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37 views

Green function integration problem [on hold]

When I'm trying to find the Green Function of Helmholtz equation for a cube $0≤x,y,z≤L$ $$\nabla^2u+k^2u=\delta(\vec{x}-\vec{x}')$$ where $u=0$ on the surface. I set to find the green function where ...
2
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2answers
84 views

Separability of a Hilbert space and its implications for the formalism of QM

In the text I'm using for QM, one of the properties listed for Hilbert space that is a mystery to me is the property that it is separable. Quoted from text (N. Zettili: Quantum Mechanics: Concepts and ...
3
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1answer
53 views

What are the definition and examples of topological excitation?

I read topological excitation in wiki, while it's too brief. What is the precise definition of topological excitation? And can give me some examples and explain why they are topological excitation? ...
0
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1answer
37 views

What are the boundary conditions for EM waves normally incident on the interface between two dielectric media?

An EM wave, amplitude $E_0$, frequency $\omega_0$, is incident upon a material with relative permittivity (dielectric function) $$\varepsilon \left( z \right) = \left\{ \begin{gathered}{\varepsilon ...
3
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1answer
107 views

Why use a particular regularization for $\int_0^\infty \mathrm{d}x\,e^{i p x}$?

There are many badly defined integrals in physics. I want to discuss one of them which I see very often. $$\int_0^\infty \mathrm{d}x\,e^{i p x}$$ I have seen this integral in many physical problems. ...
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1answer
64 views

Scalar field in a Schwarzschild metric

I have found this article recently published in Classical and Quantum Gravity giving the exact solution of a scalar field in the Kerr-Newman metric. These authors also derived Hawking radiation for ...
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0answers
55 views

What is the current situation of the Yang-Mills existence problem?

What is the current situation of the Yang-Mills existence and mass gap problem? And who are the physicists and mathematicians working in this nowaday?
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0answers
24 views

Pure Point Spectrum implies Spanning Eigenfunctions [migrated]

If $H$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$, and the spectrum of $H$ is a pure point spectrum, i.e., the spectrum consists of discrete eigenvalues (perhaps with multiplicity ...
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0answers
15 views

Free groups and their appierance/emergence/applications in physics

While trying to master my knowledge on group theory in physics from more formal point of view, i noticed an entity called free group ( http://en.wikipedia.org/wiki/Free_group ). I'm aware that it is ...
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0answers
28 views

Do all the spacelike curve terminate at the spatial infinity $i_0$ in the Penrose Diagram of a Schwarzchild black hole?

Let's restrict to the radial direction, so the metric can be expressed as $ds^2=-(1-r_S/r)dt^2+(1-r_S/r)^{-1}dr^2$ with $r_S$ the Schwarzchild radius. Expressed in Kruskal coordinates, the metric is ...
2
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3answers
209 views

About the definition of the Møller operator

The Møller operator is defined as $$\Omega_+ = \lim_{t\rightarrow -\infty} U^\dagger (t) U_0(t).$$ Does the operator $$ \lim_{t\rightarrow -\infty} U_0^\dagger (t) U(t) $$ also make sense? Is it ...
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1answer
28 views

De Donder Weyl theory

Im trying to get my head around what the difference is between a symplectic and multisymplectic manifold is. My understanding currently is that on a symplectic manifold time is the parameter that ...
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0answers
36 views

How to find out the asymptotic behavior of a Bessel function? [migrated]

How can one find out the asymptotic behavior of a Bessel function? If we start from $z= 0$, we can get a Taylor series. But in physics, we have to know the asymptotic behavior of the solution at both ...
9
votes
1answer
191 views

Why isn't the path integral defined for non homotopic paths?

Context In the Aharonov Bohm effect, there is a solenoid which creates a magnetic field. Since the electron cannot be inside the solenoid, the configuration space is not simply connected. Question ...
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0answers
36 views

Addition in linear vector spaces [migrated]

In the definition of linear vector spaces, one of the axioms is that the addition must be commutative and associative. The addition of scalars and matrices are both commutative and associate. Can ...
0
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0answers
48 views

What lives in the Hilbert Space? [duplicate]

Consider the eigenvalue equation: $$\hat{Q}\Psi = q\Psi$$ where $q$ and $\Psi$ are eigenvalues and eigenfunctions of the hermitian operator $\hat{Q}$. If the spectrum of the hermitian operator is ...
2
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0answers
31 views

Mathematical text for Astrophysics [closed]

Please recommend me some Mathematical text books and a list of Mathematical topics for Astrophysics.
4
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0answers
67 views

References to Mechanics (Classical, Quantum, Statistical) using Time-Scale calculus?

