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.
2,265
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Error in linear interpolation of $n$-dimensional curves [migrated]
Let's assume we are given an $n$-dimensional smooth curve $\gamma:[a,b] \rightarrow \mathbb{R}^n$ and $N$- sampled points $\{x_1,...,x_N\}$ of that curve.
Now we use linear interpolation (or a higher ...
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Pre-requisites for V.I. Arnold's mathematical methods for classical mechanics
I am an undergraduate, studying physics. I have studied maths courses like Groups, Linear Algebra, Real analysis, Differential geometry and probability. I wish to get into mathematical physics, ...
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Discrepency regarding term in the Wigner Characteristic Function
Consider the quantum characteristic function, $$
\chi_\rho(\mathbf{s})=\operatorname{tr}\{\hat{\rho} \hat{D}(\mathbf{s})\}
$$
The displacement operator is defined to be: $$
\hat{D}(\mathbf{r})=\exp \...
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Is the random current model tight (in the sense of probability)?
Let us consider the random current model (of the classical Ising model) on $\mathbb{Z}^d$. More specifically, we have probability measures $\mathbb{P}_L$ on the product space $\mathbb{N}^{E_L}$ where $...
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Analogue of Bargmann's theorem for Super Lie groups
Bargmann's theorem gives the criteria under which a projective representation of a Lie group $G$ can be lifted to a representation of its universal cover. More generally, if this criterion, namely $H^...
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Is the random cluster model ergodic/mixing?
Consider the random-cluster model $\mathbb{P}_G^i$ on a finite graph $G$ with parameters $p\in [0,1]$ and $q \in \mathbb{N}_+$ and boundary condition $i=0,1$ (free and wired). The main 2 references ...
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Green Function associated to a periodic Schrodinger operator
If $V:\mathbb R\to \mathbb R$ is an $L$ periodic function in $\operatorname L^{\infty}$ we can always find two independent solutions for $$\psi''(x)+V(x)\psi(x)=E\psi(x)$$
$\psi^{\pm}(x)=e^{\pm ipx}\...
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Free electromagnetic field BV action
I am trying to write down the extended BV-action of the free electromagnetic field in a physicist notation, but I don't find it anywhere. I found the following formula in example 3.1. of the paper ...
- 473
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Rigorous proof that energy level of helium is discretized [duplicate]
For the hydrogen atom, a simple separation of variables give the energy eigenvalue of the Schrodinger operator for one electron in a spherical potential. It is well known that there are no such ...
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Rigorous Theory of Path integrals [duplicate]
Does there exist a mathematical rigorous theory of the Feynman-Path-Integral in Quantum Mechanics or Quantum Field Theory?
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Questions about $C^*$-algebra approach to QM
I'm trying to read An Introduction to the Mathematical Structure of Quantum Mechanics by Strocchi, which (as I understand) takes measurements first and states second, and argues that if we allow ...
- 149
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Spectral families of commuting self-adjoint operators
I don't know if math stack exchange is more suitable for this question, but I'll try here first.
It is often stated in quantum mechanics textbooks (e.g. the first volume of Cohen-Tannoudji, Diu, Laloë,...
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Why energy-positivity?
In any relativistic quantum field theory, we require that the spectrum is bounded from below. The typical explanation is that this condition enforces the stability of the theory. However, to me this ...
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What exactly is the relation of the continuous spectra with intervals?
I've read in multiple quantum mechanics books that the name "continuous" of the continuous spectra is said continuous because in many examples it is an interval of values. But I couldn't ...
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Are band structures non-analytic only at degenerate points?
The electronic properties of (crystalline) solids is typically described in terms of the electronic band structure, which reveals many properties of the electronic structure such as the band gap, ...
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Eigenfunctions of the Dirac Hamiltonian with time-dependent electromagnetic potential
I am not looking for explicit solutions to the eigenvalue equations, but rather for general function spaces.
For free Dirac equation without potentials, the solutions of the Dirac equation, or ...
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Density of States and the Spectral Measure
The spectral theorem states that for any self adjoint operator $H$ on some Hilbert space $\mathcal{H}$, there exists a projection-valued measure $E_H$ such that
$$H= \int_{\mathbb{R}} \lambda \mathrm{...
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What "goes wrong" in a Sommerfeld expansion?
Let $f$ be the Fermi function and $H$ be a function which which vanishes as $\epsilon \to -\infty$ and which diverges at $\infty$ no worse than some power of $\epsilon$. In the Sommerfeld expansion of ...
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Is it physically relevant to restrict the solution of a nonlinear PDE to positive frequencies in the Fourier transfrom?
I would like to mention that I am a mathematician and not a physicist, so I apologize in advance if my question seems obvious.
Considering any linear PDE, it is common to understand the behavior of ...
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Discrete symmetries of Hamiltonian and Change of Representation
In quantum mechanics, one could write Hamiltonians for a given quantum system both in coordinate and momentum spaces as mentioned for example in Sakurai book. Does the discrete symmetries such as ...
