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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Is the Hamiltonian the only quantum observable with a mixed spectrum?

Let $\mathscr{H}$ be a complex separable Hilbert space of a quantum system. Assume that the Groenewold-van Hove no-go theorem did not necessarily apply and we are free to map all possible polynomial ...
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Do the topology and metric of spacetime determine whether a conformal field theory has a nonzero mass gap?

Many introductions to conformal field theory (CFT) emphasize the case where spacetime is $\mathbb{R}^n$ with the usual flat metric. In those cases, a CFT cannot have a nonzero mass gap, because that ...
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Why is the space $L^2(a,b)$ preferred over the space $C([a,b])$ of all continuous functions on $[a,b]$?

This question might be better asked on the Math.SE site but I feel it could be well placed here as well. My textbook (Sturm-Liouville Theory and its Applications , Al Gwaiz) defines the vector space $...
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How to rigorously prove that the toric code ground state has non-trivial topological order?

Consider the unique ground state $|\psi\rangle$ of Kitaev's toric code model on a sphere. Has it been rigorously proved that $|\psi\rangle$ cannot be transformed into a trivial product state by ANY ...
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When are Feynman diagrams Borel summable

I've been trying to understand Feynman diagrams more rigorously, and it seems that everything can be rigorously defined as long as the Feynman diagrams are Borel summable. However, are there any good ...
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Cauchy sequences through examples in Quantum Mechanics (at the level of the rigor of physicists)

I have just read the definition of a Cauchy sequence: A sequence ($\psi_n$) is a Cauchy sequence in a vector space $V$ when $||\psi_n-\psi_m||\to 0$ when $n,m\to\infty$. The limit of every Cauchy ...
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What is meant by the completeness of a Hilbert space? Why do we need this property for the vector space of quantum mechanics? [duplicate]

The quantum mechanical Hilbert space is defined as a complex vector space that is complete and has an inner product defined on it. Please help me understand the meaning of "complete" in this ...
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What does this definition of a Weyl star algebra in Spectral theory and QM by Moretti, 2013 mean?

I don't understand the words in boldface: Definition 11.25 Let $X$ be a (non-trivial) real vector space of arbitrary dimension (possibly infinite) and $\sigma : X \times X \to R$ a symplectic form on ...
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Can we get the discretization of $k$ due to boundary conditions by solving wave equation using Fourier transform?

I am interested in vibrations of a string in different modes with its two end fixed. But I want to use the method of Fourier transform rather than the method of separation of variables. Using the ...
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38 views

Completeness of Landau basis

We know that the Landau Hamiltonian (uniform magnetic field) is diagonalized by wavefunctions $|n,m\rangle,n,m\in \mathbb{N}$ in the symmetric gauge. However, does this set of functions form a "...
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How does flux on a lattice model translate to the continuum

Consider a square lattice with bipartite hopping terms (e.g., nearest neighbor hopping) $t_{xy}$. Then a magnetic field can be modeled by complex $t_{xy}$ so that the flux enclosed by some closed path ...
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99 views

Fourier transform in crystallography: why do the bounds of the fourier integral have to be symmetric about the origin?

When analyzing the diffraction patterns of x-rays on crystals, we utilize the formula for the scattering intensity ($I(\vec{K})$): $I(\vec{K})\propto \left|\sum_G \rho_G\int_V e^{i(\vec{G}-\vec{K})\...
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Minor details of Mermin-Wagner

In the proof of Mermin-Wagner (e.g., scholarpedia), there is a minor assumption that the average magnetization $m_\Lambda (h)$ converges in the thermodynamic limit $\Lambda \to \mathbb{Z}^d$ to some $...
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Equivalence between canonical ensemble and grand canonical ensemble

I have read that working in the grand canonical ensemble (i.e., with chemical potential $\mu$) and in the canonical ensemble (i.e., working in the $N$-particle Fock space) is equivalent in some sense, ...
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Why does centre of mass formula always lead to a vector joining origin to com point for whichever origin that one may choose?

A simple question that I had from long, the position vector from an origin to the centre of mass is given as $ \frac{ \int \vec{r} \rho dV}{M}$ where $\vec{r}$ is the position vector to the mass ...
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Lie algebra generators as rank-16 matrix spinor representations of $𝑆𝑝𝑖𝑛(10)$

A simple Lie group $𝑆𝑝𝑖𝑛(10)$ has a spinor representations of 16 dimensions, which is distinct from the vector representation of 10 dimensions (coming from standard vector representation of SO(10))...
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Slip-line fields on the the interior of a closed domain

This is a crossed post. Slip-line field theory* is a technique (essentially a change of variables, but with a great deal of physics) to solve problems in elasticity theory, in particular plasticity in ...
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Is Brillouin-Wigner (BW) perturbation correct?

