$?

Operators can be cyclically interchanged inside a trace: $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero: $${\rm Tr} ([A,B])~=~0.$$ But what about ...
1
vote
2answers
153 views

A question on a system of particles governed by laws of gravity and electromagnetic field

Consider a system of many point particles each having a certain mass and electric charge and certain initial velocity. This system is completely governed by the laws of gravitation and electromagnetic ...
4
votes
1answer
200 views

Convergence of periodic single fermion operators

First, a quick remark: I'm a mathematician, now working on some problems coming from physics (in particular Ising models on quasiperiodic chains). A few things I find rather mysterious. I would ...
0
votes
1answer
176 views

A question on smooth 1-manifolds

Consider two people living on two different smooth 1-manifolds $S$ and $T$ as shown in figure 1. The manifold $S$ is a bump function joining the points $A$ and $B$ and the manifold $T$ is formed by ...
6
votes
0answers
281 views

1-form formulation of quantized electromagnetism

In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps ...
10
votes
3answers
3k views

Integral of the product of three spherical harmonics

Does anyone know how to derive the following identity for the integral of the product of three spherical harmonics?: $\int_0^{2\pi}\int_0^\pi ...
4
votes
4answers
838 views

How to calculate the quantum expectation of frequency of a particle?

I know how to calculate the expectation of < $\Psi$|A|$\Psi$ > where the operator A is the eigenfunction of energy, momentum or position, but I'm not sure how to perform this for a pure frequency. ...
3
votes
1answer
469 views

What are Grassmann (even/odd) numbers used in superalgebras?

Are Grassmann numbers a concept of graded Lie algebras or is something specific to superalgebras? What are they (i.e: how are they defined, important properties, etc.)? Is there a reasonable ...
6
votes
3answers
618 views

Are all superalgebra's clifford algebra's

I believe the answer to be yes, but I realize that sometimes physicists place additional constraints that might not be obvious. If superalgebras are clifford algebras, why make a literary ...
2
votes
2answers
315 views

Phase Accumulation of Hankel-waves upon propagation

Hankel functions are solutions to the scalar Helmholtz-equation $$\Delta\psi + k_e^2\psi = 0$$ in cylindrical and spherical geometry (with respect to a separated angular dependence). Thus, they are ...
2
votes
2answers
581 views

Three-Dimensional Gravity

Does anyone have any references that discuss gravity in three-dimensions? I'm trying to make my way through some papers by Witten relating $SL(2,\mathbb{C})$ Chern-Simons theory and gravity in three ...
8
votes
3answers
810 views

Boundary layer theory in fluids learning resources

I'm trying to understand boundary layer theory in fluids. All I've found are dimensional arguments, order of magnitude arguments, etc... What I'm looking for is more mathematically sound arguments. ...
2
votes
1answer
138 views

potential energy of an object due to other two objects

consider object A with mass $m_{A}$ and positional vector $\overrightarrow{r_{A}}$ object B with mass $m_{B}$ and positional vector $\overrightarrow{r_{B}}$ object C with mass $m_{C}$ and positional ...
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vote
0answers
192 views

spectral function [closed]

How do you obtain the spectral function for advection equation in one dimension? 2.How about 2D? thanks a lot
6
votes
2answers
412 views

$\nabla ^2\psi$ equals $\psi -$ average value of $\psi$ at neighboring points

Let $\psi (x,y,z)$ be a scalar field. I found the following statement in Morse & Feshbach Methods of Theoretical Physics: The limiting value of the difference between $\psi$ at a point and the ...
5
votes
2answers
3k views

Limit of Lorentzian is Dirac Delta

I have a quick question that just came up in my research and I could not find an answer anywhere so I thought I'd try here. So one of the definitions of the Dirac Delta is the limit of the Lorentzian ...
15
votes
4answers
2k views

Applications of the Spectral Theorem to Quantum Mechanics

I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum ...
1
vote
3answers
498 views

Electric field at a point being an $n^{th}$ derivative of electric (or magnetic) field at some other point

This is a theoretical question for which i would like to know an answer with an example. I'd like to know if its possible to create a setup where the electric field at a point $P$ is $n^{th}$ ...
7
votes
5answers
2k views

Laplacian of $1/r^2$ (context: electromagnetism and poisson equation)

We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is ...
2
votes
1answer
539 views

What is the mathematical nature of space time quantization in string theory/super string theory?

I don't know much about string theory, apart from it being a theory of everything which brings QM, QED and nuclear forces and gravity under one single roof. I am curious to know from a mathematical ...
15
votes
3answers
897 views

Does renormalization make quantum fields into (slightly) nonlinear functionals of test functions?

Quantum fields are presented as operator-valued distributions, so that the operators in the theory are linear functionals of some test function space. This works well for free fields, giving us a ...
3
votes
3answers
398 views

How to think physically about basic “fields”

"Field" is a name for associating a value with each point in space. This value can be a scalar, vector or tensor etc. I read the wikipedia article and got that much, but then it goes it into more ...
8
votes
3answers
1k views

Group Cohomology and Topological Field Theories

I have a two-part question: First and foremost: I have been going through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Field Theories". Here they give a general definition for ...
3
votes
1answer
506 views

Math and Wormholes

Hopefully this is the correct forum for this. I felt that Physics Overflow may not be the correct place. I had a student approach me ask me what kinds of mathematics goes into the study of wormholes. ...
5
votes
3answers
526 views

Curvature of Conical spacetime

Inspired by: Angular deficit The 2+1 spacetime is easier for me to visualize, so let's use that here. (so I guess the cosmic string is now just a 'point' in space, but a 'line' in spacetime) Edward ...
11
votes
4answers
1k views

If all conserved quantities of a system are known, can they be explained by symmetries?

