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5
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2answers
128 views

What type of non-differentiable continuous paths contribute for the path integral in quantum mechanics

Consider the path integral for a 1D particle subjected to a potential $V(x)$ in imaginary time $$ \int_{x(0)=x_0}^{x(T)=x_T} [dx] \, e^{- \int_0^T d\tau \left[\frac{1}{2}\dot{x}^2 + ...
5
votes
0answers
74 views

Low Reynolds number and Gauge Theory

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 ...
3
votes
1answer
50 views

Rigorous definition of superselection sector/quasiparticle type in anyon systems

The systems I have in mind are for example Kitaev's toric code model (arXiv:quant-ph/9707021) and Kitaev's honeycomb model (arXiv:cond-mat/0506438). What I'm looking for is a mathematically rigorous ...
14
votes
0answers
881 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 ...
6
votes
2answers
236 views

Motivation to introduce von Neumann algebras in addition to $C^*$algebras?

Observables are self-adjoint elements of a $C^*$algebra. As such, this structure seems sufficient to describe physics. A theorem by Gelfand and Naimark says that a $C^*$algebra can always be ...
5
votes
0answers
57 views

Does regularity of distributions have anything to do with definiteness of their product?

Recently I've gone through some literature concerning causal perturbation theory (CPT). As is well known, it deals with UV divergences in QFT by defining products of (operator-valued) distributions ...
1
vote
2answers
33 views

A binary operator required for observing whether the particle is present in a given spatial region

Consider the wave function $\psi(x)$, I want to define an experiment using quantum mechanical rules. The experiment is to find whether the particle is in the region of space (a,b). The observable is ...
0
votes
0answers
10 views

Flow rate Calculation with Pipe Diameter and pressure?

I am using a constant Diameter pipe from a motor to a machine for oil flow.A pressure trasmitter is kept near the machine to know the pressure.So can we calculate the Flow rate using the parameters ...
2
votes
1answer
240 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.
1
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0answers
40 views

Is it possible to express continuous growth without using transcendental numbers? [closed]

Continuous growth is typically expressed using some variant on $A = Pe^{rt}$, I understand where $e$ comes from in general, it is the amount something grows in a given time interval, when continuously ...
1
vote
0answers
22 views
4
votes
1answer
93 views

How does one express a Lagrangian and Action in the language of forms?

In Lipschitzs Classical Mechanics a Lagrangian is defined as: $L(q,q',t)$ for some trajectory $q(t)$ of a particle And the action is defined as: $S:=\int^a_b L(q,q',t) dt$ How does one ...
30
votes
0answers
947 views

Linear sigma models and integrable systems

I'm a mathematician who recently became very interested in questions related to mathematical physics but somehow, I already have difficulties in penetrating the literature... I'd highly appreciate any ...
3
votes
1answer
77 views

What really are perturbation expansions?

I'm unsure if this question belongs here or at Math.SE, but since I've got to it by reading some articles about Physics I'm going to post it here anyway. In this particular article (Theoretical ...
8
votes
1answer
368 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 ...
1
vote
1answer
21 views

Complex scaling method for solving resonance states

I am now reading about the complex scaling method for solving resonance states. As far as I understand, the procedure goes like this: Let us take the 1d potential $V(x) = A e^{-x^2} x^2 $ as an ...
0
votes
0answers
51 views

Does model theory in mathematics have any usefulness in the modeling of physical systems?

The term Model Theory in mathematics seems to have a somewhat precise definition here. Reading through that reference you'll largely see discussions strictly relating to mathematical concepts, but one ...
2
votes
1answer
63 views

Causality and natural modeling of physical systems using integral forms [closed]

I posed a closely related question here but it received a tumbleweeds award. So I thought I would post it from a different angle to see if I can illicit at least some thoughtful comments if not ...
2
votes
1answer
46 views

Standard derivation of Witt algebra

I have been studying Conformal field theory for the past one week from the books by Blumenhagen and Di Francesco etal. If I understand correctly, whenever one talks of 'local (infinitesimal) ...
5
votes
1answer
144 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 ...
2
votes
3answers
170 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 ...
2
votes
1answer
29 views

Uniqueness of expression of a Lie group element

Just take the SU(2) group as an example. The three generators are $J_z$, $J_+$, and $J_-$. For an element $ g $, sometimes we want to express it as $$ g = e^{i a J_+} e^{i b J_z} e^{i c J_-} . $$ ...
0
votes
0answers
104 views

Epstein-Glaser causal perturbation theory

Why does causal perturbation theory in the sense of Epstein Glaser fall under algebraic QFT rather than heuristic QFT in renormalization?
1
vote
0answers
77 views

Divergence Theorem, mathematical approach to Gauss's Law?

Let $D$ be a compact region in $\mathbb{R}^3$ with a smooth boundary $S$. Assume $0 \in \text{Int}(D)$. If an electric charge of magnitude $q$ is placed at $0$, the resulting force field is ...
5
votes
1answer
213 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 ...
6
votes
2answers
126 views

$su(1,1) \cong su(2)$?

The three generators of $su(2)$ satisfy the commutation relations $$ [J_0 , J_\pm] = J_\pm , \quad [J_+, J_- ] = +2J_0 .$$ The three generators of $su(1,1)$ satisfy the commutation relations $$ ...
2
votes
1answer
156 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 ...
6
votes
1answer
241 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 ...
1
vote
1answer
49 views

Boundary conditions of the radial Schrodinger equation

Consider the radial differential equation $$\bigg( - \frac{d^2}{dr^2} + \frac{(\ell+\frac{d-3}{2})(\ell+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_\ell (r) = \lambda\ \phi_\ell (r),$$ which I've ...
0
votes
3answers
69 views

How can I tell if the spectrum of an operator in QM is degenerate?

I know that the collection of all the eigenvalues of an operator $\hat{Q}$ is called its point spectrum, and sometimes two or more linearly independent eigenfunctions share the same eigenvalue, and in ...
2
votes
0answers
30 views

For what values of $\lambda$ is the distribution $(x-i\varepsilon)^\lambda$ positive?

I've been reading the famous unpublished paper by Luescher and Mack "The energy momentum tensor of critical quantum field theories in 1+1 dimensions". In the proof of their main theorem, page 7 of the ...
1
vote
1answer
49 views

Lax representation of the harmonic oscillator

Peter Lax showed that the differential operators $$L=-6\partial_x^2-u,\quad B=-4\partial_x^3-u\partial_x-(1/2)u_x$$ fulfilling the Lax equation $$\dot{L}+[L,B]=0$$ is equivalent to the KdV equation ...
0
votes
1answer
193 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 ...
1
vote
1answer
96 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 ...
5
votes
2answers
477 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
129 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
votes
2answers
379 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 ...
1
vote
0answers
36 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?
-2
votes
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
186 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
132 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
116 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
148 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
54 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 ...
2
votes
1answer
77 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
1k 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
4k 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
73 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
378 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 ...