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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944 views

Zero velocity divergence for incompressible flow is derived from conservation of energy equation or conservation of mass equation?

I'm a bit confused about incompressible flow definition. In many textbooks or scientific articles, they simply claim that the incompressibility condition for Navier-Stokes equation is: $\nabla \cdot \...
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0answers
90 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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1answer
96 views

Link between dynamical algebra and symmetry group

I was wondering if there is a known link between dynamical algebra and symmetry group. In particular: Do all Hamiltonians belonging to certain dynamical algebra share the same symmetry group? Do ...
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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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1answer
117 views

Eigenvalues of a quantum field

In the book 'Quantum field theory for the Gifted Amateur", the following is stated, cf. 9.3: "A quantum field $\hat{\phi}(x)$ takes a position in spacetime and returns an operator whose eigenvalues ...
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1answer
183 views

How to understand Duhamel's principle?

I have difficulty about the explanation of Duhamel's principle on my book. Here is what's written on my book: Take wave equation as an example. Consider the equation: \begin{cases} \frac{\partial^2u}{...
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1answer
240 views

Do higher homotopy groups play any role in gauge theory?

As is more-or-less well-known, the magnetic monopoles of a gauge theory are classified by the first homotopy group of the gauge group, $\pi_1(G)$ (cf. Lubkin (1963)). The second homotopy group is ...
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66 views

How to make sense of this uncountable tensor product construction?

The reference for this discussion is this paper. It is a paper related to the Unruh effect and QFT. The problem is the following: as is well-known, when we quantize a KG field, we get a Fock space ...
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1answer
93 views

U(1) Dirac string moved to the SU(2) or SO(3) gauge theory

Dirac string describes the string connecting the U(1) magnetic monopole to the U(1) anti-magnetic monopole in the U(1) gauge theory. Since U(1) is a subgroup of SU(2) and SO(3), we may embed the U(1) ...
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50 views

Phase transition between two CFTs

If we start with a CFT (say CFT$_1$) and deform it by some relevant operator, in the IR we can get another CFT (say CFT$_2$). This is flowing from one CFT to another. I was wondering whether there ...
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1answer
81 views

Derivation of $j$ being a 4-vector in Landau-Lifschitz: Formulation with rigorous mathematical treatment?

Here on Stack exchange, there appeared the question on how to derive the 4 current actually being a Lorentz-tensor. One of the answers (How do we prove that the 4-current $j^\mu$ transforms like $x^\...
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1answer
415 views

“Weak” and “Strong” topological insulators

For translationally invariant systems, we can define some topological invariant based on the translational symmetry, which is referred to "weak" topological invariant. For example, according to Kitaev'...
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1answer
109 views

In Algebraic QFT, is the state observer dependent?

In the usual approach to QFT presented e.g., in Weinberg's book, the state of a system is dependent on the observer. Quoting this book, in page 109 we have: Notice how this definition is framed. To ...
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1answer
386 views

Schwartz's and Zee's proof of Goldstone theorem

In Refs. 1 & 2 the Goldstone theorem is proven with a rather short proof which I paraphrase as follows. Proof: Let $Q$ be a generator of the symmetry. Then $[H, Q] = 0$ and we want to consider ...
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129 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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167 views

Is the converse of Weinberg's statement on the cluster decomposition principle true?

In Weinberg's "The Quantum Theory of Fields, Vol. 1", Section 4.4, page 182, the author says: We now ask, what sort of Hamiltonian will yield an $S$-matrix that satisfies the cluster decomposition ...
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3answers
147 views

How to mathematically prove that point charge and infinitesimal volume charge are same?

In electrostatics, while deriving certain elementary equations, I have seen all the books just assuming that point charge and infinitesimal volume charge are same. How can we rigorously, ...
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1answer
131 views

Formal definition of Green function

The formal definition of a Green's function is: \begin{equation} L(\mathbf{r})G(\mathbf r,\mathbf r^\prime) = \delta(\mathbf r-\mathbf r^\prime), \tag 1 \end{equation} where L is a time linear ...
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3answers
251 views

Physical intuition behind Poincaré–Bendixson theorem

The Poincaré–Bendixson theorem states that: In continuous systems, chaotic behaviour can only arise in systems that have 3 or more dimensions. What is the best way to understand this criteria ...
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1answer
178 views

Can any vector field be decomposed into a curl-free part and a divergence-free part?

In this question, asked by @Emilio Pisanty, he says that "...the polarization can be split into a curl-free component, which is the gradient of something, and a divergence-free component, which is ...
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1answer
256 views

The Hilbert space of Chern-Simons on a torus, part one$.$

There is a key result in 2+1 dimensional Chern-Simons theory, which was first discussed in ref.1.: the Hilbert space of the theory, when quantised on $T^2\times\mathbb R$, is isomorphic to $$ \frac{\...
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2answers
135 views

Action of conjugate momentum on $TM$ and explicit form

In hamiltonian mechanics the phase space of a particle is a symplectic manifold. In the case we have a configuration space $M$, that is the manifold describing the possible positions of the particle, ...
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1answer
194 views

Green's function in Frequency Domain

I am learning some basics of Green's functions applied in physics from the article https://arxiv.org/abs/1604.02499 I am struck at equation no (23) which is said to be derived from equation (22) by ...
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57 views

Which geometry does not allow the existence of matter?

