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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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conventional matrix notation for distance interval

Why matrix notation for distance interval is represented by this? $$g_{\mu \nu}\Delta X^{\mu}\Delta X^{\nu}=(\Delta X)^Tg (\Delta X)=\Delta X^{\mu}\Delta X^{\nu}g_{\mu \nu}$$ Could you explain ...
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40 views

What is the meaning of “representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space”?

What is the meaning of "representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space"?
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1answer
49 views

Post-measurement density matrix derivation

This is something standard, by I'm trying to redo this with spectral theory. Suppose we start with the usual postulates of quantum mechanics: States are unit rays on a separable Hilbert space. In ...
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1answer
43 views

Position of a particle sliding down an arbitrary curve as a function of time

Given a curve in a frictionless environment with parameterization $\displaystyle \mathbf{r}(\theta)=x(\theta)\hat{\mathbf{i}}+y(\theta)\hat{\mathbf{j}}$ for $\theta\in[0,\theta_f]$, how can I find the ...
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1answer
69 views

Utility of the time-ordered exponential

Is the time-ordered exponential $$ \mathcal{T}\exp\left\{-i\int_{t_0}^tdt'V(t')\right\}\tag{1} $$ just a mnemonic device for the series $$ \begin{aligned} 1 + (-i)\int_{t_0}^tdt_1 \, V(t_1) +{} &...
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47 views

The role of schemes in physics

Let $X$ be a topological space, and $\mathcal O_X$ a sheaf of rings on $X$, yielding a ringed space $(X,\mathcal O_X)$. We say that $(X,\mathcal O_X)$ is a scheme if it is locally isomorphic to $\...
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1answer
152 views

Proof $\exp(-\beta H)$ trace-class operator

Let $H=\frac{p^2}{2}+\frac{x^2}{2}\, : D(H) \to L^2(\mathbb{R})$ be the Hamiltonian of the harmonic oscillator with $m=\hbar=\omega=1$. Prove that $\exp(-\beta H)$ is a trace-class operator if $\beta&...
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1answer
66 views

Trace over configuration basis

Let us take a many-body quantum system, whose phases in the configuration basis are labeled by $\mathbf {\hat q}=(q_1,\cdots, q_N)$ and momenta $\mathbf {\hat p}=\left(-i\frac{\partial}{\partial \hat ...
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Distinct choice of partition in the Path Integral

Practically all books in Quantum Mechanics and Quantum Field Theory define the non-relativistic path integral by taking one interval $[a,b]$ and breaking it up into $N$ subintervals of equal lenght. ...
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51 views

Pictures of Different Coordinate Systems in General Relativity

In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to ...
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1answer
64 views

2D global conformal transformations and the $z= \frac1w$ argument

For instance in Blumenhagen's CFT, there is a standard argument which determines that globally defined conformal transformations on the Riemann sphere where $$l_n = -z^{n+1} \partial_z$$ is an ...
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3answers
116 views

Domains of $H$ and $U(t) = \exp(-iH t )$

I am not so familiar with functional analysis. But in my impression, the Hamiltonian $H$ is often not defined everywhere on the Hilbert space. On the other hand, the time evolution operator $U(t)$, ...
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1answer
52 views

Sound wave equation: Neumann boundary conditions

In this paper it's described the solution of the damped wave equation in cylindrical coordinates $$ \nabla^2\left(c^2\rho_1+\nu\frac{\partial\rho_1}{\partial t}\right)-\frac{\partial^2\rho_1}{\...
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0answers
50 views

Motivating the Unintuitive Properties of Spinors

In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
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28 views

self-adjoint extension of the momentum operator in an infinitely deep potential

Theta parameter arises when calculating self adjoint extensions of the momentum operator of a particle in an infinnitely deep potential, what does this means physically?
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0answers
31 views

What is the magic behind Sector Decomposition?

I have a question regarding Sector Decomposition, which is briefly introduced in this paper arXiv: 0803.4177. So far I played around with a toy example and applied the Sector Decomposition method to ...
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2answers
146 views

Is this actually the rigorous definition of the path integral in Quantum Mechanics?

Let a quantum system with a single degree of freedom be given. We want to define the path integral so that we get the representation for the propagator as $$\langle q' |e^{-iHT}|q\rangle=\int_{x(a)=...
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1answer
71 views

Physical significance of no self-adjoint momentum operator on half line?