Time-Scale Calculus, is a theory which unifies ordinary (plus fractional and q-) calculus with discrete (and finite differences) calculus. In a sense, in a similar way the Lebesgue integral (or ...
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0answers
51 views

How can we prove the commutator $[F(a^{\dagger}a),a^{\dagger}a]=0$

The spin operators defined by the Holstein-Primakoff transformation are $$ S^{+}=\hbar \sqrt{2S}a^{\dagger}\sqrt{1-\frac{a^{\dagger}a}{2S}} \\\ S^{-}=\hbar \sqrt{2S}\sqrt{1-\frac{a^{\dagger}a}{2S}}a ...
2
votes
0answers
59 views

Geometric interpretation of Grassmann variable

Grassmann variables were introduced to make path-integral formalism easier to handle fermionic (anti-commutating) fields. Mathematically they represent the exterior algebra of forms (or exterior ...
3
votes
1answer
141 views

Susy QM and Atiyah-Singer index theorem

Consider maps $t\mapsto x^i(t)$ from circle to some Riemannian (spin) manifold and lagrangian $$ \mathcal L = \frac12 g_{ij}(x) \partial_t x^i \partial_t x^j + \frac12 g_{ij} \psi^j \left(\delta^i_k ...
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0answers
34 views

Any physical expressions using the integral of the Bessel function of the Second Kind? [closed]

I am currently studying the asymptotic expansions of the integral of the form $$ I(b)=\int_{0}^{\infty} \phi (x) Y_ \nu (bx) dx $$ where $ \phi $ is an element of Schwartz space and $Y_ \nu $ is the ...
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0answers
20 views

What is a Chiral Algebra for a group?

What do we mean by the Chiral Algebra for a group G (SO(3) etc )? Do you know a reference suitable for physicists? Thank you
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0answers
47 views

Is Functional Analysis used at all in String Theory? [closed]

Is functional analysis used at all in string theory? (and if so, how?)
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1answer
67 views

Quantum mechanics, operator commutes with Hamiltonian

My textbook said, if an operator $\hat{O}$ commutes with the Hamiltonian, then we can use the eigen vectors of the Hamiltonian as a basis of the Hilbert space, then express the operator $\hat{O}$ in ...
1
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1answer
42 views

Is the scattering length definitely positive if the potential is everywhere nonnegative?

Is the scattering length definitely positive if the potential is everywhere nonnegative? Intuitively, it seems reasonable. Any rigorous proof?
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0answers
32 views

Robin boundary conditions for the heat equation

Consider a 3D channel with fluid or gas with walls $\Gamma_1$, inflow part $\Gamma_2$ and outflow part $\Gamma_3$. The temperature is described by the heat equation: $$ \frac{\partial T}{\partial t} ...
3
votes
1answer
90 views

Rigorous version of field Lagrangian

In Classical Mechanics the configuration of a system can be characterized by some point $s\in \mathbb{R}^n$ for some $n$. In particular, if it's a system of $k$ particles then $n = 3k$ and if there ...
1
vote
1answer
63 views

Superfluid rotating frame of reference

I'm currently studying a text about Bose-Einstein condensates (BECs) and vortices. When they want to study whether a vortex will be formed, they look at the fact wether it's enegetically favorable. ...
6
votes
2answers
134 views

Interpreting some domain issues of (potential) momentum operators

In the context of mathematical quantum mechanics, a well known no-go theorem known as Hellinger-Töplitz tells us that an unbounded, symmetric operator cannot be defined everywhere on the Hilbert space ...
2
votes
0answers
11 views

Main differences between elastodynamic and light scattering when using S-matrix to find bound states

What are the main differences (top 5 if question is too broad), for using the S-matrix to find bound states, between elastodynamic and light scattering? (if it facilitates a higher quality ...
2
votes
3answers
53 views

How do the flow equations relate to the actual situation?

