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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Index theorem of Callias operator in physics
In the article "On the index type of Callias-type operator" (https://doi.org/10.1007/BF01896237) Anghel study the index of a Callias type operator over an odd dimensional complete ...
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Do optimal Lieb-Thirring constants have physical meaning?
In their proof of stability of matter Lieb and Thirring used a particular set of inequalities. Namely, if $H=-\Delta+V(x)$ is a Schrödinger operator, then the sum of (powers of the absolute value of) ...
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Looking for video courses on general relativity, aimed at a mathematician crowd [closed]
I am a mathematician, working in symplectic geometry.
I am looking for online avalible video recordings of courses in general relativity, which are geared towards an audience of mathematicians.
...
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The meaning of a representation in one-dimensional quantum mechanics
In many places, one reads about chosing a representation for studying a particular one-dimensional quantum system. Usual representations are the position representation, the momentum representation or ...
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Book recommendation for mathematical physics [closed]
Recently I was going online through many "Mathematical methods in physics" books.
To name two of them : Arfken, ML Boas.
I want to buy one of the above to start my undergrad physics course (...
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On which bundle do QFT fields live?
In QFT, there is a vector field of electromagnetism, usually notated by $A$, which transforms as a 1-form under coordinate changes. Since quantum fields are operator-valued, I thought it is a section ...
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Effects of Localized Medium Changes on Field Propagation
I've studied various theories related to fields. These theories often include equations describing how the activity of a source is transmitted to other locations. The properties of the medium ...
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Why we use trace-class operators and bounded operators in quantum mechanics?
The set of trace-class operators $\mathcal{B_1(H)}$ on the Hilbert space $\mathcal{H}$ is like the Banach space $l^1$, while the set of bounded operators $\mathcal{B_\infty(H)}$ is like the Banach ...
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Jensen's inequality on (super)operator exponential
Let us define the expectation value $\langle A\rangle_{\rho}$ of a superoperator $A$ over a density matrix $\rho$ as $(\rho, A(\rho))$, where the scalar product between operators reads $(O_1,O_2):= Tr[...
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Equality of Hilbert subspaces
If $A,B\in \mathscr{L_H}$ in the lattice of subspaces of a Hilbert space $\mathscr{H}$,
then is it always true that $$A\sqsubseteq B\ \&\ B\sqsubseteq A \implies A=B\ ~ ?$$
Or is there maybe an ...
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Operator systems in functional analysis & quantum mechanics: intuition
I saw this concept of operator systems in here but I am not sure if I want to get deep into it before knowing roughly what it is used for in, say, quantum information or quantum mechanics.
My very ...
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Are $\mathcal{PT}$-symmetric Hamiltonians dual to Hermitian Hamiltonians?
I was reading this review paper by Bender, in particular section VI where they show that, despite $\mathcal{PT}$-symmetric Hamiltonians not being hermitian, they can have a real spectra. They go on ...
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Angular momentum Lie algebra for infinite-dimensional Hilbert spaces
Let $V := \operatorname{span}{(J_1, J_2, J_3)}$ be a Lie algebra over the complex numbers such that
$J_1$, $J_2$, and $J_3$ are essentially self-adjoint operators on some Hilbert space $\mathcal{H}$.
...
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Electric field due to plane at constant potential and a cylinder with no flux on surface
There is a plate at a constant potential V and potential equal to zero far away. the problem is two-dimensional. For this case, the electric field lines will simply be straight lines. Now let there be ...
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Rigorous definition of the average value $\langle a^{*}(f)a(g)\rangle$ on Fock spaces for arbitrary states [duplicate]
One of the axioms of quantum mechanics states that a quantum state $\rho$ is a positive (hence self-adjoint) trace class operator with trace one. Given an observable $A$, the expected value of $A$ in ...
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Why does the finite trace of density matrix imply a discrete Schmidt decomposition? [closed]
In the paper defining an average Schmidt number for a particular entangled system, Law and Eberly say:
Because density matrices always have finite trace, the Schmidt decomposition is always discrete, ...
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What are the distinct mathematical formalisms of quantum mechanics?
Consider the physical theory called non-relativistic quantum mechanics. What are the distinct mathematical formalisms for this physical theory? That is, different mathematical frameworks for this ...
