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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1answer
35 views

Must a field approach one of its vacua to have finite energy?

I'm reading these Cornell lectures on solitons (link doesn't work right now, but it just worked yesterday), and I can't seem to prove what I thought would be a simple analysis exercise. Namely, ...
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5answers
9k views

A book on quantum mechanics supported by the high-level mathematics

I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators ...
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37 views

Relating OPE to commutator of modes

I am confused with relating commutators to operator product expansions. I am following Kac's Vertex Algebras for Beginners book and trying to relate the notions to usual CFT notions. Take $T(z) = \...
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28 views

On the definition of a stationary process

I have come across various ways one refers to a process as stationary and cannot seem to get the equivalency and the level of rigor in each of them: According to Stochastic processes in physics and ...
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3answers
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Electromagnetism for Mathematician

I am trying to find a book on electromagnetism for mathematician (so it has to be rigorous). Preferably a book that extensively uses Stoke's theorem for Maxwell's equations (unlike other books that on ...
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0answers
23 views

Intuitive Understanding of Mathematical Proof of Quantum Hall

I'm curious about the proof of the existence of a quantum Hall effect, which has recently come up because of the 40$^{\mathrm{th}}$ anniversary of the QHE problem. In particular, the paper says [The ...
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1answer
25 views

Basic doubt regarding Markov Processes

Take the Langevin equation for the position of a particle in Brownian motion. $$ m\frac{d^2x}{dt^2} = -\gamma\frac{dx}{dt} + \eta(t) $$ My professor wrote this as the following in the class: $$ \lim_{\...
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0answers
13 views

Why does this distribution function depend on time and not temperature?

When reading Sterile neutrino hot, warm, and cold dark matter I came across the following momentum distribution function for a neutrino species $\alpha$: $$\tag{5.8} f(p,t) = \frac{1}{e^{E(p)/T + \...
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54 views

Lost reference: Kähler gravity in six dimensions and three dimensional $SL(2,\mathbb{C})$ Chern-Simons theory

I've noticed that several references take for a fact that by studying Kähler gravity on a Calabi-Yau threefold one can demostrate that any lagrangian submanifold embedded in the threefold posees three ...
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4answers
3k 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 ...
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1answer
88 views

Are X and Y boson gauge mediator of X and Y gauge symmetry? With zero X and Y charge?

Recall that the electron carries U(1) gauge charge -1. the U(1) is gauged and mediated by the U(1) gauge boson which is the photon with zero U(1) gauge charge, thus $0$. Now let us take this ...
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38 views

How to prove $\operatorname{div} \mathbf{A}=\operatorname{Div} \mathbf{A} \mathbf{F}^{-\mathrm{T}}$?

I recently focus on solid mechanics and I am reading Nonlinear Solid Mechanics A Continuum Approach for Engineering by Gerhard A. Holzapfel. However, I was confused by a mathematical formula eq(2.49), ...
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1answer
630 views

Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
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2answers
488 views

What is the quantum / Berry-Pancharatnam phase for a spin-j state with z-component m?

Quantum phase arises when a spin-j state is sent through a sequence of transitions that return it to its original position. For example with spin$-1/2$, a state picks up a complex phase of $\pi/4$ ...
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0answers
29 views

What does it mean by conformal group act projectively and unitarily? [closed]

What does it mean by conformal group act projectively, unitarily, and projective unitarily?
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1answer
179 views

Can one force the octupole moments of a charge distribution (neutral and with vanishing dipole moment) to vanish using a suitable translation?

In a previous question, I noted that if you have a charge distribution with nonzero charge, then it is possible to choose an origin (at the centre of charge) which makes its dipole moment vanish, and ...
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72 views

Physical and mathematical significance of the NS-2 brane

This question is about topological string theory and it was also posted in MathOverflow. The existence of a new brane called "an NS-2 brane" is predicted in (the second paragraph in the page ...
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0answers
93 views

How to define “positive frequency” 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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67 views

How to justify the path integral derivation of the ground state projector?

