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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Finding symmetry of a part of an equation, given the group transformation property of another part

I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to ...
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120 views

What can quantum adiabatic computation provably accomplish?

Let's say I have a quantum adiabatic computer in a black-box that works perfectly, doesn't suffer from decoherence/noise problems, etc. Are there any proven bounds for an adiabatic algorithm that ...
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Relation between functional measures and states in AQFT

Let $(M,g)$ be a globally hyperbolic spacetime and $\phi$ a KG field. In AQFT we consider the algebra of observables $\mathfrak{A}$ generated by $\phi(f)$ where $f\in C^\infty_0(M)$ is a test function....
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161 views

Legal values of quantum field can take? $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

Main issue: What are the legal and possible values of the quantum field can take? Clarify by examples: (1) For example, for the spin-0 Klein Gordon field $\phi$, we may choose it to be: real $\...
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91 views

Which of the Wightman axioms are not incorporated by four dimensional quantum Yang-Mills?

I am trying to understand the quantum Yang-Mills existence problem but the best I have seen so far is the statement that there is no known interacting relativist field theory in four dimensions which ...
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137 views

Von Neumann entropy in the algebraic approach

In the algebraic approach to QM a quantum system has associated to it one $\ast$ algebra $\mathfrak{A}$ generated by its observables. A state is a positive, normalized linear functional $\omega : \...
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326 views

Rigorous treatment of Penrose diagrams

I'm looking for a rigorous exposition of Penrose diagrams (also called conformal diagrams in general relativity. By "rigorous" ("careful" is perhaps a more attractive word) I mean that it should ...
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148 views

On Seiberg-Witten curves

In page 44 of Gaiotto's article "Families of $\mathcal{N}=2$ Field Theories" on Teschner's review the author writes down the pure Seiberg-Witten curve as $$ x^2 = z^3 + 2uz + \Lambda^4z $$ with the SW ...
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68 views

Bounds on the effect of strong coupling

I am interested in bounding the effects of system-environment interaction. Suppose I have an initial state $\rho \in \mathcal{H}_S \otimes \mathcal{H}_E$ where the system and environment might be ...
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62 views

Motivation for Haag axioms and algebra of distributions

Are the Haag axioms partly motivated by the fact that in the general case, it is impossible to form a differential algebra with (basic) Schwartz distributions? I got the impression due to the ...
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70 views

Toda lattice solution for different algebras

It is well-known that Toda systems (Toda field theory) can possess different algebraic structure based on Cartan Matrix in the Hamiltonian's potential. But all solutions I have seen were written only ...
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173 views

Has the theory of fractional quantum mechanics been experimentally validated in any way?

I'm referring to the generalisation of quantum mechanics developed by Nick Laskin: http://arxiv.org/abs/quant-ph/0206098 He suggests that if a particle's trajectory is integrated over Lévy paths ...
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778 views

What is a branch cut singularity in QFT?

Peskin & Schröder say on page 216: The poles in $p^2$ come only from one-particle intermediate states, while multiparticle intermediate states give weaker branch cut singularities. In order to ...
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190 views

Interpreting the Fourier transform of a Gibbs measure

Recall that a Gibbs measure gives a probability distribution on states $x$ of the form $$ p_X(x) = \frac{1}{Z(\beta)}\exp(-\beta E(x)) $$ As I understand, the function $E$ is interpreted as the ...
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322 views

Axiomatic QFT: Time-slice Axiom vs Transformation Properties

I am studying Wightman axioms and Haag–Kastler axioms for QFT from Haag's book "Local Quantum Physics". In both axiomatic frameworks, he introduces the "Time-slice Axiom" (axiom G) as "There should ...
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303 views

Gromov-Witten invariants

I'm a mathematician studying Schubert calculus, and I'm out to compute the Gromov-Witten invariants of the complete flag manifold. Well, I actually already know how to compute them, but only in a way ...
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171 views

Intuition behind $U(1)$-gauge model of Electrodynamics in a general spacetime

As the article Electrodynamics in general spacetime greatly explains, the $U(1)$-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to ...
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188 views

References to Mechanics (Classical, Quantum, Statistical) using Time-Scale calculus?

