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1answer
102 views

Calculus of variations and string theory

In Polchinski's String theory book, Vol 1., in chapter 1, p. 18, he is deriving the Lagrangian in the light cone gauge (that's not necessary to know in order to answer this question), and he gets ...
5
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0answers
73 views

Physical intuition/interpretation of fractional derivatives/integrals?

Oftentimes, when the derivative and integral operations are introduced within the realm of physics, we are taught some physical interpretation of them: Velocity is the derivative of position ...
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0answers
39 views

Topic Suggestions - Kaehler-Einstein metrics in physics [closed]

I am hoping to get some topic suggestions for a presentation I have to give in a couple of weeks. The course the presentation is for is called Kahler-Einstein metrics. I would really like the ...
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0answers
74 views

Divergent harmonic series

Unlike the zeta regularization for the series $ \sum_{n=1}^{\infty}n^{k} $ which is the exact value for the generalized harmonic series? I mean the series, $$ \sum_{n=0}^{\infty}\frac{1}{(n+a)} $$ ...
3
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1answer
49 views

Most general separable solution of free Dirac equation

In relativistic quantum mechanics, the solution of the free Dirac equation is assumed to be $$\Psi(\textbf{r},t)=u(\textbf{p})e^{i(\textbf{p}\cdot \textbf{r}-Et)}$$ How do I know that this is the most ...
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2answers
161 views

Why isn't the path integral rigorous?

I've recently been reading Path Integrals and Quantum Processes by Mark Swanson; it's an excellent and pedagogical introduction to the Path Integral formulation. He derives the path integral and shows ...
3
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1answer
56 views

What is meant by the following divergent formula?

I have encountered the following formula a couple of times (in different physics contexts which I do not have a good understanding of) $$\int_{0}^\infty \frac{dt}{t}e^{-tx}=-\log x$$ Formally one ...
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1answer
56 views

Problem with momentum operator

Why is there no problem with the eigenfunction of the momentum operator being non-normalisable? How can it be a valid quantum state?
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0answers
48 views

A special path integral

May be $f(\vec{x}), \vec{g}(\vec{x})$ an arbitrary functions dependent on the coordinates $\vec{x}=(x,y,z)^T$. Defining the following function dependent on a 3-dimensional curve $\vec{\gamma(t)}$ ...
5
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1answer
62 views

Equivalence classes of mappings from $T^{2}$ to an arbitrary space $X$

I was reading the paper "Homotopy and quantization in condensed matter physics", by J.E Avron et al. ( http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.51). There they have classified the ...
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2answers
71 views

Distributions (generalized functions) over manifolds

I have asked a similar question on the math stackexchange website, but since this type of question might have an answer that is known to physicists better than mathematicians I'm posting the question ...
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1answer
74 views

Is my Summary of a Spinor Bundle Associated with a String Worldsheet Correct?

I've been having difficulty finding a source that lists all the properties of the spinor bundle of a string worldsheet explicitly, so I've had a go at creating my own description. I'd really ...
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0answers
44 views

Reference frames are frame fields on spacetime?

The idea of a reference frame as discussed in this question is that of a viewpoint. So that we have some phenomenon, we want to describe be able to predict things and we must specify the viewpoint ...
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2answers
61 views

When generalizing from discrete (but infinite) eigenstates to continuous eigenstates, Why do we change the definition?

The propagator function for discrete eigenstates is $$u(t)=\sum_{n=1}^{\infty}|E_n\rangle\langle E_n|e^{-iE_nt/ \hbar } \tag{1}\ .$$ But when we have continuous eigenstates, (like for the case of ...
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1answer
38 views

Reference request for supersymmetric localization

I would like to ask for some readable introduction or maybe review of the technique of supersymmetric localization for $\mathcal{N}=1,2$ SUSY theories. I would like a different one than the one people ...
2
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3answers
125 views

Expression of density operator

States in Quantum Mechanics can be thought of as density operators, i.e., positive semi-definite, normalized trace class operators on a Hilbert Space $\mathcal{H}$. In the case ...
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0answers
10 views

Is there any relation between single crystal elastic constants and plastic deformation?

The elastic constants are the tensor coefficient in Hook's Law. Which are valued only in elastic region.But, we are using elastic constants to find the hardness.The hardness is the measure of the ...
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3answers
338 views

Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?

