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42 views

Proof of the nonintegrability of the Henon-Heiles problem?

The Henon-Heiles potential is $$ U(x,y ) = \frac{1}{2} (x^2 + y^2 + 2 x^2 y - \frac{2}{3} y^3) .$$ This is a two degree-of-freedom system. The full Hamiltonian is $$ H = p_x^2 + p_y^2 + U(x,y ) ...
2
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1answer
134 views

How to prove that sum converges to integral using density of states?

Essentially, I would like to prove $$ \sum_k f(k) \to \int f(k) \rho dE \tag{1}$$ where $$ \rho = \frac{dk}{dE} \tag{2}$$ is the density of states and $k \to \infty$. The model is that there is a ...
2
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1answer
135 views

Intuition about Momentum Maps

I'm studying Classical Mechanics and there is one object that appeared recently on the book I'm not being able to get a physical intuition about it. The mathematical definition goes as follows: Let ...
0
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2answers
92 views

Examples of measure zero sets in physics [closed]

The problem with Riemann integration of functions like ${\chi }_{ℚ}(x)$ (a function which takes the value of 1 if $x\in ℚ$, and 0 otherwise) is usually used to show the need for defining the Lebesgue ...
0
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1answer
89 views

Lippmann-Schwinger equation and $T$ expansion

Lippmann-Schwinger equation, in operator form, is: $$ T=V+V\frac{1} {E-H_0+i \hbar \varepsilon} T=:V+V\Theta_0T, $$ where $H_{tot}=H_0+{V}$ is the hamiltonian ($H_0$ is the free particle hamiltonian ...
0
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1answer
77 views

How to deal with eigenvectors which are not square integrable?

In Quantum Mechanics there is one type of situation I'm still unsure on how to deal with. First of all, I want to make clear I'm trying to understand how to deal with this rigorously. What I'm talking ...
5
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1answer
88 views

Can we reconstruct 1D potentials in QM from the spectrum? [duplicate]

Knowing the potential, we can find the spectrum of the Schrödinger operator. The converse question is: Knowing the spectrum, can we reconstruct the potential? As an example, a harmonic potential has ...
2
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0answers
65 views

Flow between two infinite plates [closed]

I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks. An incompressible thermal conducting fluid is contained between ...
2
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0answers
57 views

What does complexification mean for our particles in physics

As gauge group let's consider the popular $SO(10)$ group. The fundamental representation $\pi$ of the corresponding Lie algebra $\mathfrak{so}(10)$ is $10$ dimensional $$ \pi: \mathfrak{so}(10) ...
6
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2answers
118 views

Conservation Laws and Symmetry

The toughest of topics in physics, like Quantum Mechanics, Relativity, String theory, can be explained in layman words and many have done so. Though there is no substitute to the understanding a ...
5
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1answer
106 views

State space of interacting theories

Haag's theorem states that in general, an interacting quantum field and the corresponding free field have unitary-inequivalent state space representations. I would like to have an example of a state ...
1
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1answer
207 views

How does height of a parachute affect air resistance compared to circumference or diameter?

I'm trying to find out how much a double in height (making it more ovular or oblong in shape) of the parachute affects air resistance compared to a double in circumference or diameter. Can someone ...
3
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0answers
71 views

Wightman axioms always imply triviality in 4D?

Someone mentioned to me in passing that it had been proven that the Wightman axioms are over-restrictive in four dimensions and provably always result in trivial correlators (or maybe a trivial ...
5
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0answers
109 views

Free probability in Physics

Recently I have started reading some materials on non-commutative probability. IN this area mathematicians sometimes consider quantum theory as a non-commutative version of classical probability, with ...
1
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1answer
101 views

How does one make sense of a delta function of a scalar field?

Disclaimer: Originally posted on math SE, but thought that it was better in physics SE, so deleted my post on math SE and posted here. In the classic review summary of stochastic quantization here, ...
6
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1answer
277 views

Why are the quantum observables defined on opens sets a presheaf and not a sheaf?

In local quantum field theory or AQFT one can mathematically describe over each open set $U$ of a spacetime $M$ the quantum states or observables of the theory. This structure is commonly referred as ...
3
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1answer
65 views

Effective theories and unbounded operators

If you have two operators, one the true Hamiltonian $H$ and one we call an effective Hamiltonian $H_{eff}$ and say they agree on every eigenvector with eigenvalue up to $E_{eff}.$ Above that, they can ...
0
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0answers
163 views

Definition of Hamilton operator

The Hamilton operator is by definition a self-adjoint operator $H\text{: }D\left(H\right)\to\mathcal{H}$ with $D\left(H\right)\subset\mathcal{H}$ a dense linear subspace of the Hilbert space ...
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0answers
40 views

Any good reference on Maslov index (or Morse index)?