Time-Scale Calculus, is a theory which unifies ordinary (plus fractional and q-) calculus with discrete (and finite differences) calculus. In a sense, in a similar way the Lebesgue integral (or ...
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114 views

Relation of Betti numbers to Veneziano's scattering amplitude?

I came across Veneziano's famous formula for the scattering amplitude for four tachyons written as $$A(s,t)= \sum_{n \geq0} \frac{(-1)^n}{n-1 + \alpha^{'}s}\frac{P_n(\alpha^{'}t)}{n!},\:\:\:\:P_n(x)=...
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65 views

Solutions of nonlinear systems invariant wrt. perturbations (looking for applications)

I want to ask if the following purely mathematical problem (that I'm working on) might have some applications to physics. The problem in a nutshell: describe properties of solution sets of real ...
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428 views

Lie derivative of Dirac Delta

In the setting of general relativity, I came across a source term of the wave equation of the following form: $$ \frac{1}{\sqrt{q}}\,\delta^{(3)}(p-\gamma(t)) $$ where $p\in M$ is a point in our 4d ...
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135 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 ...
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221 views

Any examples of commensurable subgroups appearing in physics?

I am a mathematician. I am studying and working on Hecke pairs which I am going to give the related definitions in the following. But first let me explain what I am looking for to learn by asking this ...
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169 views

Cauchy Problem for Boltzmann Equations

One of the first profound analysis about the solutions of the Boltzmann Equation was given by DiPerna and Lions in the late 1980s. You can find one of their main papers here: http://www.jstor.org/...
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180 views

Asymptotic limit of the two kink solution of the sine-gordon equation

I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as: $$\phi=4\arctan\left(\frac{\sinh\frac{1}{2}(\theta_1-\theta_2)}{(a_{12})^\frac{1}{2}\cosh\frac{1}{2}(\...
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219 views

K3 gravitational instanton

Could you please recommend a sufficiently elementary introduction to K3 gravitational instanton in general relativity and the problem of finding its explicit form? Under 'sufficiently elementary' I ...
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43 views

Weight $h$ as a function of the central charge $c$ in a Unitary Highest Weight Irrep of the Virasoro algebra

A highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by $(c,h)$, with $c$ the central charge and $h$ the "Witt weight" (the weight of the Witt algebra part of ...
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63 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
83 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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130 views

Why spatial infinity is a point and not an $S^2$?

First a disclaimer, this question already has been asked here, but as pointed out in comments, more detail was required. So this is a more detailed version. Let $(\mathbb{R}^4,\eta)$ be Minkowski ...
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69 views

What is meant by this variant of the euclidean plane: $\mathbb{R}^2_{\hbar}$?

I am reading some papers in mathematical physics (https://arxiv.org/abs/1006.0977) and I came across the following symbol $\mathbb{R}^2_{\hbar}$ I don't recognize nor could I find any background ...
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79 views

Comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons

I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of: Chern class (1st, 2nd), and ...
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199 views

Wess-Zumino-Witten vs. Yang-Mills-Chern-Simons and Kac-Moody$.$

There is a really nice (holographic) duality between 2d Wess-Zumino-Witten and 3d Yang-Mills-Chern-Simons models (cf. Ref.1). For example, for a given gauge group $G$, the spectrum of both theories is ...
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59 views

Finding a set of basis modes for the Klein Gordon field in Kruskal-Szekeres coordinates

Let $\phi$ be a scalar field satisfying the KG equation in the maximal extension of Schwarzschild spacetime $$(\Box+m^2)\phi=0.$$ We introduce Kruskal-Szekeres coordinates $(T,X)$ and for simplicity ...
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102 views

How can Weinberg assume that $P_b$ acts as derivative?

