# 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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### 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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### 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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### 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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### “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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### 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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### 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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### 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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### 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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### 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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### 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 ...