Any good reference on Maslov index (or Morse index)? I have some basic knowledge of differential geometry, calculus of variation. So is there any good reference for me?
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0answers
41 views

What is the first topic on mathematics,that an undergraduate physics student must study? [duplicate]

I aspire to become a theoretical physicist. I like to understand about infinite series and so on. Could you please recommend me any book?
3
votes
1answer
221 views

Can operators be argument of Dirac Delta function

In one part of Marc Bee's book on Quasielastic Neutron Scattering, he defines the pair correlation function $$ G(\textbf r,t) = \frac{1}{(2\pi)^3}\int I(\textbf Q,t)\text e^{-i\textbf Q.\textbf r}\ ...
0
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0answers
37 views

Clarification of Type D space-time

I've asked this question on MathOverflow but received no feedback, so I thought I'd try better luck here. I've read the following formulation of the Goldberg-Sachs theorem in Chandrasekhar's ...
0
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0answers
56 views

Period for small oscillations is like simple harmonic motion

In Arnold's book on mechanics there is the following problem: Consider the period of oscillations near a minimum $E_0$ of the potential energy function $U$. Then he says to compute the limit of ...
2
votes
1answer
90 views

Infinite dimensional manifolds in general relativity [closed]

In GR the concept of a manifold is very useful. However, all of these manifolds are of finite dimension. Is it possible to define a manifold with infinite dimension (ie much like Hilbert space in QM) ...
1
vote
1answer
70 views

Can the inverse operator be expressed as a series?

I've seen the claim that a function of an operator can be defined as a series. For example, say $A: H_1 \mapsto H_2$ is an operator. Then $$ e^A \equiv \sum_{n=0}^\infty \frac{A^n}{n!}. \tag{1}$$ In ...
4
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0answers
123 views

The classification of particles or fields in general spacetime- Is it still meaningful to say spin-0, 1/2 ,1 field in general spacetime? [closed]

In 3+1 dim Minkovski spacetime, the classification of particle or field, that is spin-0, 1/2 , 1..., depends on the representation of the universal covering group of $SO(1,3)$, that is $SL(2,C)$. When ...
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0answers
42 views

References on deformation quantization

I'm looking for books or introductory review papers or lecture notes on the topic of deformation quantization. (And preferably, geometric quantization as well.) I'm mainly interested in the ...
1
vote
1answer
173 views

Continuous spectrum of hydrogen atom

I wonder if there is a nice treatment of the continuous spectrum of hydrogen atom in the physics literature--showing how the spectrum decomposition looks and how to derive it.
2
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0answers
121 views

What new does geometric or deformation quantization give to physics? [closed]

What new does geometric quantization or deformation quantization give to physics? For example: prediction of new physical phenomena or just better tool for quantization. What can these schemes do in ...
2
votes
1answer
108 views

Dirac delta function definition in scattering theory

I'm studying scattering theory from Sakurai's book. In the first pages he gets to the following expression: $$\langle n|U_I(t, t_0)|i\rangle=\delta_{ni}-\frac{i}{\hbar}\langle ...
6
votes
6answers
483 views

Why does time evolution operator have the form $U(t) = e^{-itH}$?

Let's denote by $|\psi(t)\rangle$ some wavefunction at time $t$. Then let's define the time evolution operator $U(t_1,t_2)$ through $$ U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$ and ...
2
votes
1answer
91 views

Diagonal part of the configuration space of two indistinguishable quantum particles

Why is the configuration space of two indistinguishable particles given by $\frac{M^n-\Delta}{S_n}$? My question is about the $\Delta$. (Notation: $M$ is the configuration space of 1 particle. $M^n$ ...
3
votes
2answers
153 views

Square root of a matrix appears in massive gravity. How to solve $\sqrt{A+B}$ perturbatively

$A=\text{diag}\{\lambda_1,...,\lambda_n\}$, where $\lambda_i$ can be any number and not necessarily a small number, $\lambda_i>0$, $B$ is a positive definite symmetric matrix, and ...
6
votes
1answer
148 views

Where does the “Supersymmetry” in Witten's proof of the Morse inequalities come from?

