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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what is meant by completeness of Hilbert space? [duplicate]

I know definition that completeness means every Cauchy sequence of elements of the space converges to an element in the space.But what does it mean physically when we say Hilbert space is complete ...
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How to derive the $\frac{4\pi}{3}\vec{p}\delta^3(\vec{r})$ element for the dipole field, from its potential?

This might be a bit more general question about how to figure out what is the appropriate (delta) expression in singular points, but e.g. for the dipole, we can derive its potential by a taylor ...
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Why is the set of eigenfunctions of a Hermitian operator complete?

When I was studying quantum chemistry, I was told that given the time-independent Schrodinger equation $$\hat H \psi = E \psi$$ since $\hat H$ is Hermitian, the set of eigenfunctions $\{\psi_i \}_{i =...
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$\langle xp+px\rangle|_{t=0}=2\langle p\rangle\langle x\rangle|_{t=0}$ for the free particle?

Quantum Mechanics, Volume 1 by Claude Cohen-Tannoudji, Bernard Diu and Frank Laloe. Complement L-III, exercise 4 (page 342). Basically consider a free particle, and calculate the variance(uncertainty)...
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123 views

Partial deritative of $\partial_V(PV)_T$ with $PV=nRT$

The question arised from thermodynmaics. Suppose $n,R$ are positive constants, and $P,V,T$ are all positive. From $TdS=dE+PdV$, one may obtain $T\partial_V(S)_T=\partial_V(E)_T+P$ where $\partial_V()...
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Wave Equations from Decoupling Maxwell's Equations in Bianisotropic Media

For several days now, I have been trying to decouple Maxwell's equations in bianisotropic media so that I end up with a form that involves only one variable (of E, D, B, H), i.e. a so-called 'wave ...
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91 views

Electric flux over a closed surface when point charge lies on the surface

What will be the electric flux over a closed surface when point charge lies on the surface, that is neither inside nor outside? I ask this question because electric field at that point will be ...
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How do I proceed with the following coulomb integral?

I am trying to solve the $H_{2}^{+}$ ion problem using Fourier transform approach. The Hamiltonian that I am trying to solve is as follows, $$H=-\frac{\hbar^{2}}{2m_{e}}\nabla^{2}_{e}-\frac{e^{2}}{4\...
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Maslov's method of deriving the WKB approximation

For a generic one-dimensional potential, the WKB approximation yields the quantization condition $$ \oint p dq = (n + 1/2)\hbar . $$ Here, the correction factor $1/2 $ was obtained by Kramers by ...
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Complex Analysis Textbook using Fluid Dynamics

This is a literature request. I remember being told of a Complex Analysis textbook that teaches the Cauchy-Riemann equations and Potential Theory using Fluid Dynamics. Does anyone know who the ...
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222 views

How to identify higher-form symmetries?

A $q$-form symmetry is a symmetry that naturally acts on objects whose support is a $q$-dimensional surface (ref.1). For example, what we usually call a "regular" symmetry, is actually a $0$-form ...
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Is there a way to describe gravitational waves and time-dependent gravitation without tensors?

I have been reading about gravitational waves, and they fascinate me. However I struggle to follow the mathematics behind it, because they are described using tensors, index gymnastics, et cetera. Is ...
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Global conformal group in 2D Euclidean space

This is a rather naive question, but I was just wondering. I know that the local conformal algebra of 2d Euclidean space is the direct sum \begin{equation} \cal{L}_0\oplus\overline{\cal{L}_0}, \end{...
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Why do we need these two sets of modes in the gravitational collapse?

Consider the gravitational collapse spacetime: Hawking argues in his paper$^{[1]}$ about black hole radiation that the massless scalar field $\phi$ can be decomposed as $$\phi = \sum_i \{p_i b_i+...
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In what sense are solutions to the Dirac equation and solutions to the Laplace equation equivalent in string theory?

I have come across statements like elementary particles on a Calabi-Yau correspond to harmonic forms (or to cohomology classes, which is equivalent for a compact Kähler manifold, since every ...
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What are good non-paraxial gaussian-beam-like solutions of the Helmholtz equation?

I am playing around with some optics manipulations and I am looking for beams of light which are roughly gaussian in nature but which go beyond the paraxial regime and which include non-paraxial ...
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Ginzburg-Landau boundary condition in the 1D no fields case

It is commonly seen that in finding the coherence length from Ginzburg-Landau, that the following equation is found: $\frac{\partial^2 f}{\partial \eta^2} + f(1-f^2) = 0$ which is for a superconductor ...
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Can we get full non-perturbative information of interacting system by computing perturbation to all order?

As we know perturbative expansion of interacting QFT or QM is not a convergent series but an asymptotic series which generally is divergent. So we can't get arbitrary precision of an interacting ...
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Is there an intrinsic physical meaning for characteristic curves of a PDE?

For partial differential equations (such as those that govern many physical phenomena), there exist characteristic curves, along which the equations can be reduced to total derivatives and solved. The ...
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Is there any proof that any result from perturbation theory is necessary an asymptotic series?

I know that almost all the series coming from perturbation theory are divergent, such as those from eigenvalue problems or the S-matrix in quantum field theory. The lore is that the series are ...
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why is a Lagrangian submanifold a semi-classical state and not a classical state?

I read that the Lagrangian submanifold can be regarded as a semi-classical state when classical mechanics is formulated using symplectic geometry. Does anyone know why it would be a semi-classical ...
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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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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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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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88 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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110 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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174 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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217 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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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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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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3d TQFT for Fibonacci (Yang -Lee) anyons

What is the 3d TQFT whose Wilson line produces Fibonacci (Yang -Lee) anyons? I heard that 3d $SO(3)_3$ Chern-Simons theory produces the correct physics for Fibonacci anyon ($e$). How to show it? If ...
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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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79 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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“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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101 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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347 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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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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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
142 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
121 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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243 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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167 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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240 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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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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179 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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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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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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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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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 ...