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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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34 views

Does it make sense to speak in a total derivative of a functional? Part II

I am trying to derive the Noether theorem from the following integral action: \begin{equation} S=\int_{\mathbb{\Omega}}d^{D}x~\mathcal{L}\left( \phi_{r},\partial_{\nu}% \phi_{r},x\right) , \tag{1}\...
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1answer
53 views

Defining a metric on the space of paths

Imagine the following path integral $$\int_{x(0)=x_i}^{x(T)=x_f} \mathcal{D}x \, e^{\frac{i}{\hbar}S[x]}.$$ This integral is defined over the space of all paths that satisfy the boundary conditions ...
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0answers
83 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
46 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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1answer
76 views

Dirac delta function mathematical expression proof

In a discussion of the second order transitions in graphene this mathematical expression is used. $$ \left|\frac{1}{\varepsilon + i \Gamma/2}\right|^2 = \frac{2\pi}{\Gamma}\delta(\epsilon) $$ And I'm ...
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1answer
120 views
+100

What Lie group structure is used for infinite-dimensional Unitary Groups in Quantum Mechanics?

Given an infinite-dimensional Hilbert space $H$, the set $U(H)$ of all unitary operators on $H$ forms a group, known as the unitary group. Now several subgroups of this group play an important role ...
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27 views

Cantor Function and Time travel [closed]

Is it possible to build a mechanism that uses the Cantor Function to time travel? Since the mechanism can change its state even if time changes, transition from time $t_0$ to $t_1$ without changing ...
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1answer
49 views

Laplace Transform: Continuous analogue of Power series [closed]

Laplace transform is considered as the continuous analogue of the power series, $$A(x)=\sum_{n=0}^\infty a(n)x^n \rightarrow A(x)= \int_0^\infty a(t)x^t\mathbf {dt} $$ sub $\,\, x^t=e^{(ln\,x)^t} $ ...
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0answers
37 views

Riemann hypothesis and its effects in physics [closed]

I’ve heard that the Riemann hypothesis provides many benefits for us and has the potential to resolve a lot of problems. What does solving the Riemann hypothesis mean for physics? What are the ...
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0answers
39 views

What physical phenomena will happen? [closed]

A water pipe is bent into a circular, closed, horizontal position. For the water pipe to rotate 180 degrees around the East-West symmetry axis, what happens?
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0answers
51 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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1answer
97 views

What is a coordinate-free version of Noethers theorem? [closed]

What are some examples and derivations of some basic symmetries (not coordinate symmetries)? For example I remember a sufficient condition for being a symmetry of the lagrangian system is being an ...
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0answers
49 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 $Dim = 3,5$, $...
2
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0answers
63 views

Complete positivity with infinite dimensional “auxillary spaces” [closed]

The usual definition of complete positivity (e.g. Stinespring (1955)) is that a linear map between the bounded operators on some Hilbert spaces $\phi:\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{K})...
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0answers
39 views

How to prove that the limit of surface integral exists?

My book "Electromagnetic Fields" says in $\text{Section}\ 3.4$: Question Why does the limit in equation $(3.42)$ exist (convergent)? Why is the contribution from $(S-S_{\delta})$ remains ...
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0answers
34 views

Existence of a strictly a timelike curve on Lorentzian manifold

I encountered online the following exercise: Let $M$ be a Lorentz manifold of $\dim(M)=n$, and let $\psi:\Sigma\to M$ be a spacelike submanifold of dimension $n-2$ embedded into $M$. We will ...
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2answers
57 views

Calculus of Variations commutes with Integrals

I have a question about the variational calculus. Assume a function $q(t,x)$ gives rise for another function $$f(x) := \int dt q(t,x)$$ My question is why the variation $\delta$ commutes with the ...
3
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1answer
78 views

Functional derivative commutes with total derivative

I have a question about a rule from the calculus of variations. Assume we consider the space of differentiable functions on $C^1(\mathbb{R})$ (or for the sake of simplicity the smooth functions $C^{\...
5
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2answers
993 views

Facing a paradox: Earnshaw's theorem in one dimension

Consider a one-dimensional situation on a straight line (say, $x$-axis). Let a charge of magnitude $q$ be located at $x=x_0$, the potential satisfies the Poisson's equation $$\frac{d^2V}{dx^2}=-\frac{\...
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0answers
24 views

observing the conserved canonical momenta

Suppose I have a Lagrangian $\mathcal{L}[\phi]$ with $\phi$ a cyclic variable, which means that the Lagrangian is symmetric under shift of $\phi\rightarrow\phi+c\quad$. The equation of motion will be ...
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2answers
51 views

Exact solution for the perturbation of the inverse metric

So when we usually linearize general relativity with respect to metric perturbations $g_{\mu\nu}\rightarrow g_{\mu\nu}+h_{\mu\nu}$, we compute the correction to the inverse of the metric to first ...
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0answers
35 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
29 views

Piecewise solution to Euler-Lagrange equations

I would like to consider a background for a quantum field theory made up by connecting continuously two different solutions of the Euler Lagrange equations. The problem is one dimensional (let's call ...
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0answers
22 views

What is the best book to read for optimal mass transportation theory for students with physics background?

I need to read optimal mass transportation theory for my research. What is the best book to read. I am from physics background. How much mathematics and what sort of mathematics required prior to ...
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0answers
50 views

Does anyone know how to symmetrize $\gamma$-matrices?

I'm trying to construct the SO(5, 5) $\gamma$-matrices which are real and symmetric. Recently, I have 6 symmetric and 4 antisymmetric $\gamma$-matrices ($6_S + 4_A$ representation). How can I ...
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2answers
62 views

Domain of a Hamiltonian

In a recent paper (on an exactly solvable toy model and its dynamics), we studied such a toy model: $$ H = \sum_{n\in \mathbb{Z}} n |n \rangle \langle n | + g \sum_{n_1,n_2 \in \mathbb{Z}} |n_1 \...
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1answer
29 views

Is a dichotomic basis possible for 3-dimensional space?

We know that the Pauli basis for the 2-dimensional space is a dichotomic basis in the sense that every Pauli matrix has two distinct eigenvalues. Is it possible to express a 3-dimensional matrix $\...
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0answers
15 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
59 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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0answers
17 views

Book on tetrads formalism and tetradic formulation of General Relativity [duplicate]

Could anyone give me some references for mathematicians (coordinates free notation, formalism of fiber bundles etc.) about tetrads, Palatini-Cartan theory, stuff about formulation of GR with tetrads? ...
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0answers
29 views

Functions constant on boundary and topology of underlying manifold

Here are my thoughts: Say I have two manifolds $M$ (one dimensional in my thoughts) and $\mathbb{R}$. Thinking in physical terms; $M$ I imagine as my space of states: of possible configurations of my ...
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0answers
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What is the difference between real and complex instantons (mathemtically, and their physical significance), and connection to Wick rotation

I am struggling to understand the difference and physical significance between real and complex instantons- I think these are also sometimes called ghost instantons? There are also anti-instantons. ...
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0answers
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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 with $\omega = 1$, describing the ...
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1answer
35 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 ...
2
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2answers
134 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
57 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
47 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
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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
27 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
59 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
41 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
44 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
104 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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0answers
76 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 ...
2
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1answer
72 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
67 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
62 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
54 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 ...