Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [mathematical-physics]

DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com. Mathematical physics is the mathematically rigorous study of the foundations of physics, and the application of advanced mathematical methods to problems in physics. Examples include partial differential equations (PDEs), variational calculus, functional analysis, and potential theory.

0
votes
0answers
17 views

Optimising equation with 3 or more changable variables

In order not to bother you with technical details I've laid my therms in plain math. The sets are experimentally obtained mechanical characteristics, and unknowns are some empirical parameters from ...
1
vote
0answers
45 views

Wave function as a section of a complex line bundle to do QM in polar coordinates

If you want to change the coordinates of a Wave function $\Psi$ in 2D QM from cartesian to polar coordinates in a naive way one encounters a problem, namely the (naively defined) radial momentum ...
0
votes
0answers
16 views

How boundary conditions for a green function is same a original function? [closed]

I am trying to study green's function.suppose we have a differential equation, $D_x f(x) =g(x)$ , We also provided with some boundary conditions for $f(x)$, But we supply this boundary conditions in ...
0
votes
1answer
76 views

If a team of Engineers Made a Laser From Scratch When Would They Need to Use the Schrodinger Equation? [on hold]

I know the Schrodinger Equation is a key part of quantum mechanics. I am trying to understand it’s applications. Let’s say a team of engineers wants to build a laser from scratch, assuming they have ...
2
votes
0answers
26 views

Counting unitary transformations in $SU(N)$ [migrated]

I'm referring to the following article 1, in particular to section 6. The goal is to estimate the number of unitary transformations in $SU(N)$, identifying unitaries within balls of radius $\epsilon$....
2
votes
1answer
81 views

How can tempered distributions be paths?

I'm reading the Appendix A of Glimm and Jaffe book "Quantum Physics: a functional integral point of view", and there is something that I'm missing In section A.4 the authors talk in a very general ...
0
votes
2answers
26 views

Angular velocity is $\dot{g}$ carried to the identity element of the group

I was reading the example below from Arnolds book I can't really understand why the angular velocity is $\dot{g}$ carried to the identity element of the group. I would appreciate if someone who ...
1
vote
2answers
143 views

Mathematical rigorous definition for an electrical dipole

I've been reading Laurent Schwartz's Mathematics for the physical sciences, and in the chapter on distributions he makes many cool examples of ways to define in a mathematical rigorous way certain ...
1
vote
1answer
305 views

Srednicki Eqs. (6.22) and (9.6). How to get rid of $i\epsilon$ in the interaction term?

I'm studying qft from Srednicki's book. If one writes down the full $i\epsilon$ terms, passing from Eq. (6.21) (non-perturbative definition) to Eq. (6.22) (perturbative definition) yields $$\left<0|...
2
votes
1answer
73 views

Lefschetz and Witten indices$.$

I couldn't help but notice a formal similarity between the Lefschetz index $$ \mathrm{ind}(f)=\sum_k (-1)^k\operatorname{tr}(f_*|H_k) $$ and the Witten index $$ Z=\operatorname{tr}((-1)^Fe^{-\beta H}) ...
3
votes
2answers
51 views

Why do we choose the operator or supremum norm while proving unboundedness of the momentum operator?

In most sources, I've noticed that while proving the unboundedness of the momentum operator $\left(-i\hbar \frac{\partial}{\partial x}\right)$ the operator norm (or supremum norm) $\lVert\ .\rVert_\...
1
vote
0answers
41 views

How shall we show the surface integral approaches a limit (or does not blow up) at a field point near $S'$

Consider the electric field due to volume charge distribution in volume $V'$: $\mathbf{E}=\displaystyle \int_{V'} \rho' \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV'$ The integrand ...
1
vote
1answer
42 views

Discretization of path integral and linear interpolations

Consider the evaluation by discretization of the path integral $$\int e^{iS[x(t)]}\mathfrak{D}x(t),\quad S[x(t)]=\int_{t}^{t'}\left[\frac{m}{2}\dot{x}(\tau)^2-V(x(\tau))\right]d\tau.$$ One ...
4
votes
0answers
66 views

How are local observables encoded in this formulation of quantum field theory as a functor?

I've recently begun trying to understand a formulation of quantum field theory as a functor from a category of spacetimes-with-boundaries (bordisms) to a category of Hilbert spaces, as reviewed in [1]....
4
votes
0answers
122 views

Legal values of quantum field can take? $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

Main issue: What are the legal and possible values of the quantum field can take? Clarify by examples: (1) For example, for the spin-0 Klein Gordon field $\phi$, we may choose it to be: real $\...
0
votes
0answers
37 views

What, if any, is the relation between the Lie derivative and the Legendre transform?