I have seen these lectures by Fredric Schuller that discuss the obstruction theory and the role of global geometric properties in admitting a spin structure. See the video at 01:27:52 https://youtu....
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1answer
84 views

Lieb, Seiringer: Stability of Matter in Quantum Mechanics equation 2.2.7 [closed]

In page 27 of "Stability of Matter in Quantum Mechanics" by Lieb and Seiringer, it states: An application of Hölder's inequality to (2.2.4) yields, for any potential $V\in L^{d/2}(\mathbb{R}^d),\ d\...
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5answers
4k views

Newton's law requires two initial conditions while the Taylor series requires infinite!

From Taylor's theorem, we know that a function of time $x(t)$ can be constructed at any time $t>0$ as $$x(t)=x(0)+\dot{x}(0)t+\ddot{x}(0)\frac{t^2}{2!}+\dddot{x}(0)\frac{t^3}{3!}+...\tag{1}$$ by ...
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1answer
66 views

Is the string-net model Hermitian?

In Kitaev and Kong's paper, they define the Hermitian inner product on morphism spaces in Eq. (11). My question is that: Given that F symbols satisfy the pentagon identity, does that the string-net ...
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1answer
207 views

Angular Momentum Eigenvalues in Two Dimensions

Suppose we have a particle moving in a circle on the $xy$ plane. Then the angular momentum operator will be just $L_z = ε_{3jk} x_jp_k$ and $L^2 = L_z ^2$. Then if $m$ are the eigenvalues of $L_z$ we'...
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72 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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0answers
193 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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1answer
54 views

Is a displacement a vector, a line segment, or something else?

It probably seems ridiculously naive of me to ask such a basic question, but I have a need to use accurate language. Typically, I think of a displacement as a directed line segment whose end-points ...
3
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2answers
76 views

Multiplying Distributions in finite-temperature Keldysh/Thermo-field field theory

In the real-time finite temperature formalisms (Schwinger-Keldysh or Thermo-field), the free propagators are often defined with terms like: $$ \mathrm{Dirac\ Delta}\ \times \ \mathrm{Thermal\ ...
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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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1answer
189 views

Weinberg's classification of one-particle states and representations of the Poincare group

A representation of a group $G$ is a pair $(\rho, V)$ where $V$ is a vector space and $\rho : G\to GL(V)$ is a homomorphism. If $V$ is actually a Hilbert space and $\rho : G\to \mathcal{U}(V)$ maps ...
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0answers
383 views

Level-rank duality in WZW models and CS theories

Cross-posting from Physics Overflow: https://www.physicsoverflow.org/41281/level-rank-duality-in-wzw-models-and-cs-theories I know that the classical level-rank duality in the $\widehat{\mathfrak{sl}}...
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0answers
57 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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0answers
61 views

What are the applications of Weingarten calculus? [closed]

Weingarten calculus is a general and explicit method to compute the moments of the Haar measure on compact subgroups of matrix algebras1,2. My gist is that it is somehow related to random matrix ...
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0answers
77 views

What classes should i choose in order to study theoretical physics? [closed]

I am an undergraduate physics student who wants to be a theoretical physicist. The math department in my uni offers the following sequences of classes: ~Analysis 1 -> Analysis 2 -> Topology ~...
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0answers
31 views

periodic orbits in Gutzwiller's trace formula

It is said that in the Gutzwiller trace formula, one sums over the periodic orbits. I do not know how to derive the formula, but a simple question arises for me. That is, for some classical ...
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100 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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61 views

How to experimentally construct an imaginary(complex) potential, and what does it mean?

Some useful reference may be from: Finding Stagnation Points from the complex potential Imaginary potential and stationary wavefunction What is the physical interpretation of imaginary terms in the ...
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1answer
184 views

Can the fundamental laws of Physics be formalized as axioms written in First Order Logic (FOL), or any other logical system for that matter?

Is it possible to state any fundamental law in Physics as an axiom written in First Order Logic (FOL), or any other logical system (and semantic interpretation) for that matter? EDIT: in a way that ...
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1answer
124 views

Classical Green's function

I am confused about a classical calculation involving Green's functions. Suppose $\frac{d^2}{dx^2}G+k^2G=\delta(x)$. Then Fourier theory leads, apart from numerical factors, to $$G(x)=\int \frac{e^{...
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1answer
81 views

How to carry out this kind of analysis in Relativity?

In the paper arxiv.org/abs/0801.0926 the authors propose the following simple system of particles in Newtonian mechanics. Four particles of same mass placed as follows The $a(t)$ and $\theta(t)$ can ...
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2answers
426 views

Matrix derivative of a matrix with constraints

I am looking for a general method to obtain derivative rules of a constrained matrix with respect to its matrix elements. In the case of a symmetric matrix $S_{ij}$ (with $S_{ij}=S_{ji}$), one way to ...
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0answers
17 views

What is the calculation of the Hubble time based on? [duplicate]

I understand that Hubble's law states that:$\\$ $v=H_{0}D$ However, given that the Hubble time (an estimate of the age of the universe) is given by $\frac{1}{H_{0}}$, doesn't that mean... $v=H_{0}...
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1answer
133 views

Reference Request: Basis-independent formulation of tensor networks

I could not find any references for a basis-independent formulation of tensor networks: All papers I have found use pretty much (explicitly or implicitly) the canonical computational basis by defining ...
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1answer
73 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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0answers
31 views

Loop divergences and Poincare group

I was reading the paper "A Prehistory of n-Categorical Physics" by John Baez. In it he computes a single loop in a spin network, which has as gauge group $SU(2)$, and shows that it gives the dimension ...
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1answer
191 views

The DLR equations and limit of local Gibbs states

I am interested in mathematical theory of equilibrium states at positive temperature. According to the monograph by Barry Simon ...