I am dissatisfied with the presentation (not to say "definition") of "arc length", in its "Generalization to (pseudo-)Riemannian manifolds", as given in Wikipedia. (Who isn't?. But I'll sketch it here ...
2
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0answers
36 views

Asymptotic Analysis of 1-D Schrödinger Equation [closed]

I'm looking to do a small personal project regarding the time independent Schrödinger equation in 1-D: $$y'' +V(x)y=Ey$$ $$y''=Q(x)y$$ where $ Q(x):=E-V(x) $. There is obviously nothing stopping ...
2
votes
1answer
105 views

Is the Hilbert space spanned by both bound and continuous hydrogen atom eigenfunctions?

As e.g. Griffiths says (p. 103, Introduction to Quantum Mechanics, 2nd ed.), if a spectrum of a linear operator is continuous, the eigenfunctions are not normalizable, therefore it has no ...
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0answers
29 views

Solving inhomogeneous Stokes equation

I want to solve the Stokes inhomogeneous equation, i.e. $$\nabla^2 \vec v -\nabla P = \vec f(r,\theta)$$ $$\nabla\cdot\vec v=0$$ where $\vec f$ is irrotational, i.e. $\partial_y f_x - \partial_x ...
3
votes
2answers
65 views

Symplectic structure and isomorphisms

In his book Mathematical Methods of Classical Mechanics, V.I. Arnold writes To each vector $\xi$, tangent to a symplectic manifold $(M^{2n},\omega^2)$ at the point $\mathbf{x}$, we associate a ...
1
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1answer
37 views

Fractional exponent in a scalar quantum field: Is energy and momentum conserved in this case?

Assuming that I would have the following term in the Lagrangian for a scalar boson field $$L=\int d^4x g (\phi^{2-p} \phi^{\dagger 2+p}+\phi^{\dagger 2-p} \phi^{2+p}))$$ with a fractional number $p$. ...
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votes
1answer
37 views

A conjecture about the Møller operator

Consider the Møller operator $$ \Omega_+ = \lim_{t \rightarrow -\infty } e^{i H t } e^{- i H_0 t } , $$ Now, suppose a state $\psi $ is located far away from the potential $V = H- H_0$. I feel that ...
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0answers
50 views

Does this Hamiltonian have point spectrum?

Consider such a Hamiltonian $$ H = - \frac{1}{2} \frac{\partial^2}{\partial x^2} - F x + V(x) ,$$ with $F$ being some constant, and $V(x)= V(x+L)$ being some periodic potential. Does this ...
2
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0answers
38 views

What is the essential concept behind the difference in the fundamental solutions of the Stokes and Poisson equations?

The fundamental solutions, i.e., the solution with a point source, of the Poisson's equation and the Stokes equations in 3D are: $$\nabla^2 f=\delta(\boldsymbol x) \ \Longrightarrow\ G(\boldsymbol ...
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votes
2answers
157 views

Isomorphism of rigged Hilbert spaces

In connection with the statement that QM can be formulated in terms of separable complex (rigged) Hilbert spaces, the fact that all infinite dimensional separable complex Hilbert spaces are isomorphic ...
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0answers
27 views

Necessity of infinitesimal steps for reversible processes

Is there a mathematical proof for why a reversible process is one that has many infinitesimal equilibrium steps, rather than fewer large steps? Maybe something along the lines of this?: ...
2
votes
2answers
132 views

Wave functions as $x$ goes to infinity

This problem emerged when I was going through some QM exercises: I've been asked to find the commutator $[A,B]$ where $A,B$ are defined as $$A\psi(x)=x\frac{\partial }{\partial x}\psi(x),$$ ...
3
votes
0answers
92 views

Chocolate dynamics

Now I have found a possible model on how to describe chocolate when it is chewed. It has to do with geometrical transformations when a curve $\gamma$ intersects a manifold $M$. The chocolate is ...
0
votes
0answers
13 views

Are centrifugal fans subject to triboelectric effects, and if so can the increase in charge be predicted and measured?

A centrifugal blower is a device that moves gas, usually under power of an electric motor. The blower housing and impeller may be metallic, but often are made of plastic materials. My question is ...
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0answers
49 views

Is it possible to generalize quantum gauge theories? [closed]

I know that there are nonabelian gauge theories and their supersymmetric extensions. Mathematically, gauge theories basing on the fact that one can introduce a fiber bundle with a Connection. From ...
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0answers
36 views

Algebraic number theory and physics [duplicate]

I would like to ask if there are any aspects of algebraic number theory related to physics (for example in string theory or Moonshine etc). I am thinking of attending a course on algebraic number ...
2
votes
0answers
45 views

Rheological behavior of chocolate

If someone eats chocolate, the chocolate goes through the following configurations: $\chi_0:$ chocolate is solid and has a smooth Surface everywhere; the Riemann Tensor vanishes on every Point of the ...
5
votes
1answer
176 views

Noether's Theorem: Lie algebra, Lie groups

I've had a brief look through similar threads on this topic to see if my question has already been answered, but I didn't find quite what I was looking for, perhaps it is because I'm finding it hard ...
3
votes
0answers
64 views

What is a modular tensor category / functor?