In QM of finitely many degrees of freedom it is well known that due to the Stone-Von Neumann theorem, the CCR $$[Q_i,P_j]=i \delta_{ij} $$ leads to a unique representation up to unitary equivalence, ...
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141 views

Feynman Path Integral and Cardinality of, and Restrictions on, Set of Possible Paths

The Feynman path integral (for a one dimensional space such as the $x$-axis) is an infinite dimensional integral where the number of dimensions is countably infinite (it is an $n$-dimensional integral ...
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264 views

Deriving a Massive Propagator from a Massless Propagator

I'm trying compute something I already know the answer to in order to test myself and gain confidence in my QFT computational skills, but I'm not getting the right factors. The text I'm following and ...
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91 views

How to get the eigenvalue density contribution $\rho_1(x)$?

I'm studying the $1/N$ expansion beyond the planar limit in matrix models. Currently I'm trying to understand and reproduce the results of: Antisymmetric Wilson loops in $\mathcal N \geq 4$ SYM ...
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389 views

Matching Dirac/Majorana/Weyl Spinor Degrees of Freedom in Minkowski signature

Question: How do we match the real degrees of freedom (DOF) of Dirac/Majorana/Weyl Spinor in terms of their quantum numbers (spin, momentum, etc) in any dimensions [1+1 to 9+1] in Minkowski signature?...
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117 views

About Coleman's proof of the nonexistence of Goldstones in 1+1 dimensions

In his famous paper "There are no Goldstone Bosons in Two Dimensions", this link http://projecteuclid.org/euclid.cmp/1103859034, first he states that the $\delta(k^2)$ is not a distribution in two ...
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99 views

What is physical interpretation of a closed subspace of a separable hilbert space?

Can anyone tell me the physical interpretation of a closed subspace of separable hilbert space? I would like to know since quantum mechanics is best described in hilbert space and we have a physical ...
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1answer
175 views

Do translation formulae for generalised solid spherical harmonics exist?

I'm aware of the solid spherical harmonics functions, which are basically the surface spherical harmonics $Y^m_{\ell}(\theta,\varphi)$ with an additional monomial term along the radial direction: $R^...
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2k views

Time to reach steady state in transient 1-D heat conduction

Suppose that there is a slab with thickness $2L$ and other dimensions of the slab are very bigger than $L$. Both sides of the slab are kept in the constant temperature $T_0$. In time $t=0$ a heat ...
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332 views

Rigorous derivation of general relativity from first principles

What is the minimal set of axioms required to derive the mathematical formulation of General Relativity from first principles? What are these first principles? How does such a derivation go step by ...
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744 views

Mathematics of Surface Divergence and Surface Curl

While studying electrodynamics I found two functions - Surface Divergence and Surface Curl - that seem to condense the formulas for superficial discontinuities of the electric and magnetic fields ...
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61 views

Proof that fixed points of a null field are zero

Suppose we have a scalar field $V$ (which can be acoustic pressure, or a scalar electric potential) that is a solution of the wave equation $$\Box V(x,y,z,t) = 0$$ I am wondering if a fixed (non-...
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189 views

What are the implications of integrating the Poisson bracket?

Reading Ref. 1 I admit I am a little lost in some places. I was hoping that someone in this area could explain the basic premise of integrating the Lie bracket and further is it connected to nlab's ...
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252 views

From Newton to Kepler without infinitesimals

I've read some interesting calculus-free proofs of at least parts of the derivation of Kepler's Laws from Newton's gravitational force. One is of course Feyman's "Lost Lecture" (which was already ...
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100 views

Underlying C*Algebra operators in standard quantum mechanics?

Linearity in standard quantum mechanics (QM) is the key to making the math possible in this field, but the presence of nonlinear operators in QM is what is more generally dealt with. Working with the ...
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1answer
103 views

Effective theories and unbounded operators

If you have two operators, one the true Hamiltonian $H$ and one we call an effective Hamiltonian $H_{eff}$ and say they agree on every eigenvector with eigenvalue up to $E_{eff}.$ Above that, they can ...