If a system has $N$ degrees of freedom (DOF) and therefore $N$ independent1 conserved quantities integrals of motion, can continuous symmetries with a total of $N$ parameters be found that deliver ...
8
votes
2answers
960 views

Introductory texts for functionals and calculus of variation

I am going to learn some math about functionALs (like functional derivative, functional integration, functional Fourier transform) and calculus of variation. Just looking forward to any good ...
9
votes
5answers
3k views

Fourier transform of the Coulomb potential

When trying to find the Fourier transform of the Coulomb potential $$V(\mathbf{r})=-\frac{e^2}{r}$$ one is faced with the problem that the resulting integral is divergent. Usually, it is then argued ...
1
vote
1answer
1k views

Amplitude of power spectral density

Why is the amplitude of the Power spectral density higher for shorter period of time as compared to a longer period of time when calculated for any vibration data?
1
vote
1answer
255 views

Power Spectral Density of constant speed data in a car

I recently calculated the PSD of the vibration data in a car at constant speed. I would like to know what this means and what if I calculate PSD of the vibration data in a car for the total journey ...
32
votes
9answers
4k views

Rigor in quantum field theory

Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, ...
4
votes
2answers
1k views

Calculate stainless steel pole necking limit

Background Trying to determine how much weight a post can support without necking when a monitor is attached to an articulated arm: a cantilever problem. Problem There are three objects involved in ...
2
votes
0answers
384 views

An alternative, algebraic way to introduce interactions. Are there other ways out there?

An opening paragraph: The usual approach to introducing interactions in quantum field theory is to make the constraint on the amplitude of the field towards smaller values more forceful than ...
0
votes
5answers
2k views

Use of the mathematical concept 'function' in theoretical physics

The mathematical concept of function is used in physics to represent different physical quantities. For example the air pressure variation with time and space is called an acoustic wave. We use a ...
10
votes
10answers
3k views

Physics for mathematicians

How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages ...
2
votes
1answer
370 views

What does the Many-Body Problem say about mathematical physics?

I was interested to read the answers to the other Many-Body Problem questions on this site and was left with one nagging question of my own. What does the Many-Body Problem reveal about reality and ...
6
votes
1answer
575 views

There seems to be no definition of “stability” in axiomatic QFT. Is there? And, if not, is this a problem?

"stability" is invoked as the justification for the axiomatic requirement that the spectrum of the generators of the translation group must be confined to the forward light-cone. The spectrum ...
20
votes
7answers
2k views

Quantum mechanics on a manifold

In quantum mechanics the state of a free particle in three dimensional space is $L^2(\mathbb R^3)$, more accurately the projective space of that Hilbert space. Here I am ignoring internal degrees of ...
5
votes
1answer
246 views

Natural systems that test the primality of a number?

There might be none. But I was thinking of links between number theory and physics, and this would seem like an example that would definitely solidify that link. Are there any known natural systems, ...
13
votes
7answers
1k views

What are the uses of Hopf algebras in physics?

Hopf algebra is nice object full of structure (a bialgebra with an antipode). To get some idea what it looks like, group itself is a Hopf algebra, considered over a field with one element ;) usual ...
2
votes
2answers
283 views

What mechanism in string theory enforces the consistency of self-couplings of massless vector bosons?

I have been reading the stackexchange questions on enhanced symmetries in string theory, the Leech lattice, monstrous moonshine, etc. , and I have a question to ask. An astute commentator pointed out ...
5
votes
1answer
173 views

Would it be worthwhile to work out a manifestly supersymmetric superspace formalism for 16 and 32 real SUSY generators?

For 4 real SUSY generators, the superspace formalism has been worked out a long time ago. For 8 real SUSY generators, some brilliant theoreticians have worked out the details of harmonic superspace. ...
0
votes
1answer
156 views

Meaning and types of singularity in case of string or any cosmological model (Mathematical description)

What is actual meaning of singularity can we use this term for conclusion in any research paper( related to cosmological models ).what r the types .
2
votes
1answer
227 views

Is the exponential of the distribution $i\Delta^+(x)$, the 2-point function of a free quantized Klein-Gordon field theory, a distribution?

From answers to a previous question, a finite degree polynomial in the distribution $i\Delta^+(x)$, with Fourier transform $2\pi\delta(k^2-m^2)\theta(k_0)$, is a distribution, even though a product of ...
3
votes
2answers
344 views

On a principal bundle, why is the horizontal vector space not unique?

On a principal bundle, at each point you have a tangent vector space. At a given point, the vectors tangent to the fiber form the vertical vector space. Then the vector space at that point is a direct ...
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vote
2answers
462 views

Geodesics and trajectories

I'm a mathematician studying Arnold's Mathematical Methods of Classical Mechanics. On p. 83 the following definition is given. Let $M$ be a differentiable manifold, $TM$ its tangent bundle, and ...