I don't remember where I read it, or even if my memory serves me correctly, but I think that I read somewhere that the Lie derivative amounts to a Legendre transform. Is this true and, if it is, what ...
1
vote
0answers
21 views

Arranging objects by mass [closed]

I have n objects with different masses . Say mass of n1=m1, n2=m2 .... so on. I want to use concept of centrality in physics to arrange these objects in a field (computer simulation ) . Where object ...
2
votes
0answers
56 views

What justifies passing the limit to the exponent in the derivation of the path integral?

In the usual derivation of the path integral there is one strange passage. Taking Weinberg's derivation for instance from his QFT book, chapter 9, we have the following equation $$\langle q';t'|q;t\...
9
votes
2answers
199 views

Spontaneous symmetry breaking: proving the equivalence of two definitions

This question can be posed for both quantum and classical set-ups. For concreteness, let me consider a local, classical Hamiltonian $H$. The expectation values I consider are with respect to the usual ...
3
votes
3answers
118 views

Rigorous procedure of gluing together two spacetimes

There seems to exist a procedure of "gluing two spacetimes together". In particular I've seem this mentioned in the context of gravitational collapse. The examples I've seem where that of gluing ...
0
votes
1answer
93 views

Finding the maximum value of electric field

Suppose you have a surface of finite area with a fixed surface charge distribution. Does a maximum electric field magnitude $|\vec{E}|_{max}$ exist for each and every possible surface area? If yes, ...
2
votes
0answers
28 views

Can an arbitrary spin state be written uniquely in a Dicke state basis?

Consider a system of e.g. $N=3$ spin-1/2 particles. The state of the system $\vert\psi\rangle$ lives in a Hilbert space of dimension $2^N=8$. Now, consider the collective spin operator $$\mathbf{J} = ...
1
vote
0answers
46 views

Source suggestions and research topics about Mathematical foundations of QFT [closed]

i am really interested in the mathematical foundations of quantum field theory and i want to write my master thesis on some topic of this subject. However, i did a little research and found that ...
1
vote
0answers
29 views

List of Replica Symmetry results for different models?

Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have? I am aware of some of the more famous results, e....
1
vote
2answers
89 views

Value of a discontinuous function at the discontinuity

Although this is a maths related question, it is important that the answer physically makes sense, so I'm posting it here. (Btw. the problem is related to stochastic thermodynamics, and I'm using ...
3
votes
2answers
137 views

Rigorously why there should be an operator product expansion in conformal field theory?

This is probably something quite trivial I'm not getting. I'm studying CFT (conformal field theory) through David Tong's lecture notes and on page 9 he says: We now define the operator product ...
2
votes
2answers
91 views

Infinite sum: Renormalisation

Trying to do the calculation made in a physics article Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling (page 10 to go from equation 56 to 57), I ...
1
vote
1answer
40 views

Physical interpretation of Dirichlet energy for a membrane

In the following model of a membrane with a mass particle in it, why does the integral represents the elastic energy of the system? Let $\Omega$ be an open connected region (the membrane) in $\Re^2$,$...
0
votes
0answers
28 views

What is meant by surface divergence of a vector function?

My book says: If there is a surface discontinuity in a vector field $\vec{E}$, we enclose it in a thin transitional layer (of width $h$) and apply divergence theorem. If $\hat{n}_1$ and $\hat{n}_2$ ...
1
vote
1answer
56 views

What are Grassmann numbers in field theory?

I've been struggling with the use of Grassmann numbers in QFT e.g. Peskin and Schroeder. They are introduced as "numbers" whose product is antisymmetric, and associative (this isn't said, but used in ...
2
votes
2answers
113 views

Does it make sense to speak in a total derivative of a functional? Part III

In this third part of the series, I will continue the deduction of Noether's theorem initiated in the previous post - Does it make sense to speak in a total derivative of a functional? Part II. ...
2
votes
2answers
177 views

Does it make sense to speak in a total derivative of a functional? Part II

I am trying to derive the Noether theorem from the following integral action: \begin{equation} S=\int_{\mathbb{\Omega}}d^{D}x~\mathcal{L}\left( \phi_{r},\partial_{\nu}% \phi_{r},x\right) , \tag{II.1}\...
2
votes
1answer
70 views