$\DeclareMathOperator{\tr}{tr}$I am rather troubled by how BW perturbation is derived, i.e., my main concern is the assumption of intermediate normalization, i.e., $\langle \psi_0|\psi \rangle=1$, so ...
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10-dimensional and 15-dimensional matrix representations of $SU(5)$: explicit 24 Lie algebra generators

There are some previous discussions in this post Representation of the $\rm SU(5)$ model in GUT which confused me. So I want to follow up with a new question. It is easy to write down the 5-...
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Moore's function

In the dynamical Casimir effect, the Casimir force is given in terms of Moore's function R which satisfies $$R(t+L(t))-R(t-L(t))=2$$ where $L(t)$ is the trajectory of a moving mirror (while another ...
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Difficulties in proving the area-law conjecture in higher dimensions

A very famous and important open conjecture in condensed matter physics is the area law of entanglement entropy, which claims that in a locally-interacting quantum many body system, if the ground ...
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A conjecture on energy distribution of product states in locally interacting systems

Let $\hat{H}$ be a locally-interacting quantum many body Hamiltonian, for example the nearest-neighbor interacting quantum Heisenberg model or Hubbard model, and let $|\psi \rangle$ be an arbitrary ...
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Spin structures and boundary conditions for worldsheet fermions

The definition I'm aware of a spin structure is the following one: Definition: Let $(M,g)$ be a semi-Riemannian manifold with signature $(p,q)$. Let ${\cal F}M$ be the principal ${\rm SO}(p,q)$-...
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Why do we consider the Witt algebra to be the symmetry algebra of a classical conformal field theory?

In standard physics textbooks, it is usually stated that the Witt algebra is the symmetry algebra of classical conformal field theories in two dimensions. Following M. Schottenloher, A Mathematical ...
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Variation of a time-ordered exponential

Consider the time-ordered exponential (Wilson line): $$ U(t_{f},t_{i}) = \mathcal{T}\text{exp}\left(-i\int_{t_{i}}^{t_{f}}\mathcal{A}(t)dt\right)\tag{1} $$ Where $\mathcal{A}(t)$ is some matrix-valued ...
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39 views

Uniqueness of Fock space

Given a single-particle Hilbert space, it's not hard to construct a Fock space using tensor products and symmetrization/anti-symmetrization projection operators, from which we can define creation/...
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Differential forms in projective space

I am currently reading some paper about the Amplituhedron, and it is using projective geometric way to present amplitudes. How can we define forms in projective space to measure volume for a polytope?
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Magnetic monopoles in an $SU(2)$ gauge theory

I had heard from a professor saying that "Polyakov and ’tHooft discover the magnetic monopoles in $SU(2)$ gauge theory with scalar fields [Georgi-Glashow model]." And he cited two references:...
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67 views

Klein-Gordon equation on a compact, two dimensional domain

Consider the Klein-Gordon equation in two dimensions on any compact subset of $\mathbb{R}^2$ (that is, a Jordan domain). The equation is hyperbolic, and since the domain is compact it is not evident ...
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Can Feynman-Kac formula relate to partition function in a rigorous way?

Feynman-Kac formula reads: $$\psi(x,\beta) = \int e^{\int V({\bf{x}}(\tau))d\tau}\psi_{0}({\bf{x}}(\beta))d\mu_{x}({\bf{x}}). \tag{1}\label{1}$$ This is a rigorous formula, defined by means of a ...
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Rigorous Hall conductance

I have been trying to understand the rigorous argument for calculating the hall conductance by averaging over two fluxes by reading this paper {1}. I think I understand the entire derivation, except ...
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Spectrum of periodically driven Floquet operator

There is a periodically driven $XX$ model with alternating field. The piecewise Hamiltonian acts as following way \begin{equation} H_1 = \sum_{i=1}^{N-1}(\sigma^{x}_{i}\sigma^{x}_{i+1}+\sigma^{y}_{i}\...
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Bloch functions vs. Bloch state vectors

