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 ...
3
votes
0answers
53 views
Discreteness of set of energy eigenvalues
Given some potential $V$, we have the eigenvalue problem
$$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$
with the boundary condition
$$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$
If we ...
9
votes
1answer
272 views
Integral representation of Thomas-Fermi Equation
The Thomas-Fermi equation with dimensionless variables is identified as;
$$
\frac{d^2\phi}{dx^2} = \frac{\phi^{3/2}}{x^{1/2}}
$$
with the boundary conditions as
$$
\phi(0) = 1 \\
\phi(\infty) = 0.
$$
...
-2
votes
0answers
33 views
calculate distance in feet using time interval from iphone camera [duplicate]
i am working on iPhone application to achiever the below functionality.
i want the distance in feet between Two Points from the below information.
i am standing at one position with iPhone (Camera ...
9
votes
1answer
213 views
How is the Dirac adjoint generalized?
I am wondering how one can generalize the Dirac adjoint to flat "spacetimes" of arbitrary dimension and signature. To be more specific, a standard situation would be to consider 4 dimensional ...
0
votes
0answers
19 views
Specular intensity [closed]
Im currently studying for an exam and have been going through some past papers on the subject, however i have come across a question that has recursively come up each year and the notes on it are not ...
2
votes
0answers
25 views
How to numerically solve a laser driving semi-classical two-level system using Floquet formalism?
Consider the semi-classical laser driving two-level atom, where the laser is treated classically and the atom is treated quantum mechanically. The effect of laser on the atom is a dipole coupling:
$$
...
0
votes
0answers
39 views
An application of Toeplitz operators
I want to find an application of the Toeplitz operators. All I need is a known problem (not an open problem) which solution use the theory of Toeplitz operators. I don't need all the details but I ...
6
votes
2answers
376 views
$\nabla ^2\psi$ equals $\psi -$ average value of $\psi$ at neighboring points
Let $\psi (x,y,z)$ be a scalar field. I found the following statement in Morse & Feshbach Methods of Theoretical Physics:
The limiting value of the difference between $\psi$ at a point and the ...
10
votes
2answers
131 views
When are there enough Casimirs?
I know that a Casimir for a Lie algebra $\mathfrak{g}$ is a central element of the universal enveloping algebra. For example in $\mathfrak{so}(3)$ the generators are the angular momentum operators ...
6
votes
5answers
186 views
In coordinate-free relativity, how do we define a vector?
Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR).
I would normally define a vector by its transformation properties: it's something whose components change ...
6
votes
2answers
177 views
Coherent $U(N)$ intertwiners in Loop Quantum Gravity (LQG) and a measure on the Grassmannian
This is a detailed question about $U(N)$ intertwiners in LQG, and it comes from the the paper by Freidel and Livine (2011 - archive). It is very specific but related to finding a measure on a quotient ...
0
votes
0answers
33 views
What is the result of applying a fourier transform n times to a distribution?
For a function applying the fourier transform twice is equivalent to the parity transformation, applying it three times is the same as applying the inverse of the fourier transform, and applying four ...
0
votes
0answers
22 views
What is the magnetic quadrupole moment of a nucleus in cylindrical coordinates?
What is the magnetic quadruple moment of a nuclei in cylindrical coordinates?
The quadrupole moment of a nucleus is zero in spherical coordinates but in the cylindrical coordinates it can't be ...
1
vote
1answer
31 views
what is the magnetic quadrupole operator?
To find magnetic or electrical moments in quantum theory we must calculate the expectation value of an appropriate operator. the dipoles operator are similar and is easy to find but the magnetic ...
2
votes
1answer
90 views
Energy Functional
I am a graduate student in pure mathematics, during my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, ...
26
votes
7answers
2k 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 ...
6
votes
2answers
621 views
EM wave function & photon wavefunction
According to this review
Photon wave function. Iwo Bialynicki-Birula. Progress in Optics 36 V (1996), pp. 245-294. arXiv:quant-ph/0508202,
a classical EM plane wavefunction is a wavefunction (in ...