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Non-orientability in electromagnetism
I'm currently studying E&M and I have a question related to the mathematical formalism of the theory. Electrodynamics depends heavily on the divergence and Stokes's theorem which in their ...
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Concrete statement about QFT not being mathematically rigorous [duplicate]
It is often mentioned that QFT is ill-defined mathematically. I have seen this as stated that QFT can be defined on a lattice, but that it breaks down if the lattice spacing goes to zero. ...
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Elasticity theory: homogeneous deformations of a perfect lattice
I want to understand how the macroscopic (linear) elasticity theory emerges from the microscopic properties of matter. My question is about the model of the "perfect lattice", which is used ...
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Trouble understanding the experssion of gravity on a cube shaped earth
I'm a high school student working on a maths/geophysics project of which my goal is to try to mathmatically expressthe forces that apply on fluids, and then put them together to express geostrophic, ...
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Question about the identity operator and the bosonic ladder operators
Consider a self-adjoint operator $B$, such that for each mode $a_1,...,a_n$ [of a quantum bosonic system with Hilbert space $\cal H$ given by the corresponding Fock space] we have $B a_i B^\dagger = ...
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Determining Bound States from Møller Operator
Hello I came across an interesting property of the Møller operator, which I summarize below:
The Møller operator $\Omega^{(+)}$ maps in-states that belong to the continuum spectrum of the free ...
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Equivalent definitions of Wick ordering
Let $\phi$ denote a field consisting of creation and annihilation operators. In physics, the Wick ordering of $\phi$, denoted $:\phi:$, is defined so that all creation are to the left of all ...
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Scattering matrix vs. unitary transformations
In quantum optics, the input/output bosonic modes at a beam splitter transform according to the scattering matrix
$$
\begin{pmatrix}
a_1 \\
a_2 \\
\end{pmatrix} = \dfrac{1}{\sqrt{...
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Multimode squeezed operator
Given CCR (bosonic) algebra, with creation / annihilation operators $a_{i}^{\dagger}, a_i$ acting on a single particle Hilbert space $\mathbb{h}$, let's introduce the multimode squeezed operator for $...
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Why does a singularity imply the need for a distribution?
I am following Section 11 of Prof. Etingof's MIT OpenCourseWare notes on "Geometry And Quantum Field Theory" in which he says:
...for $d = 1$, the Green's function $G(x)$ is continuous at $...
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When do two state functions represent the same quantum state?
According to the standard quantum mechanics, quantum states are one-dimensional subspaces of a separable Hilbert space. In practice, this Hilbert space is $L^2(M)$ where $M$ is the classical ...
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Correlation length in a 3d Ising slab with one dimension much smaller than the other two
Suppose I have a 3d Ising model on a cubic lattice, but one of its dimensions is much smaller than the other two. That is, I have an $L$ by $L$ by $L'$ slab with $L' << L$; in particular, $L'$ ...
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Why is $H = J \sum_i (S^x_i S^x_{i+1} + S^y_iS^y_{i+1})$ always gapless for any spin $S$?
In the following I have in mind antiferromagnetic spin chains in periodic boundary conditions on a chain of even length $L$.
Consider the spin-$S$ spin chain
$$H = J \sum_{i=1}^L (S^x_i S^x_{i+1} + S^...
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Holonomic constraints as a limit of the motion under potential
In Mathematical Methods of Classical Mechanics, Arnold states the following theorem without proof in pages 75-76:
Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local coordinates where
$...
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(Time-)Orientability in the Language of Fiber Bundles
I'm currently studying spin geometry through Hamilton's book Mathematical Gauge Theory. At a given point, Hamilton considers a pseudo-Riemannian manifold, which I'll take to be Lorentzian in $d=3+1$ ...
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Azimuthal coordinate operator: Hermition or not? Self-adjoint or not?
I am told that the azimuthal coordinate operator $\hat{\phi}$ is not self-adjoint. I am told this by people who I am sure know much more about this stuff than I do. To my unsophisticated mind, "...
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Einstein's gravity Lagrangian invariance under the change of differential structure
I came across an article claiming the appearance of singularities in the energy-momentum tensor $T_{\mu \nu}$ as a result of changing the differential structure:
I wonder what symmetry or current (in ...