I am watching a quantum mechanics lecture by professor Schuller. He mentioned that there does not exist any self-adjoint momentum operator defined on the half line. What is the physical significance ...
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1answer
77 views

Is this a “good enough” statement of Wigner's theorem from Quantum Mechanics?

I posted this on math StackExchange and got no replies, so I'm trying my luck here! I'm a fourth year physics and math student who is writing up a report on some quantum mechanical symmetries and ...
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1answer
66 views

Domain of symmetric momentum operator vs self-adjoint momentum operator

Is there an example of a function that is not in the domain of the 'naive' symmetric (but not self-adjoint) momentum operator $p:=-i\frac{d}{dx}$ but is in the 'true' self-adjoint momentum operator $p:...
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34 views

Will the expressions for magnetic vector potential and Biot-Savart law hold at points on line current?

If we have a line current, will magnetic field be defined at points on the line? Will the expressions for: $(1)$ Magnetic vector potential $$\vec{A}=k\int^l_0 \dfrac{\vec{dl}}{r}+\nabla f(x,y,z)$$ $(...
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Advection diffusion equation in the limit of fast exchange

Consider the following advection diffusion equation for some material with concentration $c(x,t)$ $$\frac{\partial c(x,t)}{\partial t}=v \frac{\partial c(x,t)}{\partial x}+D \frac{\partial^2 c(x,t)}{\...
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1answer
98 views

Question about characteristics and classification of second-order PDEs

I know this question is quite maths-focused; however, it relates closely to numerical methods that are used to solve Physics problems (for example in Fluid Dynamics/CFD). I asked the same question ...
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1answer
62 views

Joint Spectral Measure theorem

I want to gain an intuition to understand the joint spectral measure theorem. In the case that operators involved in this theorem have purely discrete spectrum, the theorem should be reduced to the ...
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2answers
182 views

Weyl anomaly in 2d CFT (string theory lectures by D.Tong)

In his lectures on String Theory (http://www.damtp.cam.ac.uk/user/tong/string.html), Tong gives a proof of the Weyl anomaly, using equation $(4.36)$. It seems wrong to me. Here he uses the OPE ...
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Notation for Young tableau in Hubsch book

I'm looking at the notation given for Young tableau in Hubsch's book "Calabi-Yau Manifolds: A Bestiary for Physicists" - see Chapter 3, p. 96. A Young tableau (for a $U(n)$ representation) is denoted ...
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1answer
70 views

Chern-Simons and framing dependence$.$

According to ref.1, the correlation functions of a Chern-Simons theory are topological invariants, up to the so-called framing, that is, the trivialisation of $TM\oplus TM$. The origin of this framing ...
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1answer
36 views

Finding the unit of Pressure/Volume

Me and my friend are trying to find the exact unit of Power=Pressure/Volume. We know that the unit has to be W(J/s) so we try from the equation above: (Kgm²/s³)=(N/A)/V The right side will be: (Kgm/s²)...
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1answer
32 views

Inverse second derivative of a Legendre transformation

I'm trying to find the legendre transformation of $$ f(x)=x^3 $$ I have calculated it using the approach we learned in class: 1 - Find the derivative of function => $y(x) = f'(x)$ 2 - Take ...
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1answer
68 views

Taylor Expansion of a Vector Field in Stokes Flow About a Sphere's Surface

So I am going through what could be a pretty simple identity: $\frac{1}{4 \pi a^2} \int_S \vec{u}(\vec{x}) dS=\vec{u}(\vec{0})+\frac{a^2}{6} \nabla^2\vec{u}(\vec{x})|_{\vec{x}=\vec{0}}$ where S is ...
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1answer
41 views

Will the flux through an arbitrary closed surface be finite or infinite when a plane charge intersects the Gaussian surface?

Let's consider a closed Gaussian surface (in red). The white line and the white shaded part lies inside the Gaussian surface and the black line and the portion above it lies outside the Gaussian ...
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1answer
58 views

Formal Connection Between Symmetry and Gauss's Law

In the standard undergraduate treatment of E&M, Gauss's Law is loosely stated as "the electric flux through a closed surface is proportional to the enclosed charge". Equivalently, in differential ...
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0answers
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Extensors in mathematics and in physics [closed]

Could someone explain in a simple but accurate manner what extensors are as mathematical entities and how they are used? How do extensors essentially differ from tensors? Are there or could there be ...
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0answers
38 views

How to make sense of these zero frequency creation operators and corresponding states?