Defining a metric on the space of paths

Imagine the following path integral $$\int_{x(0)=x_i}^{x(T)=x_f} \mathcal{D}x \, e^{\frac{i}{\hbar}S[x]}.$$ This integral is defined over the space of all paths that satisfy the boundary conditions ...
1
vote
0answers
102 views

Transient solution system of differential equations obtained from master equation

I have to solve the following equation (or at least obtain an approximate estimate) for the diagonal terms of the density matrix. We consider that the initial state is a coherent state $\rho_{n,n}(0)=...
1
vote
1answer
52 views

Domain of the infinite square well hamiltonian

I am reading the book by Gitman et al. 'self-adjoint extensions in quantum mechanics'. In the book, they give a precise definition of the domain of the hamiltonian of an infinite square well. For ...
1
vote
1answer
78 views

Dirac delta function mathematical expression proof

In a discussion of the second order transitions in graphene this mathematical expression is used. $$ \left|\frac{1}{\varepsilon + i \Gamma/2}\right|^2 = \frac{2\pi}{\Gamma}\delta(\epsilon) $$ And I'm ...
7
votes
1answer
176 views

What Lie group structure is used for infinite-dimensional Unitary Groups in Quantum Mechanics?

Given an infinite-dimensional Hilbert space $H$, the set $U(H)$ of all unitary operators on $H$ forms a group, known as the unitary group. Now several subgroups of this group play an important role ...
0
votes
0answers
12 views
1
vote
0answers
56 views

Is there a useful relationship between connection on space coordinates and material derivative?

I am referring to an important part of the question Relationship between Connection and Material Derivative. Here is a paste and cut of the relevant part. That is the directional derivative along $...
2
votes
1answer
107 views

What is a coordinate-free version of Noethers theorem? [closed]

What are some examples and derivations of some basic symmetries (not coordinate symmetries)? For example I remember a sufficient condition for being a symmetry of the lagrangian system is being an ...
1
vote
0answers
49 views

Can real numbers dimensions exist?

This title may not explain my question right but I could not think of any better short explanation. My question is, if there is a possibility of a structure (or space) with the dimension $Dim = 3,5$, $...
2
votes
0answers
66 views

Complete positivity with infinite dimensional “auxillary spaces” [closed]

The usual definition of complete positivity (e.g. Stinespring (1955)) is that a linear map between the bounded operators on some Hilbert spaces $\phi:\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{K})...
0
votes
0answers
41 views

How to prove that the limit of surface integral exists?

My book "Electromagnetic Fields" says in $\text{Section}\ 3.4$: Question Why does the limit in equation $(3.42)$ exist (convergent)? Why is the contribution from $(S-S_{\delta})$ remains ...
0
votes
0answers
36 views

Existence of a strictly a timelike curve on Lorentzian manifold

I encountered online the following exercise: Let $M$ be a Lorentz manifold of $\dim(M)=n$, and let $\psi:\Sigma\to M$ be a spacelike submanifold of dimension $n-2$ embedded into $M$. We will ...
0
votes
3answers
86 views

Calculus of Variations commutes with Integrals

I have a question about the variational calculus. Assume a function $q(t,x)$ gives rise for another function $$f(x) := \int dt q(t,x)$$ My question is why the variation $\delta$ commutes with the ...
3
votes
1answer
91 views

Functional derivative commutes with total derivative

I have a question about a rule from the calculus of variations. Assume we consider the space of differentiable functions on $C^1(\mathbb{R})$ (or for the sake of simplicity the smooth functions $C^{\...
5
votes
2answers
1k views

Facing a paradox: Earnshaw's theorem in one dimension

Consider a one-dimensional situation on a straight line (say, $x$-axis). Let a charge of magnitude $q$ be located at $x=x_0$, the potential satisfies the Poisson's equation $$\frac{d^2V}{dx^2}=-\frac{\...
1
vote
2answers
57 views

Exact solution for the perturbation of the inverse metric

So when we usually linearize general relativity with respect to metric perturbations $g_{\mu\nu}\rightarrow g_{\mu\nu}+h_{\mu\nu}$, we compute the correction to the inverse of the metric to first ...
1
vote
0answers
38 views

Size of quantum corrections at infinity

Suppose we have a one dimensional field theory for the field $\phi(r)\;r\in[0,\infty]$ and that the solution for the background (Euler Lagrange equations) give a function $\phi_0$ that goes to a ...
0
votes
0answers
51 views

Piecewise solution to Euler-Lagrange equations in effective field theory

I would like to consider a background for a quantum field theory made up by connecting continuously two different solutions of the Euler Lagrange equations. The problem is one dimensional (let's call ...