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2
votes
0answers
79 views

Reference for stochastic processes which helps moving from a basic level to a measure theory one

I'm looking for a reference (books, notes, lectures) which helps a physicist to understand the language of measure theory in the context of stochastic processes (in particular markov chains). I've ...
11
votes
1answer
119 views

Triality and charge

I have a few questions about triality for the representations of $SU(3)$. (I have seen the wikipedia page, but it does not make the connection with physics.) What is triality, how can you compute ...
6
votes
2answers
110 views

Yang-Mills existence and mass gap

In the Clay institute problem description of the Yang-Mills existence and mass gap problem it states that the quantum Yang Mills needs to be formulated in $\mathbb{R}^4$ space. I was wondering whether ...
7
votes
1answer
126 views

Recommendation on mathematical physics book of Symplectic geometry

I want to learn the applications of symplectic geometry in physics. Which mathematical physics textbook will have a detailed and heuristic explanation of this aspect?
6
votes
0answers
71 views

Topology-dependent groud state degeneracy of $B \wedge F + B \wedge B$ and $B \wedge F + B \wedge B \wedge B$

There are some examples of topological BF theory with extra terms allow it still being topological. See this Ref. paper In 4d (3+1D), we have the trace of: $$ \int\frac{k}{2\pi}\text{Tr}[B \wedge F + ...
3
votes
1answer
160 views

Reducibility of tensor products of Lorentz group representations

Consider the statement: (34.29 in Srednicki's QFT text) $$\tag{34.29} (2,1)\otimes(1,2)\otimes(2,2)~=~(1,1)\oplus\ldots$$ Where of course, $(a,b)$ label representations of Lorentz group in the usual ...
4
votes
1answer
149 views

How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group?

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...
0
votes
0answers
20 views

Connecting global conservation laws to local features of a function

I have a heuristic question about using global constraints of a problem to make local estimates of geometric features of a curve, such as its local slope. Consider a suitably well behaved function ...
0
votes
0answers
27 views

Section of Phase space

Suppose $\mathfrak{X}$ is the configuration space of some system on a Riemannian manifold $\mathcal{M}$. Then, the phase space is $\mathrm{T}^{*}(\mathfrak{X})$. What is a section of ...
0
votes
1answer
36 views

Is $H_0^1$ something reasonable for the electric field for a perfect conductor?

I'm trying to pull over some concepts that were derived for Navier-Stokes like equations to Maxwell's equations for the perfect conductor. At a certain point, I am about to assume that the electric ...
8
votes
1answer
181 views

Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction

The Schrodinger equation in Hilbert space is expressed as : $$\frac{\partial}{\partial t} \psi(t) = \frac{-i}{\hbar}H\psi(t). $$ Here $\frac{\partial}{\partial t} \psi(t) \equiv \psi'(t) \equiv\lim ...
5
votes
1answer
121 views

What is the relationship between Cyclic Coordinates and Killing Vector Fields?

My question is related to this question. There are three or four other questions on Killing Vector Fields here, however none of them that I have seen address my question. $\\$ I've been studying ...
0
votes
1answer
63 views

Magnetic field of a Herzian dipole antenna

If I am given the dipole moment of very short dipole antenna as $P = P_0 sin (\omega t)$, what will be the magnetic field and polarization of far field radiation? Do I need to consider the time ...
5
votes
0answers
95 views

Meaning of conformal field theory [closed]

Can anyone summarize what conformal field theory is actually about,i.e. 1) what are its goals? (for example, to study such-and-such fields/functions/maps/etc. to see whether they have such-and such ...
-3
votes
1answer
118 views

Magnitude and Direction of Magnetic Field Homework Question [closed]

A $150~\text{V}$ battery is connected across two parallel metal plates of area $28.5~\text{cm}^2$ and separation $0.00820~\text{m}$. A beam of alpha particles (charge $+2e$, mass $6.64 \times ...
13
votes
1answer
274 views

Self-dual Maxwell equations, the second homology group, and topological invariants of a four manifold

In Witten's paper Quantum Field Theory and the Jones Polynomial, he mentioned that: Geometers have long known that (via de Rham theory) the self-dual and anti-self-dual Maxwell equations are ...
2
votes
0answers
48 views

What is the Levi--Civita connection of a Wick rotated metric?