In quantum field theory a soft particle is a low energy particle. If I'm not mistaken one introduces a energy treshold $\Lambda$ and calls soft any particle with energy $\omega < \Lambda$. If $a^\...
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1answer
41 views

Adding solutions to get a new solution?

Recently I've stumbled upon the same question as here: What happens to the position function when an oscillator is overdamped and does not have angular frequency? And the answer that I preferred (the ...
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0answers
44 views

Can you do gauge theories over topological groups?

Quantum gauge theories involve (functional) integration over a Lie group. Is there any meaningful generalisation to (non-manifold) topological groups? Consider for example the Whitehead tower $$ \...
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1answer
85 views

Proving a Mathematical hypothesis using Physics [closed]

I've asked the question below on mathexchange here about 2 weeks ago. while I did not satisfied with the comments and answer there specially because the lack of examples and references that I was ...
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1answer
184 views

Is there a function which is square integrable and doesn't tend to zero at infinity but it belongs in the domain of the momentum operator?

Is there a function which is square integrable and doesn't tend to zero at infinity but it belongs in the domain of the momentum operator? There are some counterexample for functions that are square-...
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2answers
37 views

Confusion in calculating electric field due to infinite plane

(I) Electric field at a point on positive $x$-axis: Let us consider Cartesian coordinate system with infinitely large circular plane at $y$-$z$ plane. Let $P$ be any point where we want to measure $\...
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1answer
29 views

Additive constant in Hamilton-Jacobi theory?

In Hamilton-Jacobi theory Hamilton's principal function S is a function of n+1 constants , But we take one of the n+1 constants as an additive constant . I don't get this step?
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0answers
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Have fractional order differential models been explored as an alternative to standard gravitational field theory?

Since Einstein introduced his field equations and general theory of relativity, experimental evidence, at least on the cosmic scale has repeatedly supported the theory. Nevertheless, many seeking to ...
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1answer
68 views

Auxiliary Grassmann variables in supergeometry

I was reading on super geometry and how it is used to model fermions and supersymmetry in classical field theory. In texts like [1] or [2] the authors introduced auxiliary Grassmann odd variables to ...
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2answers
255 views

Commutator expectation value in quantum mechanics

Suppose $A$ and $B$ are operators, $A$ is Hermitian, $B$ anti-hermitian, and their commutator is the identity, i.e. $$[A, B] = I \, .$$ Denoting the eigenvectors of $A$ as $\lvert a \rangle$, so that $...
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0answers
49 views

Fixing the Poisson equation to match the deformation of elastic sheet with experimental observation

I am working on the calculation of the deformation of a circular elastic sheet with radius $R=1.2~m$ when a plate with mass $M$ and radius $r_0 = 4~cm$ is sitting in the center of the sheet. I used ...
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0answers
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what is meant by completeness of Hilbert space? [duplicate]

I know definition that completeness means every Cauchy sequence of elements of the space converges to an element in the space.But what does it mean physically when we say Hilbert space is complete ...
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1answer
85 views

How to derive the $\frac{4\pi}{3}\vec{p}\delta^3(\vec{r})$ element for the dipole field, from its potential?

This might be a bit more general question about how to figure out what is the appropriate (delta) expression in singular points, but e.g. for the dipole, we can derive its potential by a taylor ...
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2answers
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Why is the set of eigenfunctions of a Hermitian operator complete?

When I was studying quantum chemistry, I was told that given the time-independent Schrodinger equation $$\hat H \psi = E \psi$$ since $\hat H$ is Hermitian, the set of eigenfunctions $\{\psi_i \}_{i =...
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1answer
143 views

$\langle xp+px\rangle|_{t=0}=2\langle p\rangle\langle x\rangle|_{t=0}$ for the free particle?

Quantum Mechanics, Volume 1 by Claude Cohen-Tannoudji, Bernard Diu and Frank Laloe. Complement L-III, exercise 4 (page 342). Basically consider a free particle, and calculate the variance(uncertainty)...
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1answer
120 views

Partial deritative of $\partial_V(PV)_T$ with $PV=nRT$

The question arised from thermodynmaics. Suppose $n,R$ are positive constants, and $P,V,T$ are all positive. From $TdS=dE+PdV$, one may obtain $T\partial_V(S)_T=\partial_V(E)_T+P$ where $\partial_V()...
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35 views

Wave Equations from Decoupling Maxwell's Equations in Bianisotropic Media

For several days now, I have been trying to decouple Maxwell's equations in bianisotropic media so that I end up with a form that involves only one variable (of E, D, B, H), i.e. a so-called 'wave ...