Let $$\hat{H}=\frac{\hat{\mathbf{p}}^2}{2m} + V_L(\mathbf{r})$$ where the lattice potential $V_L(\mathbf{r})=V_L(\mathbf{r}+\mathbf{R})$ for any lattice vector $\mathbf{R}$, and let $\hat{T}_{\mathbf{...
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105 views

Reference Request: Mathematical Foundations of Physics

I am looking for reading on examples, or preferably a comprehensive summary on how the foundations of mathematics are related to physical theory. I would like to know whether basic set-theoretic and ...
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115 views

Infinite number of primaries in CFT

I want to prove the fact that there are infinite number of primary operators in CFT by Conformal bootstrap. However, for that I need to show that the crossed conformal blocks $g_{\Delta,\ell}(1-z,1-\...
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168 views

Quiver Mechanics

What do you suggest as an essential and introductory set of references in Physics literature for learning quivers? Any textbook?
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2 magnet bars are placed in same plane and one of them can rotate freely. The relation as system is balanced [closed]

Each magnet is in same plane. $$ \ell \ll r $$ The magnet1 has been fixed. The magnet2 can rotate with center of the magnet itself. $$ \theta_{1} ~~,~~ \theta_{2} :=\text{each angle between the ...
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Is there a foundation of mathematical logic? [duplicate]

As Mathematics has its foundations in logic and set theory in the sense that you can derive all of mathematics from such theories, does mathematical physics have such foundations? A theory or theories ...
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What is the difference between the Interaction picture (Dirac Picture) and a rotating reference frame?

In David McIntyre's Quantum Mechanics, we examine an electron within a magnetic field $$\vec{B}=B_o \hat{z}+B_1[\cos(\omega t)\hat{x}+\sin(\omega t)\hat{y}]$$ The Hamiltonian is then time-dependent ...
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Electrostatics: the field between a wedge-type protrusion and parallel plate

I am interested in modeling the electric field between a wedge-type protrusion and a parallel plate, as shown in the diagram below. There exist many solutions to similar problems for field emitters ...
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Why the Minkowski and Euclidean spinors need to be fermions?

Minkowski spinors are the spinor representations of the spin group $Spin(1,d)$ of spacetime rotational symmetry. Euclidean spinors are the spinor representations of the spin group $Spin(1+d)$ of ...
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Literature recommendations for the relationship between fundamental physics and pure mathematics?

So, I have been reading up on the works by Kenneth Wilson, mainly his 3 statements that he concluded to be true about our universe. His first: 'There exists a hierarchy to our universe'. From this i ...
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How does the fractional Fourier transform apply to an out-of-focus imaging system? Do we use the fractional distance to the focal plane?

In Fourier optics it is sometimes convenient to think of lenses as "Fourier transformers". For an imaging system between two planes with a pupil in the center, the amplitude in the pupil is ...
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Michio Kaku: General relativity action is not bounded from below (?)

In p.9 of Michio Kaku book Introduction to Superstrings and M-Theory-Springer (1998), he said General relativity (GR) is also plagued with similar difficulties. The GR action is not bounded from ...
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Expansion with respect to a uncountable set of eigenvectors

Let $\mathscr{H}$ be a Hilbert space. If $\{e_{\alpha}\}_{\alpha \in I}$ is a Hilbert (orthonormal) basis, one can write every element $\psi \in \mathscr{H}$ as: \begin{eqnarray} \psi = \sum_{\alpha \...
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Logarithm term in light-cone expansion

I'm cross-posting from Maths site here because I don't know where this question fits better. I'm trying to understand how the logarithm in eq. (3.25) from this paper appears from the equation above eq....
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Topological Field Theory for Physicists [duplicate]

I was wondering if anyone knows good resources for Topological Field Theories aimed at physicists. In particular, I am looking for references which are full of examples, starting with simple toy ...
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1answer
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Clustering of correlations in extremal thermal states

Background Consider a quantum system described by an algebra $\mathcal{A}$ of local observables, which are supported on subsets of the lattice $\Lambda = \mathbb{Z}^{d}$. Given an observable $A$, let $...
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Why is it possible to find electrostatic energy of a conductor by integration of $\int_{space} \vec{E}^2 dV$?

In Griffith's Introduction to Electrostatics, International 4th ed, pg-94 these two equations are given for calculating energy of continous charge distributions: $$ W= \frac12 \int \rho V d \tau \tag{...
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Pseudo vector rotation

I originally posted this in the math stack exchange but after no one answered I remembered that tensors are way more used in physics than in math. in my textbook, it says that if we have 2 vectors, a ...

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