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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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 ...
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35 views

What is the best book to read for optimal mass transportation theory for students with physics background?

I need to read optimal mass transportation theory for my research. What is the best book to read. I am from physics background. How much mathematics and what sort of mathematics required prior to ...
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117 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 ...
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44 views

What is the difference between real and complex instantons (mathemtically, and their physical significance), and connection to Wick rotation

I am struggling to understand the difference and physical significance between real and complex instantons- I think these are also sometimes called ghost instantons? There are also anti-instantons. ...
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68 views

Prove that the electric field produce by a punctual charge is isotropic and radial

I would like to prove mathematically that the electric field produced by a punctual charge is isotropic and radial, i.e. $$\vec{E}(r,\phi,\theta)=E(r)\vec{e}_r\tag{1}$$ I think that this statement ...
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65 views

Why is the singularity not taken into account?

In this article "Reflections on Maxwell’s Treatise", Section 4.2, it says: He replaces $\mathbf{m}$ with a volume element of magnetization $\mathbf{M}\ dV$ , integrates over $V$ , and lets the same ...
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34 views

Is the disposition of $1$s and $0$s when writing orthogonal ket vectors purely conventional?

If I want to define the basis in the form of $4$-vectors, how do I proceed to make sure they are orthonormal with one $1$ and three $0$ in each vector? Is it just by convention? Does it matter if I ...
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36 views

Meaning and Origin of an Expression which Involves Virtual Displacement

As an additional point of confusion related to the answer given here: Confusion with Virtual Displacement I have encountered the following expression in my study of virtual displacements. $$\delta{...
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85 views

Maslov's method of deriving the WKB approximation

For a generic one-dimensional potential, the WKB approximation yields the quantization condition $$ \oint p dq = (n + 1/2)\hbar . $$ Here, the correction factor $1/2 $ was obtained by Kramers by ...
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107 views

Complex Analysis Textbook using Fluid Dynamics

This is a literature request. I remember being told of a Complex Analysis textbook that teaches the Cauchy-Riemann equations and Potential Theory using Fluid Dynamics. Does anyone know who the ...
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164 views

General proof of independence of TM and TE modes in a waveguide

In electromagnetic field analysis for a typical waveguide that has a uniform cross section along its axial direction (say $z$), we often describe the E and H fields conveniently in terms of their ...
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89 views

Time dependence of Hamiltonian in Schrodinger picture in open quantum system

Many questions in StackExchange and papers that I see consider the following von Neumann equation for describing evolution of density matrix $\rho$ in time $t$ given a Hamiltonian $H$: $$ \frac{d\rho}...
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52 views

For closed circuits, why can't we have more than one $f(r)$?

Force between current elements depends on a function of angles [$f(\eta, \theta, \theta^{\prime})$] and also on a function of distance between them [$f(r)$] . For closed circuits, there are more ...
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50 views

What is a convolution transform?

In quantum optics, on talks about the nonclassicality of a field in terms of the $P$, $Q$ and $R$ functions. The the $P$ and $R$ functions are related as $R(z,\tau) = \frac{1}{\pi \tau} \int d^2w ~\...
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71 views

How to properly understand the residue in the LSZ theorem?

The LSZ theorem for a scalar field reads $$ \mathcal M=\lim_{p^2\to m^2}\left[\prod_{i=1}^n(p^2-m^2)\right]\tilde G(p_1,\dots,p_n) $$ where $G$ is the $n$-point function, to wit, $$ G(x_1,\dots,x_n)=\...
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31 views

Do you know dynamical formulas for hydraulic cylinders, motors and pumps?

I seeking dynamic differential equations for hydraulic cylinders, motors and pumps. I have one differential equation for a hydraulic cylinder, but I'm not sure if it's correct. Assume a hydraulic ...
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121 views

Movement of a random walk in the limit (a particle in diffusion)

I asked this question in Math Exchange and MathOverflow and obtained no answer. This question may lack of mathematical rigorous, but I would like to understand why this type of reasoning is sometimes ...
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76 views

Physical meaning of the eigenfunctions and eigenvalues of the Fractional Laplacian

What is the physical meaning of the eigenfunctions and eigenvalues of the Fractional Laplacian?
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41 views

Is there a sensible way to model a long-range, lower-dimensional force?