2
votes
0answers
50 views
About deriving the multi-trace index in terms of the single-trace index
This question is in reference to this paper
Combining their equations 5.2, 5.3, 5.6 and 5.7 one seems to be looking at the integral/partition function,
$Z(x) = \prod_{n=1}^{n =\infty}\left [ \int ...
6
votes
1answer
76 views
Motivation for the Deformed Nekrasov Partition Function
I have recently been doing research on the AGT Correspondence between the Nekrasov Instanton Partition Function and Louiville Conformal Blocks (http://arxiv.org/abs/0906.3219). When looking at the ...
3
votes
0answers
96 views
Holonomy twisting
There is Witten's topological twist of standard SUSY QFTs with enough SUSY into Witten-type TQFTs. What is a holonomy twist?
0
votes
1answer
172 views
Expansion in solid spherical harmonics on the lattice
I'm interested in calculating scattering processes (e.g. Coulomb scattering of an electron beam by a single ion) in the context of lattice quantum field theory, and wonder if there is something like ...
7
votes
1answer
73 views
What exactly is meant by the conformal group of Minkowski space?
This is sort of a silly question because I'm a total beginner, and I debated whether it was better to ask here or on Math.SE. I decided on here because it's about how physicists use terminology, even ...
2
votes
1answer
63 views
Consequences of Compactness in Physics
If we understand spacetime as a $4$-dimensional manifold $M$, from the point of view of physics what are the consquences of a subset of it being compact? My point here is simple: in math we usually ...
3
votes
3answers
355 views
Equations of fluid dynamics and differential geometry
Where can I look for equations of fluid motion written in terms of nifty things from differential geometry like exterior derivative, Hodge dual, musical isomorphism?
Preferably both with and without ...
0
votes
1answer
75 views
Higher order covariant Lagrangian
I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at ...
2
votes
2answers
150 views
In quantum mechanics(QM), can we define a high-dimensional “spin” angular momentum other than the ordinary 3D one?
Inspired by my previous question Questions about angular momentum and 3-dimensional(3D) space? and another relevant question How to define angular momentum in other than three dimensions? , now I get ...
13
votes
3answers
340 views
Homotopy $\pi_4(SU(2))=\mathbb{Z}_2$
Recently I read a paper using $$\pi_4(SU(2))=\mathbb{Z}_2.$$ Do you have any visualization or explanation of this result?
More generally, how do physicists understand or calculate high dimension ...
12
votes
8answers
1k views
Why $\displaystyle i\hbar\frac{\partial}{\partial t}$ can not be considered as the Hamiltonian operator?
In the time dependent Schrodinger equation $\displaystyle, H\Psi = i\hbar\frac{\partial}{\partial t}\Psi$ , the Hamiltonian operator is given by
$\displaystyle H = -\frac{\hbar^2}{2m}\nabla^2+V$
...
2
votes
1answer
105 views
Phys.org Spectral geometry to unite relativity and quantum mechanics, restate in laymens terms?
Lingua Franca links relativity and quantum theories with spectral geometry
Could someone give me a short synopsis of this article in laymens terms? What implications does this have in the physics ...
1
vote
1answer
126 views
Validity of the Multi-Species Navier-Stokes Equations for real gases
I'm wondering what are the validity limits of Multi-Species Navier-Stokes equations. I'm aware of the limit for rarefied gases. But is there any new limit that arises in the context of real gases?
I ...
7
votes
4answers
664 views
How can a point-particle have properties?
I have trouble imagining how two point-particles can have different properties.
And how can finite mass, and finite information (ie spin, electric charge etc.) be stored in 0 volume?
Not only that, ...
2
votes
0answers
49 views
Helicity for Zero Rest Mass Field Equations
I'm trying to reconcile the usual definition of the helicity operator, namely
$$ h = \hat{p}.S$$
with the definition of a massless helicity $n$ field as a symmetric spinor field $\phi^{A\dots B}$ ...
1
vote
1answer
80 views
Spin(n) group SO(n) relation
Is it correct to state that the elements of Spin(n) fulfill a Clifford algebra and that the Lie group generators of Spin(n) is given by the commutator of the elements?
If not, then what is the ...
