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 this definition of the Fourier Transform of a quantum field operator rigorous?
Let there be a a quantum field operator $\hat\phi(t,\vec{x})$ which, because it acts (pointwise) on a separable Hilbert space, I expect I can write as
$$\hat\phi(t,\vec{x}) = \sum_n\sum_m\phi^n_m(t,\...
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Why are scattering states orthogonal in general?
A similar question was asked here. Unfortunately, I don't think the question was well-formulated/explained enough for people to actually care about the answer, so let me provide some reason to why I'm ...
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Relation between the residues of the correlation functions of field redefinitions
The aim of this post is to prove the equivalence theorem. In the post Equivalence Theorem of the S-Matrix, they treat the subject but they do not prove the relation of the residues.
The LSZ reduction ...
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What is the current status (October 2021) of the Yang-Mills existence and mass gap problem?
One of the famous "millennium prize problems" is the "Yang–Mills existence and mass gap" problem, which in its official description by E. Witten and A. Jaffe has the following form:...
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Question regarding Fourier transform of Green functions
The quasiparticle Green's function is defined,
\begin{equation}
G^+_{\lambda \lambda'} (\tau) = -i (1-n_\lambda) \delta_{\lambda \lambda'} \begin{cases} -i \ \text{exp} (-i \epsilon_\lambda \tau) , &...
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Maximal entropy distribution in a finite doman with random endpoints
I am trying to solve for the distribution of a random variable $x$, which will maximise my entropy in a finite domain, let's say $[0, R]$.
$$
S = -\int_0^Rdxp(x)\ln p(x)
$$
The distribution that ...
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Canonical Invariance in the Context of Thermodynamics of the Quantum Harmonic Oscillator
I'm reading a paper titled "Thermodynamical analysis of a quantum heat engine based on harmonic oscillators". Section $C$, is dealing with a subject called "Canonical invariance". ...
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Defining the functional integral measure from the generating functional
In standard QFT we define the generating functional from the functional integral as $${\cal Z}[j]=\int\mathfrak{D}\phi e^{-S[\phi]+i\int d^Dx j(x)\phi(x)}\tag{1}.$$
On the other hand, intuitively ...
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Equivalence between Hamiltonians of different dimensions
Consider a 2D lattice Hamiltonian $H_2$ of symmetry class A and a 1D ladder Hamiltonian $H_1$ of class AIII having the same number of bands and the same TKNN number for each band.
Can $H_2$ and $H_1$ ...
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How to justify $\int D\phi\exp[-\frac{1}{2} \int d^4 x'\int d^4 x\phi(x')M(x',x)\phi(x)]$
It's related to a homework and exercise. The homework was more complicated, but I needed this to figure out the convention that was used.
Consider the integral
$$\int D\phi\exp[-\frac{1}{2} \int d^4 x'...
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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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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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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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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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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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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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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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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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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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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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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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