For example, in our universe, gravity and electromagnetism obey an inverse-square law due to the dimensionality of space; in higher dimensions, they would drop off faster (with the consequence that ...
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114 views

Why do the solutions of the Schrödinger equation form a basis for $L^2$?

So I want to investigate the kind of convergence the basis function obtained from the solutions to the Schrödinger equation (SE) offer. But before I look at this, I'd like to learn why we say that the ...
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84 views

Modeling the dissolution of an oil droplet with detergent

How to model the dissolution of an oil droplet on the surface of water when a detergent is (gradually) added to the water? I believe this includes an interplay of coalescence keeping the droplet ...
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352 views

Transformer universal EMF equation derivation

At this Wikipedia page, we've that the 'Transformer universal EMF equation' looks like: $$\text{E}_{\text{rms}}=\frac{2\pi\times\text{f}\times\text{n}\times\text{a}\times\text{B}_{\text{peak}}}{\sqrt{...
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86 views

Wannier's solution of the Landau-Zener problem

It is said that Wannier had a very simple solution of the Landau-Zener problem. His paper is on Physics 1, 251 (1965). Apparently this journal is closed and I cannot find the paper online. Does ...
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43 views

Mathematical Description of Time Speeding Up?

People are able to experience time speeding up or slowing down. This is confusing to me on a mathematical level because dT/dT is 1. Is there some way that makes sense for this not to be 1? The speed ...
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83 views

Application of the uniform boundedness principle to Physics

The uniform boundedness principle is the following result from functional analysis: Let $X$ be a Banach space and $Y$ a normed linear space. Suppose that $F$ is a collection of continuous linear ...
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67 views

How does one define a constant of motion in algebraic qft?

Among the postulates of AQFT there is no dynamical evolution principle postulated, i.e. there is no analog to a postulated Heisenberg equation of motion. How does one define a constant of motion in ...
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184 views

Norm of Classical (Poissonian) Hamiltonian Operator

In the Poissonian formulation of classical mechanics, one finds that the time evolution of the phase space vector $\eta = (q_1,q_2\cdots q_n ; p_1, p_2\cdots p_n)^T$ can be put in terms of the ...
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142 views

Does this divergent series appear anywhere in physics?

Question I recently managed to analytically continue certain divergent series. I was hoping if anyone could tell me if this expression appeared somewhere in physics: $$ \implies \lim_{k \to \infty} ...
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53 views

How many sheets for the Green function?

The Hamiltonian of a particle in a 1D potential is $$H = H_0 + V(x) . $$ Here $H_0 = p^2/2m$ is the free part. It is known that the Green function $$ G_0(E) = \frac{1}{E - H_0 } $$ has a cut ...
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163 views

Extending projection operator to infinite-dimensional case

Hi I have a basic question regarding bra-ket notation. Given that $\{|e_n \rangle \}$ is a discrete orthonormal basis, $$\langle e_m | e_n \rangle = \delta_{mn}$$ then $$\sum_{n}|e_n \rangle \langle ...
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155 views

Are there applications of $L_p$ spaces in quantum mechanics?

In quantum mechanics, there a lot of emphasis on $L^2$ spaces since Hilbert spaces describe states in quantum mechanics, so we have $$ \langle \psi | \psi \rangle = \int |\psi^2(x)|\, dx$$ Even ...
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55 views

How to determine time of dephasing?