4
votes
2answers
112 views
Twistor Function for Coulomb Field
In an article by Penrose in Hughston and Ward "Advances in Twistor Theory", it is claimed that the twistor function
$$ f(Z^\alpha) = \log{\frac{Z^1Z^2 - Z^0Z^3}{Z^2Z^3}}$$
produces an anti-self-dual ...
1
vote
1answer
61 views
How to assign coordinates to the elements of a flat metric space
Consider the metric space $(M, d \,)$ where set $M$ contains sufficiently many (at least five) distinct elements,
and consider the assignment $c_f$ of coordinates to (the elements of) set $M$,
$c_f ...
5
votes
2answers
159 views
A question on the existence of Dirac points in graphene?
As we know, there are two distinct Dirac points for the free electrons in graphene. Which means that the energy spectrum of the 2$\times$2 Hermitian matrix $H(k_x,k_y)$ has two degenerate points $K$ ...
1
vote
0answers
49 views
Geometry for Physics [duplicate]
I am currently a high school student interested in a research career in physics. I have self taught myself single variable calculus and elementary physics upto the level of IPHO . And I am comfortable ...
1
vote
1answer
115 views
How to use Legendre polynomials in order to determine the (an)isotropy of an on-lattice cluster aggregate?
I am currently testing various models of on-lattice (square lattice in two dimensions) cluster growth for anisotropy.
I end up with a cluster, the boundary of which, in case of a truly isotropic ...
9
votes
1answer
322 views
False vacuum in axiomatic QFT
There is an elegant way to define the concept of an unstable particle in axiomatic QFT (let's use the Haag-Kastler axioms for definiteness), namely as complex poles in scattering amplitudes. Stable ...
7
votes
2answers
2k views
Some Korean researchers saying that they solved Yang-mill existence and mass gap problem
Today, Korean media is reporting that a team of South Korean researchers solved Yang-Mill existence and mass gap problem. Did anyone outside Korea even notice this? I was not able to notice anything ...
2
votes
1answer
79 views
Physics Applications of Fredholm Theory:
I find Fredholm theory beautiful, especially the Liouville-Neumann series for solving Fredholm integral equations of the second kind. There seems to be a consensus that these equations are quite ...
3
votes
3answers
520 views
Are Mathematical Physics and Occam's Razor compatible?
Occam's Razor and mathematical beauty appear to be compatible when reviewing Michael Atiyah's video.
But are the high levels of complexity associated with mathematical physics compatible with ...
8
votes
2answers
954 views
Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?
The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
0
votes
0answers
81 views
Quantum uncertainty can explain the Riemann Hypothesis?
In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of ...
3
votes
1answer
79 views
Transformation law for fermionic measure in functional integral
I am reading the paper "Bosonization in a Two-Dimensional Riemann-Cartan Geometry", Il Nuovo Cimento B Series 11
11 Marzo 1987, Volume 98, Issue 1, pp 25-36, ...
20
votes
4answers
757 views
In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?
A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential.
Is there a one to one correspondence between the potential and its spectrum?
If the ...
5
votes
1answer
136 views
Is the Hilbert space of $\phi^4$ theory known?
Consider free, real scalar field theory in $d=1+3$ dimensions: $H = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi + \frac{1}{2} m^2 \phi^2$. The Hilbert space of this theory is known; it is just ...
6
votes
1answer
68 views
Are observables associated to spacetime regions?
In the Haag-Kastler approach to axiomatic quantum field theory, it is assumed that observables are 'associated' to spacetime regions. What this actually means is that there is a map $\mathcal{A}: R ...
3
votes
1answer
153 views
Calculating Riemann Tensor Using Tetrad Formalism
I was trying to calculate the Riemann Tensor for a spherically symmetric metric:
$ds^2=e^{2a(r)}dt^2-[e^{2b(r)}dr^2+r^2d\Omega^2]$
I chose the to use the tetrad basis:
$u^t=e^{a(r)}dt;\, ...
2
votes
0answers
87 views
Finite or ∞ set of masses & ∃ gravity center?
Any finite & non empty set of masses has a computable center of gravity:
$\vec{OG} = \frac{\sum_i m_i \vec{OM}_i}{\sum_i m_i}$ .
Does the contrapositive permits to conclude that a mass system ...
