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 ...

learn more… | top users | synonyms

1
vote
0answers
142 views

Question on energies obtained via WKB approximation

Suppose we are given an ODE problem $$ y''(x)+V(x)f(x)=E_{n} y(x) $$ with boundary conditions $ y(0)=y(\infty)=0$. Here $V(x)$ is a potential function. Then is it always true that (for $n ...
12
votes
2answers
407 views

The entropic cost of tying knots in polymers

Imagine I take a polymer like polyethylene, of length $L$ with some number of Kuhn lengths $N$, and I tie into into a trefoil knot. What is the difference in entropy between this knotted polymer and ...
5
votes
2answers
337 views

Proof that Statistical Mechanics is a model of Themodynamics

The laws of thermodynamics are essentially four axioms of a mathematical theory. The expectation values of a statistical ensemble are supposed to satisfy the axioms of thermodynamics (under the ...
7
votes
1answer
390 views

The role of metric in the Wave Equation

The wave equation is often written in the form $$(\partial^2_t-\Delta)u=0,$$ involving the Laplace-Beltrami operator $\Delta$. However, the Laplace-Beltrami operator $\Delta$ is defined only in the ...
6
votes
1answer
470 views

Boundary Conditions Invariant Under Conformal Transformations in Electrostatics?

in two dimensional electrostatics it is assumed that the whole physical system is translationally invariant in one direction. Here, the two-dimensional Laplace equation $$\Delta \phi(x,y) = ...
2
votes
1answer
189 views

Did classical applications of density functional theory precede its use as an electronic structure method?

Density Functional Theory (DFT) is usually considered an electronic structure method, however a paper by Argaman and Makov highlights the applicability of the DFT formalism to classical systems, such ...
11
votes
3answers
1k views

What are the mathematical problems in introducing Spin 3/2 fermions?

Can the physics complications of introducing spin 3/2 Rarita-Schwinger matter be put in geometric (or other) terms readily accessible to a mathematician?
6
votes
1answer
437 views

Why Zeta regularization is not valid for multiple-loops?

Why zeta regularization only valid at one-loop? I mean there are zeta regularizations for multiple zeta sums. Also we could use the zeta regularization iteratively on each variable to obtain finite ...
2
votes
2answers
232 views

What is sector decomposition

What is sector decomposition and how can it be used to 'disentangle' UV and IR divergences? I have read about it in the paper SecDec: A general program for sector decomposition, but I have no idea ...
2
votes
2answers
518 views

Operator relation involving the logarithm of an operator?

Dirac gives the relation: $\exp(iaq)f(q,p) = f(q, p - a\hbar)\exp(iaq)$ where $\hbar$ is Planck's constant. Can anybody give me the corresponding relation when the $\exp$ function is a $\ln$?
10
votes
2answers
989 views

Are there books on Regularization and Renormalization in QFT at an Introductory level?

Are there books on Regularization and Renormalization, in the context of quantum field theory at an Introductory level? Could you suggest one? Added: I posted at math.SE the question Reference ...
3
votes
2answers
780 views

Why was the truncated icosahedron (i.e. soccer ball) geometry chosen for the implosive lenses in the “Fat Man” atomic bomb?

Quoting from Wolfram Mathworld: " It is the shape used in the construction of soccer balls, and it was also the configuration of the lenses used for focusing the explosive shock waves of the ...
4
votes
10answers
2k views

Addition of different physical quantities

We all know the "apples and oranges" rule which says that it's meaningless to add or subtract two different quantities like apples and oranges. But the same rule doesn't hold for the multiplication ...
6
votes
1answer
823 views

Gelfand-Yaglom theorem for functional determinants

What is the 'Gelfand-Yaglom' Theorem? I have heard that it is used to calculate Functional determinants by solivng an initial value problem of the form $Hy(x)-zy(x)=0$ with $y(0)=0$ and ...
8
votes
1answer
573 views

Iterated dimensional regularization

Given a 2-loop divergent integral $\int F(q,p)\,\mathrm{d}p\mathrm{d}q$, can it be solved iteratively? I mean I integrate over $p$ keeping $q$ constant Then I integrate over $q$ In both iterated ...
6
votes
2answers
381 views

Lagrangians combining terms with 1 and 2 derivatives

How are field theory Langrangians treated when some terms have 2 derivatives but others have only 1? Because the number of derivatives in a Lagrangian term is more easily even than odd, the ...
11
votes
5answers
1k views

What does a frame of reference mean in terms of manifolds?

Because of my mathematical background, I've been finding it hard to relate the physics-talk I've been reading, with mathematical objects. In (say special) relativity, we have a Lorentzian manifold, ...
2
votes
6answers
3k views

real world applications of Mathematics which use functions with singularities, not just as a matter of mathematical taste but for conceptual reasons

EDIT [for aptness of this question to this site, read 'real world applications' as 'applications in Physics'] The concept of function (of the form $f : \mathbb{R} \to \mathbb{R}$ ) has been used in ...
4
votes
2answers
232 views

Is the Assumption That Space-time Has to Be a Continuum Just a Matter of Mathematical Taste?

Is the assumption that space-time has to be a continuum just a matter of mathematical taste? Isn't there any physical significance associated with it?
2
votes
2answers
80 views

In a gas of particles, how is the displacement vector related to the number density?

Suppose I have a gas of particles that is initially uniformly distributed so that the number density is $n_0$ (number of particles per unit volume), and then I displace the particles by the vector ...
2
votes
1answer
201 views

Church–Turing Thesis

Can the Church–Turing Thesis be proved assuming classical mechanics, how is the proof or disproof? Edited: I was looking for a proof of "everything computable by a device obeying CM is computable by ...
2
votes
3answers
147 views

Length of a curve in D dimensional euclidean space

In a book I am reading on special relativity, the infinitesimal line element is defined as $dl^2=\delta_{ij}dx^idx^j$ (Einstein summation convention) where $\delta_{ij}$ is the euclidean metric. Next, ...
2
votes
2answers
444 views

How do you find conserved quantities for linear second order ODEs?

I have a differential equation of the form $ \frac{d^2 y}{dt^2} + f(t) \frac{dy}{dt} + g(t) y = 0 $ where $f$ and $g$ are known functions of time. Is there a systematic (or otherwise) way of ...
2
votes
2answers
291 views

a question on Lagrange's equation when the time derivative of the generalized co-ordinates is constant

Consider a system whose generalized co-ordinates are $q_i$ and is under the constraints $\dot{q_i} = K_i \forall i = 1,2,3,...$ where $K_i$ are constants. I have a problem in writing the Lagrange's ...
2
votes
1answer
336 views

significance of maxima and minima of time varying kinetic energy of a system

Consider a system of particles where the kinetic energy of the system is varying with time. I'd like to know the significance (or meaning) of the time derivative of the kinetic energy being zero at a ...
4
votes
1answer
765 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 ...
1
vote
0answers
386 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
652 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
1k 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 $$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$ ...
33
votes
4answers
3k 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
148 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
195 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
266 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 ...
9
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
728 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. ...
2
votes
1answer
363 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 ...
5
votes
3answers
568 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
286 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
518 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
674 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
135 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 ...
1
vote
0answers
190 views

spectral function [closed]

How do you obtain the spectral function for advection equation in one dimension? 2.How about 2D? thanks a lot
3
votes
4answers
3k views

How many digits of Pi are required in physics?

In other words: which physics experiment requires to know Pi with the highest precision?
6
votes
2answers
408 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 ...
14
votes
4answers
1k 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
492 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 $n^{th}$ ...
5
votes
6answers
1k 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
478 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 ...