This question might seem silly but I'll try to make it clear. It's a question (I think) about partial differential equations systems in general, but since currently I'm studying fluid mechanics I'll ...
3
votes
1answer
266 views

Teach me Wick's theorem the honest way

Generally speaking the average guy marginally acquainted with quantum field theory or advanced combinatorics describes Wick's theorem as some sort of correspondence between higher order differential ...
0
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0answers
40 views

Action principles and covariant equations [duplicate]

Can every physically sound differential equation, that is covariant, deterministic etc. be derived by extremising a suitable action using a suitable lagrangian, that may be arbitary. Is this a ...
6
votes
0answers
69 views

Is there a null incomplete spacetime which is spacelike and timelike complete?

Geodesic completeness, the fact we can make the domain of the geodesic parametrized with respect an affine parameter the whole real line, is an important concept in GR. Especially, because the lack of ...
7
votes
3answers
704 views

Mathematical understanding of Quantum Mechanics

Assuming that $\phi(r) = F (\psi(r))$ for some operator $F$ in Quantum Mechanics. Then, in our lecture today, we said that $$\phi(r) = \langle r|F |\psi\rangle = \int_{\mathbb{R}} \langle r |F| r' ...
1
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2answers
86 views

Good Fiber Bundles reference for Physicists

I'm a student of Physics and I have interest on the theory of Fiber Bundles because of the applications they have in Physics (gauge theory for example). What are good books to learn the theory of ...
4
votes
3answers
919 views

Why does Law of Large Numbers work?

Often I see books and professors reasoning that, in order to make a good experiment, many measurements are necessary because then the average value of a quantity is closer to the expected value ...
2
votes
0answers
72 views

Several Complex Variables in QFT

After reading the very interesting quote about several complex variables in QFT: "The axiomatization of quantum field theory consists in a number of general principles, the most important of ...
2
votes
2answers
80 views

Domain of simple quantum harmonic oscillator

When discussing the spectral theory of unbounded operators, one often starts with an operator defined on a densely defined subspace of your Hilbert space, and then proves that the operator is ...
2
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0answers
32 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 ...
1
vote
1answer
47 views

Correspondence between one-parameter subgroups of $G$ and $T_eG$

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
1
vote
0answers
50 views

Learning Roadmap to Mathematical Physics [duplicate]

Currently, I am a graduate student specializing in algebraic geometry. On the other hand, I have also become extremely interested in the mathematical physics. However, I am not sure what steps I ...
1
vote
1answer
77 views

Momentum Representation vs Position Representation

We are given an operator $g$ from $\mathcal{l}^2(\mathbb{Z})$ to $\mathcal{l}^2(\mathbb{Z})$, i.e., the space of functions that are square summable over $\mathbb{Z}$ such that ...
26
votes
6answers
634 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 ...
10
votes
0answers
282 views

A Theorem Due to Hodge: Hawking/Ellis

This is probably quite an obscure question but hopefully somebody has a simple answer. I'm studying the proof of the topology theorem on black holes due to Hawking and Ellis (Proposition 9.3.2, p. 335 ...
1
vote
1answer
86 views

How to choose the Correct Green's Function?

In order to solve the Green’s function of the Helmholtz operator $$(\nabla^2+k^2)G(\vec r-\vec r’)=\delta^{(3)} (\vec r-\vec r’)$$ one can obtain four different Green’s functions corresponding to four ...
2
votes
1answer
86 views

Floquet quasienergy spectrum, continuous or discrete?

I haven't got a feeling about Floquet quasienergy, although it is talked by many people these days. Floquet theorem: Consider a Hamiltonian which is time periodic $H(t)=H(t+\tau)$. The Floquet ...
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0answers
30 views

Guidance regarding research in Mathematical Physics [duplicate]

I am currently a Master's student in Mathematics. The main focus of my undergraduate programme was on Mathematics. However as a part of the course, I have done 8 Theoretical Physics courses(2 courses ...
1
vote
1answer
47 views

Finding expectation value of $p^2$ without integrals

So the expectation value of momentum, if you know the expectation value of position is $$\langle p \rangle = m \frac{d\langle x \rangle}{dt}$$ Is there a nice formula like this for $\langle p^2 ...