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4
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1answer
63 views

Cayley transform to von Neumann theorem

Self-ajointness of an operator can be found using the Cayley transform of the operator, if its unitary, $$ U = (A - i I)(A + i I)^{-1} $$ From this we can go about finding the deficiency subspaces ...
1
vote
1answer
67 views

What really is the self-adjoint extension?

Going through the Quantum mechanics book by Capri, am time and again held with some stupid doubts on this topic of self-adjointness. We have for the momentum operator in finite domain, $$ p = ...
3
votes
1answer
59 views

Eigenvectors of $p_x$ in a particular domain

Defining the $p_x$ operator for the problem of particle in a infinite well. In the book by Capri on Quantum mechanics, the domain of the operator is given by, $$ p = -i\hbar \frac{\partial ...
5
votes
1answer
81 views

Ground state for interacting field thoeries

Are there references where the ground state of an interacting quantum field theory is explicitly written in terms of states of the underlying free theory? For example, let us suppose to have a self ...
6
votes
0answers
93 views

What is the physical interpretation of the automorphism on bounded operators induced by an S matrix?

In a QFT, the S-matrix $S$ is a unitary operator, that fixes the vacuum and commutes with the unitary operators implementing the action of the Poincare group on an appropriate Hilbert space $H$. ...
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2answers
140 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 ...
0
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0answers
93 views

Integrability in classical mechanics

An integrable system in classical mechanics is defined by action-angle variables and closed loop trajectories in phase space. I have also heard that the flow lines of an integrable system are ...
5
votes
4answers
123 views

Physical intuition on $\mathbf{v}\otimes \mathbf{w}$

On Physics there's one very clear intuition on what a vector $\mathbf{v}$ is: they represent things with direction and magnitude (although when no metric is available there's no clear concept of ...
3
votes
2answers
175 views

Why do some bound states disappear in a discontinuous way?

Generally, we have the picture that as the parameter (say, the depth of a trap) of a system varies, the bound state gets more and more extended and disappears eventually at some critical parameter ...
55
votes
14answers
16k views

Best books for mathematical background?

What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory? Some subjects off the top of my head that probably need covering: ...
2
votes
1answer
61 views

When can an autonomous system be written using a Hamiltonian?

If I have an autonomous series of differential equations $$\tag{1} \frac{dx_i}{dt} ~=~ A_i(x_1,...,x_n)$$ with the condition that $$\tag{2} \sum_{i=1}^n\frac{\partial A_i}{\partial x_i}~=~0$$ in all ...
15
votes
1answer
509 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 ...
7
votes
4answers
374 views

Coadjoint orbits in physics

I am looking for some application of coadjoint orbits in physics. If you know some of them please let me know.
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5answers
1k views

Hydrogen radial wave function infinity at $r=0$

When trying to solve the Schrödinger equation for hydrogen, one usually splits up the wave function into two parts: $$\psi(r,\phi,\theta)= R(r)Y_{l,m}(\phi,\theta).$$ I understand that the radial ...
1
vote
2answers
72 views

Particle in a one dimensional box conditions

Why does the wave function have to be $C^1(\mathbb{R})$ for a finite square well but not for an infinite square well? For an infinite square well with boundaries at $x=0$ and $x=L$, we have ...
2
votes
0answers
39 views

What physical effects cause materialization of a system of particles for a short time?

It is well-known from physics that a photon with enough energy creates a pair of particles: one electron and one positron. This pair of particles can only exist for a short time. This process is ...
20
votes
7answers
1k views

Tensor Operators

Motivation. I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. ...
1
vote
2answers
67 views

Metric and the Lagrangian [duplicate]

Does the Lagrangian formalism require a metric on the configuration manifold $Q$ in order to define a Lagrangian $L$ on the tangent bundle $TQ$? Further, if we specify a metric on the tangent bundle ...
1
vote
1answer
58 views

Product of deltas in kinetic second quantization hamiltonian

I am trying to derive the result for a kinetic hamiltonian in second quantization in term of the fields, that is: $\hat{H} = \int - \Psi^\dagger (r) \frac{\hbar^2\hat{\nabla}^2}{2m} \Psi(r)$ I start ...
4
votes
1answer
84 views

What algebraic structure does the collection of all physical quantities form?