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Why are light rays expressed or generally shown as a sine wave? [duplicate]
Whether it be a YouTube video or book lights rays i.e(the visible specturm,uv rays) are expressed as a sine wave. Why is it so?
What is the plot of its against? Why does it oscillate? Why does it ...
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Limit for big system size of Fokker-Planck eigenfunctions
I am learning how to use diagonalization methods applied to Fokker-Planck equations with Gardiner's book and these notes. The idea is to find the probability density, $ P[X_t\in[x,x+dx]]=\rho_t \, dx$,...
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What does it mean for classical mechanics to be based on the category of sets?
It is quite common[1][2] in the study of physics in the context of category theory to say that one of the fundamental difference between classical mechanics and quantum mechanics is that classical ...
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The limit of time evolution operator
Through reading Nenciu's rigorous proof on the Adiabatic Theorem and Gell-Mann-Low Theorem, I found:
Since the limit $t_0\to-\infty$ does not exist for $U(t,t_0,\epsilon)$, in order to make use of ...
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Is integral of energy-momentum tensor in QFT over a region $R$ self-adjoint?
Consider a quantum field theory in flat 1+1D spacetime for simplicity. Let $T_{\mu\nu}$ be the conserved symmetric stress tensor. One writes operators by integrating the tensor over the whole space, ...
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Mathematical explanation of the extinction paradox
I am trying to learn properly about scattering. For this I was pointed to Wave Propagation and Scattering in Random Media by Ishimaru.
I got a bit stuck in section 2-2 General properties of the Cross ...
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Analytical continuation as regularization in Quantum Field Theory, the remaining questions
There is an old question posted (Regularization) which did not get an answer, about the validation of analytic continuation as regularization. It did get some discussion in the comments, referring to ...
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Mathematical references for gauge theory in condensed matter physics
I am currently trying to go through some literature on the classification of symmetry protected topological phases. Primarily, I am interested in the classical of topological phases using mathematical ...
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Why are these unbounded operators (essentially) self-adjoint?
Can anyone provide exact mathematical reasoning as to why the following fundamental unbounded symmetric operators are essentially self-adjoint? I.e. on, their natural domains, they admit a unique ...
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Use of mathematical structure on physics [closed]
I want resources for studying in detail the connection between the mathematical structures of physical theories and said physical theories.
For example, i know what a Hilbert space or a principal ...
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Charge Density Of A Conducting Strip
I am currently doing a project which requires me to figure out the charge density of a strip. Assume that the strip is isolated in a vacuum.
Assume the strip is 1 dimensional, kind of like a rod. What ...
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How do we know Schwinger functions exist?
Let $\mathcal{D}'(\mathbb{R}^n)$ denote the dual of $C^\infty_C(\mathbb{R}^n)$, that is distributions on the set of infinitely differentiable functions with compact support. If $d\mu$ is a probability ...
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Conformal Transformation of Torsion
It is well known that under a conformal transformation, we have
$$\tilde{g}_{\mu \nu}=\Omega^2 g_{\mu \nu}, \; ; \tilde{w}_{\mu}=w_{\mu}-\frac{1}{\alpha} \partial_{\mu} \log(\Omega^2),$$
where $\...
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Outer product as an operator in an infinite dimensional Hilbert space
The outer product between a bra-ket $|a\rangle\langle a|$ where if $|a\rangle\in\mathcal{H}$ and $\langle a|\in\mathcal{H}_{dual}$ is a vector in the tensor vector space formed by the Hilbert space ...
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How does one rigorously define two-point functions?
Let $\mathscr{H}$ be a complex Hilbert space, and $\mathcal{F}^{\pm}(\mathscr{H})$ be its associated bosonic (+) and fermionic (-) Fock spaces. Given $f \in \mathscr{H}$, we can define rigorously the ...
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Thermal ground state?
Consider a system of $N$ fermions in a periodic box $\Lambda \subset \mathbb{R}^{d}$, described by the Hamiltonian
$$H_{N} = \sum_{k=1}^{N}(-\Delta_{x_{k}}-\mu) + \lambda \sum_{i< j}V(x_{i}-x_{j}) \...
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