A Wick rotation is a transformation that allows to change from a Lorentzian manifold to a Riemaniann manifold. In the cases when this is possible, is the Levi-Civita connection of the Riemaniann ...
5
votes
3answers
320 views

Is there any relationship between gauge field and spin connection?

For a spinor on curved spacetime, $D_\mu$ is the covariant derivative for fermionic fields is $$D_\mu = \partial_\mu - \frac{i}{4} \omega_{\mu}^{ab} \sigma_{ab}$$ where $\omega_\mu^{ab}$ are the spin ...
7
votes
0answers
317 views

When can we take the Brillouin zone to be a sphere?

When reading some literatures on topological insulators, I've seen authors taking Brillouin zone(BZ) to be a sphere sometimes, especially when it comes to strong topological insulators. Also I've seen ...
2
votes
4answers
95 views

Infinite series of derivatives of position when starting from rest

Suppose you have an object with zero for the value of all the derivatives of position. In order to get the object moving you would need to increase the value of the velocity from zero to some finite ...
0
votes
1answer
81 views

About Hilbert and Physics [duplicate]

Was one of Hilbert questions regarding physics to make an axiomatic foundation for physics? Regardless of Godels work could some Physics principles that are 'basic' and 'presently verifiable' be ...
4
votes
1answer
95 views

The properity of $\mathbb{R}^4$ that has infinitely many differential structures is related to Yang-Mills field?

I heard a saying that $\mathbb{R}^4$ having infinitely many differential structures which are not diffeomorphic to each other has a relationship with Yang-Mills field. Does anyone can explain it, and ...
6
votes
1answer
162 views

Does Feynman path integral include discontinuous trajectories?

While reading this derivation of relation of Schrödinger equation to Feynman path integral, I noticed that $q_i$ can differ form $q_{i+1}$ very much, and when the limit of $N\to\infty$ is taken, there ...
4
votes
1answer
122 views

Which is the role of Algebraic Geometry in String Theory? [closed]

Could someone sketch me what algebraic geometry has to do with string theory? Are there other mathematical disciplines that are interwoven with string theory? I'm aware of a similar question on ...
7
votes
1answer
126 views

The curvature of the space of commuting hermitian matrices

This is a question that I asked in the mathematics section, but I believe it may get more attention here. I am working on a project dedicated to the quantisation of commuting matrix models. In the ...
0
votes
0answers
56 views

PDE from London's Equation with Cylindrical Symmetry

The question is from ISSP by Kittel and as follows: (a)Find a solution of the London equation that has cylindrical symmetry and applies outside a line core. In cylindrical polar coordinates, we want ...
1
vote
1answer
106 views

Do generalized Pauli Operators generate SU(n)?

A commonly used generalization of Pauli Operators is the "clock" and "shift" operators summarized here: http://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices Pauli Operators are generators ...
0
votes
0answers
63 views

Ricci/Tensor Calculus and Matrix Calculus

What are the differences in the types of problems Ricci/Tensor Calculus and Matrix Calculus can solve? One formulation is generally used by the physics community and the other by statisticians and ...
1
vote
0answers
47 views

Conformal mapping for calculating magnetic reluctance

I work currently on the calculation of magnetic resistance for a air-gap of an electrical machine. For this I am using conformal transformations. I am doing this on a 2D slice. As the machine is ...
3
votes
3answers
344 views

Why Hausdorff and Paracompact manifold in GR?

What can we say about the transition map if the manifold is a Hausdorff space? Why do we need the manifolds to be Hausdorff and paracompact in General Relativity?
6
votes
2answers
203 views

When can we assume that the wavefunction is separable

While working out the stationary states of a single particle in a 3d infinite potential box ($V=0$ inside a cuboid of known dimensions, $V=\infty$ everywhere else), I realized I had to assume the ...
3
votes
1answer
174 views

Rigorous mathematical formalism of particle physics

Can anyone provide me with a rigorous mathematical definition of the fundamental particles (all fundamental bosons and fermions), reflecting the analogy of action of groups with interaction of ...
4
votes
1answer
97 views

Some question about symplectic transformation

I read Arnold's book Mathematical Methods of Classical Mechanics and come across with three problems in page 229. 1.Let $\lambda$ and $\bar{\lambda}$ be simple (multiplicity 1) eigenvalues of a ...
0
votes
0answers
59 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 + ...
1
vote
2answers
177 views

Can we describe Quantum Mechanics using filters and matrices? [closed]

Can mathematical filters or ultrafilters be used to predict quantum physics 'events' as accurately as using matrices like Schrodinger did? Is there a way to explain some of the predictive power of ...
5
votes
0answers
82 views

Intuitively what's the relationship between forces and connections?

