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837 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 ...
9
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6answers
1k 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 ...
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0answers
648 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 ...
12
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2answers
334 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 $...
13
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1answer
935 views

Is there a “covariant derivative” for conformal transformation?

A primary field is defined by its behavior under a conformal transformation $x\rightarrow x'(x)$: $$\phi(x)\rightarrow\phi'(x')=\left|\frac{\partial x'}{\partial x}\right|^{-h}\phi(x)$$ It's fairly ...
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0answers
69 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 ...
7
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2answers
370 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 ...
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0answers
124 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?
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1answer
1k 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 ...
3
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2answers
175 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 ...
6
votes
2answers
780 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 ...
3
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2answers
231 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 ...
2
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0answers
90 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 $\phi^{A\dots B}$ ...
2
votes
1answer
337 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
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2answers
152 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 ...
3
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1answer
135 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, ...
8
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1answer
384 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 ...
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0answers
82 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 ...
9
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2answers
7k views

Some Korean researchers saying that they solved Yang-Mills 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 ...
9
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1answer
318 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 ...
1
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0answers
295 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
170 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, http://link.springer.com/article/10.1007%...
7
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1answer
348 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 ...
5
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3answers
697 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 Occam'...
7
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1answer
107 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 \...
9
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2answers
962 views

Nonseparable Hilbert space

What kind of things can go wrong if we try to do quantum mechanics on a nonseparable Hilbert space? I have heard that usual mathematical manipulations that we take for granted will no longer hold. ...
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0answers
448 views

Noether currents for the BRST tranformation of Yang-Mills fields

The Lagrangian of the Yang-Mills fields is given by $$ \mathcal{L}=-\frac{1}{4}(F^a_{\mu\nu})^2+\bar{\psi}(i\gamma^{\mu} D_{\mu}-m)\psi-\frac{1}{2\xi}(\partial\cdot A^a)^2+ \bar{c}^a(\partial\...
4
votes
3answers
296 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 ...
4
votes
2answers
413 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 Klein-...
4
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0answers
128 views

Question about the HVZ theorem

In this paper1 the authors cite the HVZ theorem2 saying that it follows from the method used by M. Reed & B. Simon without modifications; I don't really understand this point. Is there anyone who ...
7
votes
3answers
2k 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 $K$ ...
8
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2answers
511 views

How to understand worldsheet fermion as a section?

I am reading Witten's paper on topological string, and I found some mathematical notation is hard to understand for me. Consider the nonlinear sigma model in 2 dimensions governed by maps $\Phi : \...
3
votes
2answers
458 views

Schmidt basis: Entanglement

I do not understand how any state in Hilbert Space $\mathcal{H}=\mathcal{H}_A\otimes\mathcal{H}_B$ of dimension $\text{dim}(\mathcal{H}_A)\times\text{dim}(\mathcal{H}_B)$ can be decomposed in the ...
4
votes
1answer
1k views

Change of coordinates from an arbitrary frame to a locally inertial frame in General Relativity

If I have the following metric: $$ds^2=(1-2\phi)c^2 dt^2 - (1-2 \phi)(dx^2+dy^2+dz^2)$$ $\phi$ being the gravitational potential with $|\phi| << 1$ everywhere. How do I find a coordinate ...
12
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1answer
524 views

Reference Request: Classical Mechanics with Symplectic Reduction

I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie ...
5
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3answers
448 views

Why is this identity an if, rather than if and only if?

A recent question (Product of exponential of operators) asked who to proved that the exponentials of operators multiply in same manner as those of scalars if and only if the commutator of the ...
15
votes
3answers
2k views

What is a dual / cotangent space?

Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never ...
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2answers
996 views

How much pure math should a physics/microelectronics person know [duplicate]

I do condensed matter physics modeling in my phd and I was struck up learning quite an amount of physics. But while having done lot of physics courses, I see that if I learn pure math I would ...
7
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0answers
224 views

Electric potential of a spheroidal gaussian

I'm looking for results that compute the electrostatic potential due to a spheroidal gaussian distribution. Specifically, I'm looking for solutions of equations of the form $$ \nabla^2\Phi=N\exp\left({...
3
votes
3answers
358 views

Results of Statistical Mechanics first obtained by formal mathematical methods

I have a question that seems natural in Physics and Mathematics mainly in Statistical Mechanics of Equilibrium. Results that are proven by formal mathematical methods that were already seem intuitive ...
4
votes
2answers
268 views

Infinitely many planets on a line, with Newtonian gravity

(I apologize if this question is too theoretical for this site.) This is related to the answer here, although I came up with it independently of that. $\:$ Suppose we have a unit mass planet at each ...
9
votes
2answers
662 views

Algebraic/Axiomatic QFT vs Topological QFT

Can anybody please tell me a good source investigating the relation between Algebraic/Axiomatic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT)? Or is there none?
1
vote
1answer
732 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 model,...
3
votes
1answer
874 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;\, u^r=e^{b(...
1
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3answers
284 views

Is a quantum system mandatory for generating true random sequence?

Is a quantum system necessary if we want to generate true random sequence? The mathematical framework used for classical mechanics doesn't involve any random value. But the mathematical framework of ...
2
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1answer
158 views

Does a constant of motion always imply a Hamiltonian formulation?

If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and ...
1
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1answer
160 views

Can I prove boundedness of an operator without checking it for its whole domain?

(I don't have a direct reference so this is a little fishy and I'll delete it if nobody recognises what I'm talking about, but I though for starters I'll ask anyway) I've heard at university that if ...
0
votes
1answer
437 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 ...
4
votes
1answer
332 views

Tangent bundles and $\mathbb{C}P^n$ and $\mathbb{C}^n$

As discussed here the complex projective space $\mathbb{C}P^n$ is the set of all lines on $\mathbb{C}^n$ passing through the origin. It would seem natural to assume that any $\mathbb{C}P^n$ can be ...
3
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2answers
3k views

Greens function in EM with boundary conditions confusion

So I thought I was understanding Green's functions, but now I am unsure. I'll start by explaining (briefly) what I think I know then ask the question. Background Greens are a way of solving ...