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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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7
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1answer
206 views

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 ...
1
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1answer
70 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 ...
15
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0answers
169 views

Is the converse of Weinberg's statement on the cluster decomposition principle true?

In Weinberg's "The Quantum Theory of Fields, Vol. 1", Section 4.4, page 182, the author says: We now ask, what sort of Hamiltonian will yield an $S$-matrix that satisfies the cluster decomposition ...
1
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1answer
88 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 ...
2
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1answer
108 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 ...
2
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2answers
2k views

Constructing the exponential form of a unitary operator

I think I've got this figured out but wanted to make sure I'm doing this right. Working with operators that satisfy bosonic commutation relations $$[b,b^\dagger] = 1,$$ I define a very general ...
12
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1answer
522 views

Is it possible to show a diffraction caustic as a home experiment / lecture demonstration?

This nice paper contains a neat demonstration of the fact that optical caustics, when seen from a wave-optics perspective, contain a bunch of interesting interference terms which can be calculated ...
1
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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 $...
2
votes
2answers
707 views

Relationship between Connection and Material Derivative

Suppose $D\subset \Bbb R^3$ contains a fluid and that $f : D\times \mathbb{R}\to \mathbb{R}$ is a time dependent function defined on the fluid region. In that case, the material derivative is defined ...
2
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0answers
72 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})...
1
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0answers
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 $Dim = 3,5$, $...
0
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0answers
42 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 ...
0
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0answers
37 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 ...
3
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1answer
162 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
1k 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{\...
17
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3answers
8k views

Why the statement “there exist at least one bound state for negative/attractive potential” doesn't hold for 3D case?

Previously I thought this is a universal theorem, for one can prove it in the one dimensional case using variational principal. However, today I'm doing a homework considering a potential like this:$...
1
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2answers
75 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 ...
1
vote
1answer
112 views

Surjectivity of momentum mapping

I have to show that the following mapping of momenta is surjective. The mapping $\{p_i^{\mu},p_j^{\mu},p_k^{\mu}\}\rightarrow\{\tilde{p}_{ij}^{\mu},\tilde{p}_k^{\mu}\}$ is given by $$ \tilde{p}_k^{\...
1
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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 ...
0
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0answers
28 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 ...
2
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2answers
72 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 \...
8
votes
4answers
4k views

Binomial expansion of non-commutative operators

I would like to determine the general expansion of $$(\hat{A}+\hat{B})^n,$$ where $[\hat{A},\hat{B}]\neq 0$, i.e. $\hat{A}$ and $\hat{B}$ are two generally non-commutative operators. How could I ...
0
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1answer
33 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 $\...
1
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0answers
25 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 ...
1
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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 ...
1
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0answers
19 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? ...
0
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0answers
30 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 ...
0
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0answers
33 views

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. ...
1
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0answers
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 ...
2
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0answers
62 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 ...
30
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8answers
8k views

Introduction to string theory

I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I ...
2
votes
2answers
159 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 ...
0
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1answer
55 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 ...
1
vote
1answer
59 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 ...
0
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0answers
25 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: $${\bf E}({\bf r})=k\int_V \frac{\rho({\bf r'})}{R^2}\hat{\bf R}\, \mathrm dV$$ ...
1
vote
1answer
136 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 $:...
1
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0answers
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(...
0
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0answers
67 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 ...
102
votes
6answers
5k views

What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis?

This question was listed as one of the questions in the proposal (see here), and I didn't know the answer. I don't know the ethics on blatantly stealing such a question, so if it should be deleted or ...
2
votes
1answer
80 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 ...
5
votes
0answers
99 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
votes
1answer
159 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 ...
1
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0answers
60 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 ...
-1
votes
1answer
85 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 ...
2
votes
0answers
101 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 ...
0
votes
0answers
63 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 ...
5
votes
2answers
844 views

Lagrange multiplier and constraint force

The Lagrangian with Lagrange multiplier in the form $$L= T- V + \lambda f(q, \dot{q},t).$$ But there are different ways of writing the constraint $f = 0$. Will that lead to different EOMs? Let me ...
38
votes
6answers
1k views

Motion described by $m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}=-k\frac{\mathrm{d}^{\frac12 }x}{\mathrm{d}t^{\frac12}}$

What kind of motion would a (preferably dimensionless for simplicity) body do if the force acted on it was proportional to the semi-derivative of displacement, i.e. $$m \frac{\mathrm{d}^2 x}{\mathrm{...
5
votes
0answers
185 views

What are the applications of hyperbolic $3$-manifold theory to cosmology?

I am a pure mathematician specialized in hyperbolic $3$-manifold topology. That has been an incredibly active field of research in the past few decades due to the seminal work of Thurston, as many of ...