In Einstein's General Relativity we relate the effects of gravity with the curvature of the Levi-Civita connection on the spacetime manifold. Also, when we get the electromagnetic tensor $F = dA$ ...
4
votes
1answer
270 views

Mandelstam variables 1 positive 2 negative

The three Mandelstam-variables are defined as: $$s=(p_A+p_B)^2=(p_C+p_D)^2,$$$$t=(p_A-p_C)^2=(p_B-p_D)^2$$$$u=(p_A-p_D)^2=(p_B-p_C)^2.$$ Where A and B are the incoming particles and C and D are the ...
3
votes
1answer
274 views

Diagonalization of Hamiltonian

Typically, one way of understanding the physics of an interacting quantum system is by diagonalizing the Hamiltonian. In principle, can we always diagonalize a Hamiltonian, such that it is expressed ...
7
votes
2answers
281 views

Lie algebra and Lie group about quantum harmonic oscillator

We know that in the quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ are the annihilation and creation operators, and $H$ is the ...
4
votes
2answers
105 views

Defining quantum effective action (Legendre transformation), existence of inverse (field - source)?

Given a Quantum field theory, for a scalar field $\phi$ with generic Action $S[\phi]$, we have the generating functional $$Z[J] = e^{iW[J]} = \frac{\int \mathcal{D}\phi e^{i(S[\phi]+\int d^4x ...
3
votes
1answer
102 views

References on $C^{*}$-algerbas, $W^{*}$-algebras and Quantum Theories

I would like to know some references regarding $C^{*}$ and $W^{*}$-algebras and quantum theories. I'm interested in concrete physical applications, models and problems. Here it is the list of ...
2
votes
1answer
102 views

Mathematical Physics SUSY QM Resource Recommendation

I want to study SUSY QM. I found some excellent physically motivated articles on Arxiv. Despite, I am especially interested in the mathematical structure behind SUSY QM. Does anybody know whether ...
4
votes
1answer
138 views

A question about canonical transformation

I have posted this question in math.stackexchange before with no answer till now. It may be more suitable to post here. There is a problem in Arnold's Mathematical Methods of Classical Mechanics ...
1
vote
2answers
186 views

Trace in non-orthogonal basis?

Physicists define the trace of an operator $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal basis, and this quantity is ...
2
votes
0answers
98 views

Solving the Schrodinger equation with appropriate symmetry

In the paper Markov Fields by Edward Nelson the introduction section claims that analytically continuing a Markov process with appropriate symmetry properties yields the solution of the Schrodinger ...
3
votes
2answers
85 views

Statistical Mechanics - Distribution of Energies

Consider a state space $\mathbb{X}$. The probability density function under a canonical ensemble is given by the Boltzmann distribution $$\pi_{\mathbb{X}}(x)=\frac{e^{-\beta ...
8
votes
0answers
239 views

p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)

I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be a close relation between p-adic string theory and the refinement of the superstring ...
0
votes
2answers
233 views

Why is physics consistent? [closed]

It seems like it's a given that the laws of physics that govern our universe are consistent, and that inconsistency somehow is a reason to doubt an explanatory theory (such as famously Godel's ...
2
votes
0answers
34 views

Lie algebra and BPS stats

i would know what is a charges Lattice and the relationship between it and roots lattice ? and if there is relationship between mutation of quiver and weyl group? and how the PBS stats correspond ...
4
votes
1answer
62 views

Why is the Legendre transformation an application of the duality relationship between points and lines?

When I read the Wiki about Legendre transformation, there is a statement The Legendre transformation is an application of the duality relationship between points and lines. What's the meaning of ...