What algebraic structure -- by which I'm referring to abstract algebra theoretic ones such as ring, field, module, etc. -- does the collection of all physical quantities form? An related and/or ...
1
vote
1answer
76 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 ...
1
vote
1answer
52 views

Description of charged sphere with Heaviside function in cylindrical coordinates

I need to describe density of charge of uniformly charged sphere (radius R, total charge Q, position of centre (0,0,0)) with Dirac delta function and Heaviside step function. The hard part is to ...
1
vote
1answer
152 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 ...
9
votes
1answer
165 views

What are the implications for the AdS/Cft program if AdS is unstable?

To my understanding recent progress in the study of the non linear stability of AdS spacetime suggest that $AdS$ might be unstable. If this is true, what are the physical and mathematical ...
3
votes
2answers
385 views

Physical significance of Taylor and Maclaurin series - What is the significance of defining a Maclaurin series in Mathematical Physics?

In physics, usually Taylor series is used to express a quantity which keep changes with coordinate. For example the potential energy of a molecule changes with coordinate, so we express the potential ...
1
vote
2answers
146 views

Good Fiber Bundles and Differential Geometry references 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 ...
1
vote
1answer
142 views

Help needed in finding the integral curves given by orbits of one-parameter groups

Equip $\mathbb{R}^2$ with standard symplectic structure and inner product. Consider a Hamiltonian $$H=(x,y)A(x,y)^t.$$ I have to determine orbits of one-parameter groups acting by isometries of ...
7
votes
5answers
2k views

Laplacian of $1/r^2$ (context: electromagnetism and poisson equation)

We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is ...
5
votes
4answers
1k views

Intuitive meaning of the exponential form of an unitary operator in Quantum Mechanics

I'm an undergraduate student in Chemistry currently studying quantum mechanics and I have a problem with unitary transformations. Here in my book, it is stated that Every unitary operator ...
0
votes
0answers
81 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
votes
1answer
248 views

Percolation and number of phases in the 2D Ising model

Update. As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive. After a long time I came back to try to understand an article on the Ising model. ...
2
votes
1answer
3k views

Plane wave expansion in cylindrical coordinates

I am trying to solve scattering problem in 2D and got to expand the wave function in cylindrical system which comes out to be Hankel function. Can you tell me how to expand the plane wave $\exp(i ...
3
votes
1answer
97 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 ...
0
votes
1answer
62 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 ...
18
votes
5answers
4k views

A pedestrian explanation of conformal blocks

I would be very happy if someone could take a stab at conveying what conformal blocks are and how they are used in conformal field theory (CFT). I'm finally getting the glimmerings of understanding ...
6
votes
2answers
267 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
votes
1answer
70 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 ...
0
votes
1answer
65 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?
0
votes
0answers
57 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
votes
1answer
79 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 ...
2
votes
1answer
142 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 ...
3
votes
2answers
149 views

Symplectic geometry in thermodynamics

There seems to be analogues between Hamiltonian dynamics and thermodynamics given the Legendre transforms between Lagrangian and Hamiltonian functions and all of Maxwell's relations. Poincarè tried to ...
1
vote
1answer
720 views

Proof of equality of the integral and differential form of Maxwell's equation

Just curious, can anyone show how the integral and differential form of Maxwell's equation is equivalent? (While it is conceptually obvious, I am thinking rigorous mathematical proof may be useful in ...
1
vote
1answer
84 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 ...
4
votes
1answer
77 views

Tip of a spreading wave-packet: asymptotics beyond all orders of a saddle point expansion

This is a technical question coming from mapping of an unrelated problem onto dynamics of a non-relativistic massive particle in 1+1 dimensions. This issue is with asymptotics dominated by a term ...
0
votes
2answers
72 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 ...
0
votes
1answer
462 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 ...
0
votes
3answers
357 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 ...
10
votes
5answers
2k views

Hubble's law and conservation of energy

If all distances are constantly increasing, as Hubble's law say, then lots of potential energies of form ~$\frac{1}{r}$ changes, so how is the total energy of the Universe conserved with Hubble's ...
23
votes
6answers
2k views

Is the converse of Noether's first theorem true: Every conservation law has a symmetry?

Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Is the converse true: Any conservation law of a physical ...