Let's assume that I have an oscillating value A. After some time the oscillations are being damped so the diagram of A is like on the picture below: Now how to determine when does the A is reduced by ...
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297 views

Topology of Anti-de Sitter manifold with black hole

I'm interested in understanding the topology of space-time with a black hole. In other words how does having a black hole affect quantities such as the fundamental group, de-Rham cohomologies, Euler ...
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472 views

Definition of Hamilton operator

The Hamilton operator is by definition a self-adjoint operator $H\text{: }D\left(H\right)\to\mathcal{H}$ with $D\left(H\right)\subset\mathcal{H}$ a dense linear subspace of the Hilbert space $\mathcal{...
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126 views

References on deformation quantization

I'm looking for books or introductory review papers or lecture notes on the topic of deformation quantization. (And preferably, geometric quantization as well.) I'm mainly interested in the ...
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49 views

Is there a physical interpretation of the alternating property?

A map from a vector-space to its base field is called "alternating" if each vector with repeated elements is mapped to zero. I've read that symplectic geometry is an important representation of ...
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191 views

Quantized Banach Spaces

Reading the paper here, it mentions on the very first page that "The requirement of 'closed'-ness is imposed because we want to think of operator spaces as 'quantized (or non-commutative) Banach ...
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177 views

Regarding Ampere's Circuital Law

If I am to show that Ampere's Circuital law holds true for any arbitrary closed loop in a plane normal to the straight wire, with its validity already established for the closed loop being a circle of ...
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97 views

Integral of absolute value of spin angular momentum of $N$-body system

There are $N$ particles moving freely in a plane. Let $J(t)$ be the spin angular momentum of the system of particles about its center of mass. (even center of mass keeps changing with time as ...
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136 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 ...
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81 views

A special path integral

May be $f(\vec{x}), \vec{g}(\vec{x})$ an arbitrary functions dependent on the coordinates $\vec{x}=(x,y,z)^T$. Defining the following function dependent on a 3-dimensional curve $\vec{\gamma(t)}$ ...
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77 views

Does this Hamiltonian have point spectrum?

Consider such a Hamiltonian $$ H = - \frac{1}{2} \frac{\partial^2}{\partial x^2} - F x + V(x) ,$$ with $F$ being some constant, and $V(x)= V(x+L)$ being some periodic potential. Does this ...
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Can we avoid singularities by embedding the manifold on a bigger space, maybe Euclidean or Riemannian?

Can singularities be avoided by embedding Riemannian manifold on bigger space, or more specifically black hole singularities can be avoided or not by embedding in any other manifold. We don't have ...
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132 views

Symplectic Structure without predefined Hamiltonian

Here there is a link which has helped me understanding the relationship between symplectic geometry and classical mechanincs. In short, the symplectic form transforms the derivative of the Hamiltonian,...
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127 views

Do all the spacelike curve terminate at the spatial infinity $i_0$ in the Penrose Diagram of a Schwarzchild black hole?

Let's restrict to the radial direction, so the metric can be expressed as $ds^2=-(1-r_S/r)dt^2+(1-r_S/r)^{-1}dr^2$ with $r_S$ the Schwarzchild radius. Expressed in Kruskal coordinates, the metric is ...
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477 views

Robin boundary conditions for the heat equation

Consider a 3D channel with fluid or gas with walls $\Gamma_1$, inflow part $\Gamma_2$ and outflow part $\Gamma_3$. The temperature is described by the heat equation: $$ \frac{\partial T}{\partial t} -...
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140 views

Examples of application of detour matrices in physics?

Are there any good examples of application of detour matrices in physics?
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87 views

Charge and current density fields

The charge and current density fields in classical electromagnetism are scalar real number fields on space time manifold. But these fields diverge/become infinite in case of point charges, how is this ...
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99 views

Is assuming spacetime to have |2^N| points overkill?

The physical continuum is commonly assumed to have a mathematical continuum of points, that is, with a cardinality equinumerous with the Power Set of the set of natural numbers. However, since [ZFC + ...
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What's the local law of propagation of disturbances

In Vladimir I. Arnold's Lectures on Partial Differential Equations, Chapter 3 Huygens' Principle in the Theory of Wave Propagation, which is devoted to the proof of Huygens principle (original one by ...

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