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5
votes
1answer
336 views

Time-ordered operator in Srednicki

On page 51 Srednicki states, "Note that the operators are in time order...we can insert $T$ without changing anything". This I agree with. But then on the next paragraph he states "The time order ...
0
votes
0answers
40 views

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

What new does geometric quantization or deformation quantization give to physics? For example: prediction new physical phenomena or just better tool for quantization. What can these schemes by ...
0
votes
0answers
19 views

Are $T^2/Z_2$ orbifolds just ironed spheres? [migrated]

The only $Z_2$ symmetries I can think of the torus are reflection on plane, whose quotient should form an open cylinder, and rotation by 180 degrees, which becomes the "branched covering" of the ...
2
votes
1answer
55 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 ...
1
vote
0answers
58 views

Quadratic Casimir Operator of $SO(5)$ [on hold]

In the article A Four Dimensional Generalization of the Quantum Hall Effect, arXiv:cond-mat/0110572, by Zhang and Hu Quadratic Casimir operator for $SO(5)$ is given as $$p^2/2+q^2/2+2p+q .$$ When ...
5
votes
1answer
171 views

Functional derivatives as distributions

I have asked this on math stack exchange, due to its mostly mathemtical content, but aside from one upvote and minimal views it has not garnered any attention, so I am trying here as well. This isn't ...
5
votes
6answers
244 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 ...
3
votes
2answers
117 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 ...
16
votes
9answers
4k views

Crash course on algebraic geometry with view to applications in physics

Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an ...
2
votes
1answer
42 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. $M^n$ ...
15
votes
2answers
323 views

Geometric quantization of identical particles

Background: It is well known that the quantum mechanics of $n$ identical particles living on $\mathbb{R}^3$ can be obtained from the geometric quantization of the cotangent bundle of the manifold ...
4
votes
1answer
99 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? ...
4
votes
1answer
289 views

Resources for theory of distributions (generalized functions) for physicists

I am looking for tutorials, articles or books containing theory of distributions in context of mathematical physics. Please suggest.
22
votes
6answers
3k views

Are there two theories that are mathematically identical but ontologically different?

I have no background in physics but there is a question that has been bothering me, so I'm asking you. Are there at least 2 physical theories that are : Mathematically identical, which means that ...
1
vote
1answer
102 views

Path dependent phase in quantum mechanics

In elementary treatments of quantum mechanics, we are taught that the wavefunction of a single particle is complex valued ($\Psi : \mathbb{R}^3 \to \mathbb{C}$). In particular, the wavefunction has a ...
4
votes
2answers
232 views

Why there is no Gibb's phenomenon in QM?

Why we don't see any Gibb's phenomenon in quantum mechanics? EDIT At sharp edges (discontinuities), we usually find ringing. This can be observed in many physical phenomenon (eg. shock waves). ...
0
votes
1answer
60 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 ...
8
votes
1answer
394 views

Reference Request: Classical Mechanics with Symplectic Reduction

I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie ...
0
votes
2answers
33 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 ...
2
votes
2answers
119 views

Detailed conditions for symmetries of Lagrangian

Edit: To clarify the question, I am asking why we are justified in calling a continuous symmetry a symmetry of a system when it changes the Lagrangian by a total derivative of a function of $t, q(t)$ ...
9
votes
1answer
628 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 ...
6
votes
1answer
267 views

How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group?

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...
2
votes
1answer
193 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 ...
1
vote
1answer
60 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 $U(1)$?
1
vote
1answer
123 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 ...
6
votes
2answers
105 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 ...
0
votes
0answers
29 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 ...
11
votes
1answer
1k views

Rigged Hilbert space and QM

Are there any comprehensive texts that discuss QM using the notion of rigged Hilbert spaces? It would be nice if there were a text that went through the standard QM examples using this structure.
1
vote
1answer
113 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 ...
2
votes
1answer
85 views

How is Green function in many-body theory introduced?

Normally, for a (linear) operator $L$ and a DE $$ Lu(x) = f(x) $$ the Green function is defined as $$ LG(x,s) = \delta(x-s) $$ and it is found that $$ u(x) = \int G(x,s) f(s) ds $$ is the ...
15
votes
1answer
1k views

Motivation for the use of Tsallis entropy

Every now and again I hear something about Tsallis entropy, $$ S_q(\{p_i\}) = \frac{1}{q-1}\left( 1- \sum_i p_i^q \right), \tag{1} $$ and I decided to finally get around to investigating it. I haven't ...
0
votes
0answers
21 views

why is it legal to use “separation of variables” method? [migrated]

heyho, i am using the seperation of variables method for quite a while now, but what was always bothering me a bit, is why is it possible to do those operation. I'll give a concrete example (source ...
0
votes
1answer
225 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 ...
5
votes
2answers
102 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 ...
1
vote
0answers
32 views

quantum determinant calculation [closed]

Heyho, i've got problems understanding a certain calculation of the trace of an operator valued matrix right now. We've got the Matrix $T(\lambda)= \begin{pmatrix} A(\lambda) && B(\lambda) ...
0
votes
2answers
70 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 ...
-6
votes
0answers
48 views

Does Godel's incompleteness theorem deny theory of everything? [duplicate]

Ironically, Hawking's movie is named "Theory of Everything" when in fact he denied TOE based on Godel's incompleteness theorem Some people will be very disappointed if there is not an ultimate ...
3
votes
2answers
155 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 ...
3
votes
1answer
166 views

Is there any physically relevant example of constructing series solution about infinity of an ordinary differential equation?

I was reading about how to test if a given second order ordinary differential equation has singularity at infinity from Arfken and Weber. I understood the steps mathematically but I could not find its ...
1
vote
0answers
48 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 ...
10
votes
10answers
3k views

Physics for mathematicians

How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages ...
3
votes
1answer
83 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 ...
4
votes
0answers
107 views

Why the field strength tensor for locomotion at low Reynolds number may be written like that?

I've been studying locomotion at low Reynolds number for some time now and it has been a quite tough problem. I've already asked two questions about the problem here, and now there is this question ...
0
votes
0answers
39 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 ...
1
vote
0answers
45 views

Is the Cauchy Horizon of Anti deSitter spacetime stable?

The Cosmic Censorship Conjecture are two mathematical conjectures about the structure of spacetime. In particular the so called Strong Cosmic Conjecture asserts heuristically that generically, ...
1
vote
1answer
56 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 ...
3
votes
2answers
908 views

Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
0
votes
0answers
42 views

Extension of vector field and fluid velocity

I've been studying locomotion at low Reynolds number with gauge theories reading this paper and on pages 567 and 568 we find the explanation on how to compute the field strength tensor. For simplicity ...
1
vote
1answer
65 views

What is the required differentiability of the solutions to do QFT?

Given a real scalar field satisfying: $$P\psi=(\square_{g}+m^{2})\psi=0$$ on a globally hyperbolic spacetime ($M,g_{ab}$). One can construct a $C^{*}$-algebra $A(M,g)$ ("the minimal algebra") which ...
2
votes
1answer
61 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 ...