Where does the "Supersymmetry" in Witten's proof of the Morse inequalities (original paper and outline of proof for mathematicians) come from? Hopefully someone can provide an intuitive understanding? ...
0
votes
1answer
110 views

How to Derive Atomic Hamiltonian and Cavity Hamiltonian?

In a Fundamental of Quantum Optics and Quantum Information book which I am reading, it states without explanation that, in a two-level atomic configuration in a cavity system, the Atomic ...
0
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2answers
41 views

How to express a convex function of a Hermitian operator in terms of its eigenvalues and eigenvectors?

The Hermitian operator $\hat O$ can be expressed as $$\hat{O}=\sum_i O_i|O_i\rangle\langle O_i|.$$ How to prove that a convex function $f(\hat O)$ can be expressed like $$f (\hat O)=\sum_i ...
1
vote
1answer
99 views

Is EM interpreted in a principal or vector bundle?

I've read in a few places that EM is a $U(1)$-principal bundle; but is this correct? Isn't it rather an associated vector bundle using the adjoint representation of $U(1)$?
0
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0answers
64 views

How does the functional Bethe Ansatz work?

This question is a bit general and less specific. I've understood the algebraic Bethe Ansatz so far, but since we have a completeness problem Sklyanin suggested a different method, the so called ...
0
votes
2answers
95 views

Uniqueness of solution of Laplace's equation

Consider steady-state temperature in a rectangular plate; we get solution when we solve Laplace's equation. In the midway of solving this equation, we take linear combination of basic solutions and ...
1
vote
1answer
221 views

How to calculate the functional derivative of the functional integral?

I study by myself with the QFT, in the page 197 of book of Lewis H. Ryder (2nd edition), The author wrote that he take the functional derivative of equation 6.69: $$\frac ...
12
votes
2answers
314 views

Does geodesic incompleteness in Penrose-Hawking theorems imply curvature blow up?

The singularity theorems in General Realtivity roughly stated say that given: A global causal condition An energy condition The existence of a closed trapped surface then spacetime must be ...
4
votes
2answers
202 views

Is the Legendre transformation a unique choice in analytical mechanics?

Consider a Lagrangian $L(q_i, \dot{q_i}, t) = T - V$, for kinetic energy $T$ and generalized potential $V$, on a set of $n$ independent generalized coordinates $\{q_i\}$. Assuming the system is ...
1
vote
0answers
101 views

The fate of Poincaré recurrence with the Big Rip

Recently, there has been a lot of talk in the media about the "Big Rip". It most certainly resulted from the paper by Marcelo M. Disconzi and Thomas W. Kephart where they have figured out a ...
3
votes
1answer
136 views

Can the math for physics be expressed without any uncountable sets at all?

I am wondering about this and have wondered about it for a short while. Usually physics is modeled using things based off of the Real Number Line $\mathbb{R}$, which is uncountable. (E.g. we may use ...
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0answers
41 views

Is there a physical interpretation of the alternating property?

A map from a vector-space to its base field is called "alternating" if each vector with repeated elements is mapped to zero. I've read that symplectic geometry is an important representation of ...
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0answers
86 views

Does Cauchy horizons in AdS have dual picture in the dual Cft?

The AdS/Cft correspondance has kindle interest in anti-de Sitter and asymptotically AdS spacetimes which are non globally hyperbolic. That means Cauchy horizon forms in these spacetimes. Moreover, ...
1
vote
1answer
77 views

Gauge freedom in tetrad

I asked the question in the MathOverflow, but didn't get any response. I thought maybe better luck here. I'm reading the following paper about Petrov type D space times called "Type D vacuum ...
0
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0answers
33 views

Can anybody recommend a book on quantum physics for a mathematically minded student? [duplicate]

Basically what the title says. I'm a math/physics double major going into my second year, but I'm much more math oriented. I'll be taking a quantum class in the fall and I'd like a book that ...
3
votes
1answer
120 views

Separability of the Hilbert space: countable orthonormal basis vs. continuous spectrum

Hilbert spaces are mostly assumed to be separable. A Hilbert space is separable if and only if it admits a countable orthonormal basis. How does this fit together with the possible existence of the ...
0
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0answers
53 views

Quantized Banach Spaces

Reading the paper here, it mentions on the very first page that "The requirement of 'closed'-ness is imposed because we want to think of operator spaces as 'quantized (or non-commutative) Banach ...