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0
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1answer
29 views

Realistic interacting QFT construction

May I ask is it true that all the interacting 4 dimension qft couldn't be constructed and defined consistently and rigorously? If we are able to rigorously constructed lower dimension qft, what are ...
3
votes
0answers
40 views

Locomotion of deformable bodies at low Reynolds number

I've been studying some approaches with gauge theory to some problems in Mechanics and I've found the problem of self propulsion at low Reynolds number a quite complicated one. The approach I'm asking ...
5
votes
2answers
463 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
votes
1answer
127 views

How do we know there are dimensions above the third?

Excluding the interpretation of time being the forth dimension, what mathematical evidence is there that suggests it exists? When talking about string theory they say how there is 12 dimensions and I ...
2
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2answers
374 views

p-adic quantum mechanic [closed]

i got a degree on physics so my question is ?as a physicist could i learn P-adic analysis or p-adci quantum mechanics ?? is there any good book on the subject ? as an introductory level How the ...
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0answers
34 views

Role of Hilbert Space in physics [duplicate]

Why in quantum mechanics do we require that the state space be a Hilbert space rather than just an inner product? Does the completeness of the norm have any clear physical role?
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0answers
24 views

p-adic and quantum mechanics [duplicate]

In physics we call integer based quantity as “quantization”, so you have energy, momentum etc taking quantized values. But in math, this kind of situation is really the age old problem of number ...
3
votes
2answers
180 views

What exactly implies the need of quantum mechanics for self-adjoint and not only symmetric operators? [duplicate]

We know that quantum mechanics requires self-adjoint operators, not only symmetric. Can we say that this follows ONLY from the two following axioms of quantum mechanics, namely that each observable ...
2
votes
0answers
36 views

Refined diffusion/heat models [duplicate]

I'm a math student myself, but as is natural I encounter some physics along the way too. My question concerns the heat/diffusion model. As the reader will recall this is that the heat function ...
5
votes
2answers
122 views

Momentum is a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
2
votes
1answer
107 views

About states, observables and the wave functional interpretation in QFT with gauge fields

First of all, I'm a mathematician, so forgive me for my possible trivial mistakes and poor knowledge of physics. In a QFT, we just start with a field (scalar, vectorial, spinorial, gauge etc), so I ...
3
votes
1answer
143 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 ...
4
votes
2answers
50 views

Amplitude-phase decomposition as a canonical transformation

I am studying a classical dynamical system defined on $\mathbb{CP}^2$: the phase space is parametrized in terms of three complex coordinates $\psi_i$ ($i=1,2,3$) and Hamilton's equations of motion ...
3
votes
1answer
69 views

What is the most general definition of a coordinate system?

What is the most general definition of a coordinate system? Specificly: given a suitably general metric space $(\mathcal S, s)$ consisting of a set $\mathcal S$ of elements (for instance: a set ...
38
votes
2answers
994 views

Can we infer the existence of periodic solutions to the three-body problem from numerical evidence?

I recently found out about the discovery of 13 beautiful periodic solutions to the three-body problem, described in the paper Three Classes of Newtonian Three-Body Planar Periodic Orbits. Milovan ...
39
votes
9answers
3k views

Is there something similar to Noether's theorem for discrete symmetries?

Noether's theorem states that, for every continuous symmetry of a system, there exists a conserved quantity, e.g. energy conservation for time invariance, charge conservation for $U(1)$. Is there any ...
0
votes
1answer
65 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 ...
5
votes
5answers
369 views

Infinite series of derivatives of position when starting from rest

Suppose you have an object with zero for the value of all the derivatives of position. In order to get the object moving you would need to increase the value of the velocity from zero to some finite ...
1
vote
1answer
57 views

When is the spectrum of the Hamiltonian (purely) continuous?

Given a quantum hamiltonian $H = \frac{1}{2m}\vec{p}^2 +V(\vec{x})$ in $n$-dimensions, the general rule-of-thumb is that the energy will be discrete for energies $E$ for which $\{ \vec{x} | ...
1
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0answers
39 views

What insights does category theory offer in terms of grand unified theories?

What insights does category theory offer in terms of grand unified theories? Any references to books or papers that give categorical descriptions of any of the common grand unified theories would be ...
0
votes
0answers
22 views

Why must the Gamow state be exponentially increasing?

Why must the Gamow state be exponentially increasing? Why cannot it be exponentially decreasing? The energy is $E- i \Gamma$, but the square root of $E- i \Gamma $ can have either positive or negative ...
3
votes
1answer
35 views

Necessary and sufficient conditions for a function to be the Wigner function of state

For any quantum state defined with a continuous position, the Wigner function is a quasiprobability distribution on phase space. It has many properties, such as that its marginal are probability ...
4
votes
1answer
56 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
56 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 = ...
2
votes
1answer
228 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.
3
votes
1answer
57 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 ...
4
votes
1answer
67 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 ...
8
votes
1answer
347 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
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0answers
36 views

Jets and vertical differential [migrated]

For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $x\in M$ and $u=\phi (x)$. The vertical differential of the section $\phi$ at point $u\in E$ is the map: ...
5
votes
1answer
131 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 ...
6
votes
0answers
86 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 $H$. ...
1
vote
2answers
130 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
votes
0answers
85 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
109 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 ...
1
vote
0answers
32 views

Normal matrix is diagonalizable [migrated]

If $[A,A^*]=0$ ($A^*$ is a conjugate transpose of $A$), that is, $A$ is a normal matrix, How is $A$ diagonalizable? Or, this is just a definition of normal matrix?
3
votes
2answers
172 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 ...
53
votes
14answers
15k 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
57 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 ...
5
votes
1answer
186 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 ...
2
votes
1answer
139 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 ...
3
votes
0answers
83 views

Expansion of gauge potential on infinite dimensional manifold

I'm studying geometrical approaches to locomotion at low Reynolds number by reading the article Geometry of self-propulsion at low Reynolds number by Alfred Shapere and Frank Wilczek and found a ...
15
votes
1answer
497 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 ...
0
votes
0answers
22 views

Existance of observables trace orthogonal to Hamiltonian

I need to understand under which conditions there exists an observable (hermitian matrix) which solves $Tr(B \ U(t,s) \ H_c \ U(s,t)) = 0$ for all $s \in [0,t]$ where $t>0$. I am only interested ...
7
votes
4answers
336 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.
6
votes
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
64 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 ...
1
vote
1answer
83 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
1answer
230 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 ...
0
votes
1answer
173 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 ...
2
votes
0answers
32 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 ...