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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.

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2answers
135 views

Global conformal group in 2D Euclidean space

This is a rather naive question, but I was just wondering. I know that the local conformal algebra of 2d Euclidean space is the direct sum \begin{equation} \cal{L}_0\oplus\overline{\cal{L}_0}, \end{...
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0answers
46 views

When do the solutions of combinatorial Dyson-Schwinger equations generate a Hopf subalgebra?

Say I have a set of combinatorial Dyson-Schwinger equations of the form $$\begin{align} X_1 &= \mathbb{1} + \alpha B_+^a (f_1(X_1,...X_N)) \\ & ... \tag{1} \\ X_N &= \mathbb{1} + \alpha ...
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0answers
36 views

Why shouldn't I choose my boundary limits corresponding to the direction I'm integrating?

I have a question regarding the choice of boundary limits when it comes to vector integrals. Why shouldn't I always choose the boundary limits corresponding to the direction I'm integrating. I.e why ...
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0answers
19 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 ...
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3answers
117 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 ...
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0answers
64 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 ...
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4answers
3k views

Gelfand-Yaglom theorem for functional determinants

What is the 'Gelfand-Yaglom' Theorem? I have heard that it is used to calculate Functional determinants by solving an initial value problem of the form $Hy(x)-zy(x)=0$ with $y(0)=0$ and $y'(0)=1$. ...
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1answer
79 views

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

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
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1answer
94 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 ...
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2answers
155 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 ...
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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 ...
2
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1answer
95 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}) ...
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0answers
326 views

Minimal strings and topological strings

In http://arxiv.org/abs/hep-th/0206255 Dijkgraaf and Vafa showed that the closed string partition function of the topological B-model on a Calabi-Yau of the form $uv-H(x,y)=0$ coincides with the free ...
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0answers
160 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 $\...
5
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1answer
403 views

Schwartz's and Zee's proof of Goldstone theorem

In Refs. 1 & 2 the Goldstone theorem is proven with a rather short proof which I paraphrase as follows. Proof: Let $Q$ be a generator of the symmetry. Then $[H, Q] = 0$ and we want to consider ...
3
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2answers
63 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_\...
6
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1answer
743 views

SUSY QM and Atiyah-Singer index theorem

Consider maps $t\mapsto x^i(t)$ from circle to some Riemannian (spin) manifold and Lagrangian $$ \mathcal L = \frac12 g_{ij}(x) \partial_t x^i \partial_t x^j + \frac12 g_{ij} \psi^j \left(\delta^i_k \...
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0answers
45 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
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1answer
48 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 ...
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2answers
94 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 ...
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0answers
40 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 ...
12
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1answer
952 views

Classical mechanics: Generating function of lagrangian submanifold

I have a short question regarding the geometrical interpretation of the Hamilton-Jacobi-equation. One has the geometric version of $H \circ dS = E$ as an lagrangian submanifold $L=im(dS)$, which is ...
18
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3answers
4k views

Symmetrizing the Canonical Energy-Momentum Tensor

The Canonical energy momentum tensor is given by $$T_{\mu\nu} = \frac{\partial {\cal L}}{\partial (\partial^\mu \phi_s)} \partial_\nu \phi_s - g_{\mu\nu} {\cal L}. $$ A priori, there is no reason to ...
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6answers
1k views

Which QFTs were rigorously constructed?

Which QFTs have mathematically rigorous constructions a la AQFT? I understand there are many such constructions in 2D, in particular 2D CFT has been extensively studied mathematically. But even in 2D ...
13
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1answer
2k views

Understanding and deriving ellipsoidal coordinates geometrically

If one were to read old texts on mathematical physics, like Maxwell, Morse & Feshbach, Hilbert and Courant, Jacobi, etc... they'd find ellipsoidal coordinates popping up, but the authors derive ...
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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 ...
49
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2answers
2k 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 ...
9
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2answers
220 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 ...
16
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3answers
2k views

Global vs. local gauge group in mathematical sense - physics examples?

Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group: Local gauge group $G$. Corresponds to the fibers of the $G$-...
0
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0answers
52 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 ...
8
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2answers
283 views

Calculating the potential on a surface from the potential on another surface

The question is short: If a charge (or mass) distribution $\rho$ is enclosed by surface $S_1$, I can calculate the electrostatic (or gravitational) potential on that surface by integrating $\rho(r') \ ...
10
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2answers
752 views

Spin-Statistics Theorem (SST)

Please can you help me understand the Spin-Statistics Theorem (SST)? How can I prove it from a QFT point of view? How rigorous one can get? Pauli's proof is in the case of non-interacting fields, how ...
0
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0answers
113 views

Does anyone know how to symmetrize $\gamma$-matrices?

I'm trying to construct the SO(5, 5) $\gamma$-matrices which are real and symmetric. Recently, I have 6 symmetric and 4 antisymmetric $\gamma$-matrices ($6_S + 4_A$ representation). How can I ...
3
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2answers
168 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
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0answers
44 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
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0answers
49 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 ...
2
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0answers
37 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....
5
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1answer
434 views

Operator product expansion (of free field theory) in functional quantization approach

(Note: Those who are already familiar with the notion of OPE's might prefer to skip directly to the Question section below) Setup: Consider, for simplicity, the quantum theory of the Euclidean K-G ...
2
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2answers
184 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}\...
16
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4answers
841 views

Can physics get rid of the continuum?

Almost every physical equation I can think of (even though I don't actually feel comfortable beyond the scope of classical mechanics and macroscopic thermodynamics, as that's enough for dealing with ...
2
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2answers
117 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
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2answers
93 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
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1answer
54 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
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0answers
46 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
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1answer
74 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 ...
13
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5answers
617 views

Why must the field equations be differential?

In Landau–Lifshitz's Course of Theoretical Physics, Vol. 2 (‘Classical Fields Theory’), Ch. IV, § 27, there is an explanation why the field equations should be linear differential equations. It goes ...
1
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0answers
112 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)=...
2
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1answer
79 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 ...
7
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1answer
206 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 ...
1
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1answer
70 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 ...