I have reads several answers here about this notion, especially regarding topological order, see e.g. this answer, but this notion sounds completely new for me. Also, I found nothing really helpful on ...
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1answer
71 views

How is this a gauge choice mathematically?

I've been reading an article about the "square cat", which is described as the system bellow Such system is a deformable body that can change $a$ and $\theta$ but has $b$ fixed. The article uses ...
0
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1answer
37 views

A series of bound states covering an interval

Generally, the bound states (normalizable eigenvectors) of a Hamiltonian have discrete eigenvalues. Is it possible for the eigenvalues to cover an interval? Say, $(a,b)$? That is, for each $E \in ...
1
vote
1answer
108 views

About the orthogonality of the Hamiltonian eigenstates for the the continuous energy spectrum

I would like first to describe a strange case that I encountered. $ \ \ - $ I solved the Schrodinger equation with a potential barrier (a potential well limited by a finite height wall which decrease ...
2
votes
2answers
123 views

If path integrals aren't well-defined, how can they have any physical meaning?

I am confused about a particular point about the nature of path integration. According to what I've read, what we really mean when we say functional integration is \begin{equation} ...
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0answers
68 views

Non Hermitian Quantum Mechanics

I was just reading about Non-Hermitian Quantum Mechanics dealing with Hamiltonians $H$ that are not Hermitian operators. Then it is unclear that we get orthonormal eigenstates. Now, I was reading a ...
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0answers
36 views

Deriving general boundary conditions from first principles for elastodynamic scattering

It seems that most of the relevant books only give the linear case and the rest say something along the lines of "here are common examples of boundary conditions." What are the most general boundary ...
7
votes
2answers
95 views

Is there a reason why the subset of our Hilbert space that corresponds to a particle is a vector subspace?

I'm trying to gain some intuition behind the definition that states a particle is an irreducible unitary representation of the restricted Poincare group (or more specifically, its double cover). ...
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0answers
46 views

Can we avoid singularities by embedding the manifold on a bigger space, maybe Euclidean or Riemannian?

Can singularities be avoided by embedding Riemannian manifold on bigger space, or more specifically black hole singularities can be avoided or not by embedding in any other manifold. We don't have ...
0
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1answer
49 views

Simulating of Fraunhofer Diffraction of Zigzags by FFT

I tried to study the diffraction pattern of the following zigzag grating by Matlab(FFT of this image).. And the result showed like this(please ignore the scale bar in this img) I think the ...
4
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0answers
65 views

Derivation of the Lippmann-Schwinger equation

I was trying to understand the derivation of the Lippmann-Schwinger equation in Sakurai's Modern Quantum Mechanics, Section 6.1. Our teacher presented a much simpler derivation, similar to that on ...
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1answer
37 views

How does the Hermiticity of an operator imply that functions have an expansion in in multiple bases?

In Shankar QM it is stated that since the $\boldsymbol K$ operator is Hermitian, vectors, which are expanded in the $\boldsymbol X$ basis with components $f(x) = \langle x | f \rangle$, must have an ...
3
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1answer
64 views

Is there a general theorem stating why the restricted Lorentz group's exponential map is surjective?

The exponential map for the restricted Lorentz group is surjective. An outline of why is shown on the wiki page Representation Theory of the Lorentz Group. Is there a more general theorem that states ...
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0answers
38 views

Legendre Transformation for multiple variables

I need to show that for $F(x_1, .., x_n)$, the Legendre transformation is, $$G(s_1, ..., s_n) = \sum_{i}^{N} x_i s_i - F$$ where $$s_i = \frac{\partial F}{\partial x_i}$$ and has the property that ...
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0answers
18 views

QM scattering in a finite-sized box

Background Consider a non-relativistic particle in a one-dimensional box of length $L$ with (for definiteness) an attractive delta function at the origin: $H = \frac{P^2}{2m} -|c|\delta(x), \qquad ...