Usually one argues that the euclidean path integral is able to recover the ground state of a system along the following lines: Take the time evolution operator $U(t,t_0)=e^{-iH(t-t_0)}$. Transform to ...
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1answer
277 views

Wick renormalization

I'm trying to understand the Wick renormalization in the framework of the Ito integral. I saw the Wick theorem as presented on Wikipedia in a QFT course and I would like to understand how that is ...
3
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2answers
207 views

Global conformal group in 2D Euclidean space

This is a rather naive question, but I was just wondering. I know that the local conformal algebra of 2d Euclidean space is the direct sum \begin{equation} \cal{L}_0\oplus\overline{\cal{L}_0}, \end{...
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2answers
137 views

Physics interpretation of Sobolev space

What is the physics interpretation of Sobolev space? $H^{s,p}:=\left\{u\in L^p(\mathbb{R}^n):\mathcal{F}^{-1}((1+|\cdot|^2)^{s/2}\mathcal{F}(u))\in L^p(\mathbb{R}^n)\right\}$, $s\geq 0,\, 1<p<\...
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14 views

Bloch Component under Lorentz Condition

Suppose we have Gaussian state in the momentum representation $$ a(\textbf p) = (2\pi)^{-3/4}w^{3/2}exp(-\textbf{p}^2/2w^2) $$ and a state $$b(\textbf p) = K\text{sinh}(\frac{\alpha}{2}) q_z a(\...
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11answers
5k views

Does it make sense to say that something is almost infinite? If yes, then why?

I remember hearing someone say "almost infinite" in this YouTube video. At 1:23, he says that "almost infinite" pieces of vertical lines are placed along $X$ length. As someone who ...
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0answers
57 views

Measure of Feynman path integral

Feynman path integral for non-relativistic case is defined as: $$\int\mathcal{D}[x(t)]e^{iS/\hbar}$$ where $$\int \mathcal{D[x(t)]}=\lim_{N\rightarrow\infty}\Pi_{i=0}^{i=N}\bigg(\int_{-\infty}^{\infty}...
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1answer
279 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, ...
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0answers
33 views

What happened to positive energy theorem for $n\geq 8$?

I was flipping through Geometric Relativity by Dan A. Lee where on chapter $8$ the book proved spacetime positive energy theorem for $3\leq n <8$. What happened to positive energy theorem for $n\...
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225 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 ...
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48 views

Mean value of observable in non-normalizable state

If $|\psi\rangle$ is a normalizable state in the Hilbert space of a quantum system and if ${\cal O}$ is some observable we can always evaluate the mean value of ${\cal O}$ on the state $|\psi\rangle$ ...
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2answers
157 views

Questions about BRST formalism and BV formalism

This is from Pierre J. Clavier and Viet Dang Nguyen's paper Batalin-Vilkovisky formalism as a theory of integration for polyvectors. In section 2.3, it states: A symmetry is said to be open when it ...
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0answers
11 views

How are coefficients of equation of sideslip angle in the paper “A general Solution to the Aircraft Trim Problem” calculated?

I am sure many of you guys(Aerospace related) must have read the paper, "A General solution to the Aircraft Trim Problem" by Marco, Duke and Bernt. I am working with the turning of the Aircraft and I ...
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2answers
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In quantum mechanics (QM), can we define a higher-dimensional “spin” angular momentum other than the ordinary 3D one?

Inspired by my previous question Questions about angular momentum and 3-dimensional(3D) space? and another relevant question How to define orbital angular momentum in other than three dimensions? , ...
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3answers
195 views

Line integral of a point charge

I am trying to teach myself Electrodynamics through self-study of Griffiths' Introduction to Electrodynamics, and I am having difficulty with a calculation that involves a line integral of a point ...
3
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2answers
136 views

Velocity-Dependent Potential and Helmholtz Identities

I'm currently working through the book Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized ...
3
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1answer
974 views

Killing vectors for $\rm SO(3)$ (rotational) symmetry

I am reading a paper$^1$ by Manton and Gibbons on the dynamics of BPS monopoles. In this, they write the Atiyah-Hitchin metric for a two-monopole system. The first part is for the one monopole moduli ...
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0answers
43 views

Expression for sum over paths

In an introductory lecture on the path integral formalism, I came across the following. Suppose that $\gamma$'s are paths such that a particle travelling along any of them reaches the position co-...
2
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1answer
59 views

Can a null hypersurface be foliated by spacelike sections?

Let $(M,g)$ be a $d$-dimensional Lorentzian manifold and let $\Sigma \subset M$ be a null hypersurface, which therefore has dimension $(d-1)$. We know that its normal vector $k^\mu$ is null and since ...
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0answers
34 views

What are infinitesimal formally? [duplicate]

Can any problem in physics involving infinitesimal be converted to a rigorous epsilon Delta argument? Say for example finding the moment of inertia of a continuous body? Another is what is the ...
3
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1answer
228 views

Hopf Algebras in Quantum Groups

In the theory of quantum groups Hopf algebras arise via the Fourier transform: A third point of view is that Hopf algebras are the next simplest category after Abelian groups admitting Fourier ...
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1answer
50 views

Index on a compact manifold

How can the integral of a topological term (like the Nieh-Yan term) on all of a compact manifold be nonzero whereas it's a total derivative and the manifold has no boundary? I assume the manifold can ...
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3answers
443 views

Combining rotations for orbital and rotational motion of a planet

If I have a planet orbiting the Sun (assuming circular orbit) at angular velocity $\Omega$ and rotating about its axis at $\omega$. I also have a normal to the surface of the planet $\vec{n}_{\rm surf}...
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1answer
46 views

Difference between two derivative operator given in Jackson's book

As I was reading Jackson (3rd edition), On page 543 I see two different types of derivatives. they are given, (11.76) $$ {\partial^\alpha} {\equiv} \frac{\partial}{\partial x_\alpha} = (\frac{\...
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2answers
566 views

Matrix derivative of a matrix with constraints

I am looking for a general method to obtain derivative rules of a constrained matrix with respect to its matrix elements. In the case of a symmetric matrix $S_{ij}$ (with $S_{ij}=S_{ji}$), one way to ...
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2answers
61 views

Can 'distance' be mathematically described as the convolution of velocity and time, in time domain?

I have phrased the question as such, to confirm that convolution of the two functions raises the dimensionality of the convolution product. So, if I do convolution of velocity and time, then the ...
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2answers
66 views

Linking the de Rham bundle/complex over spacetime to the gauge bundle

In some textbooks, the Maxwell equations are stated in a very simple mathematical form (up to multiplicative constants coming from the system of units being used): $$ \begin{array} \mbox{d}F =0, \\ \...
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36 views

Interpretation of random matrix eigenvectors in physics

Random matrices may be used in physics to replace Hamiltonian of complex system, for instance in nuclear physics. Eigenvalues of these matrices are simply interpreted as the energy levels (even if we'...
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7answers
5k views

Mathematical rigorous introduction to solid state physics

I am looking for a good mathematical rigorous introduction to solid state physics. The style and level for this solid state physics book should be comparable to Abraham Marsdens Foundations of ...
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1answer
364 views

Transformer universal EMF equation derivation

At this Wikipedia page, we've that the 'Transformer universal EMF equation' looks like: $$\text{E}_{\text{rms}}=\frac{2\pi\times\text{f}\times\text{n}\times\text{a}\times\text{B}_{\text{peak}}}{\sqrt{...
13
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1answer
2k views

String theory from a mathematical point of view

I have a great interest in the area of string theory, but since I am more focused on mathematics, I was wondering if there is any book out there that covers mathematical aspects of string theory. I ...
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1answer
47 views

Average quantity over Keplerian orbit

I have been working through some lecture notes and am quite confused on something. I am trying to understand how to average a quantity over an orbit (Keplerian) but I am struggling to get a clear idea ...

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