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Projective representations and central extensions- classification in terms of $H^2(G,\mathbb{C}^{\times})$
In quantum mechanics, unitary projective representations play a crucial role. To be more general, I want to pose the question in sense of projective representations, then everything would follow as a ...
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Spectrum and Schrödinger equation [duplicate]
The time-evolution in quantum mechanics is given by Schrödinger's equation. For time-independent Hamiltonians, one searches for solutions of the problem $\hat{H}\psi = E\psi$.
The physicist point of ...
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What's the ladder operator of Duffing Oscillator?
I know the ladder operator for harmonic oscillator can be obtained by factorization method, can the same method be applied to oscillators with potential $V(x)=x^{4}$ (the Duffing case) or higher ...
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2D CFT from sigma models
$X$ is a closed manifold with a positive-definite metric $g$.
$M_2$ is a 2D oriented closed manifold with a positive-definite metric $G$ and a compatible volume form $\omega$.
We can then consider the ...
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Generalization of Bargmann's theorem
Bargmann's theorem is usually stated for a simply connected Lie group with vanishing second Lie algebra cohomology $H^2(\mathfrak{g},\mathbb{R})$. I found a generalization of this result in a thesis ...
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Projective representations and cental extensions based on Schottenloher's book
In Schottentloher's book, a theorem is stated:
Later on, a remark is made:
My confusion comes: Is E always of the form made in the remark, namely are the following to sets equal(up to set theoretic ...
- 473
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Second-order Trotter error involving an unbounded Hamiltonian
I have an Hamiltonian of this form:
\begin{equation}
H = \frac{p^2}{2m} + V(x),
\end{equation}
I would like to approximate the time evolution for a time $\tau$ of a known initial Gaussian state $|\...
- 121
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1
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Hilbert space of free theory vs interacting theory
In view of Haag's Theorem, it seems the Hilbert spaces of a free theory and an interacting theory are not the same. Though it seems very believable, I could not find a result that states that this is ...
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591
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Mathematical definition of states in Quantum Theory
I was reading Valter Moretti's book on Spectral Theory and Quantum Mechanics, and saw 2 definitions of a quantum state:
1.Let $\mathcal{H}$ be a Hilbert space. A positive, trace-class linear map $\rho:...
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Is linear momentum quantized in quantum harmonic oscillator
I'm self-studying QM and have a basic question on quantum harmonic oscillator. The Hamilton is certainly quantized under this model, that is $E_n=(n+1/2)\hbar \omega$, for $n=0,1,2,...$. But is linear ...
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Experiment design: can one actually measure the speed of non-local light in curved spacetime
The equivalence principle tells us that in some local neighborhood, every free-falling observer in a general relativistic spacetime will measure the speed of light to be $c$; this literally means at a ...
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Adjoint of the Quantum Momentum Operator
I'm studying quantum mechanics and I have a question about the momentum operator. We have that the momentum operator is given by
\begin{equation*}
p = -i\hbar\nabla
\end{equation*}
and so its adjoint ...
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HaMiDeW coefficients - recursive calculation of the coincidence limits
In his book Aspects of Quantum Field Theory in Curved Spacetime Stephen Fulling calculates the coincidence limit $[a_1]$ and gives an idea of how $[a_n]$ with $2 ≤ n$ can be found recursively.
Since ...
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The spectrum of the Hamiltonian in quantum mechanics
Consider the Hilbert space $\mathscr{H} = L^{2}(\mathbb{R}^{d})$ and a Hamiltonian:
$$H = -\frac{\hbar^{2}}{2m}\Delta + V(x)$$
for some potential function $V$. States of well-defined energy $E$ are ...
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Holley and FKG Lattice Conditions
There's an interesting exercise (page 13, Exercise 11) in Hugo Duminil-Copin's Lectures on the Ising and Potts models on the hypercubic lattice, which states that the following 2 statements are ...
- 1,924
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Unitary and Self-adjoint superoperators for the Hilbert-Schmidt product
What are the necessary and sufficient conditions for a linear superoperator to be Unitary or Self-adjoint with respect to the Hilbert-Schmidt inner product $\left(\hat{A},\hat{B}\right)=Tr\left(\hat{A}...
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Finding solutions for the free electron model with Born-Von-Karman boundary conditions
I'm trying to solve the Schrodingers equation for the free electron model with the Born-Von-Karman boundary conditions.
I'm aware that at least a possible solution of the problems are plane waves ...
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Why is the Lattice of a crystal required to have at least as much symmetry as its motif?
I know that a crystal structure is formed by the addition of a motif to a lattice (crystal structure = lattice + motif). I also know that an arbitrary lattice will in general exhibit certain ...
- 1,547
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Is the operator $P=-i\hbar\frac{d}{dx}$ self-adjoint given the Hilbert space of the problem of particle in a box?
The operator $P=-i\hbar\frac{d}{dx}$, is a symmetric operator in the domain $$D(P)=\left\{f(x) \big|f\in L_2[0, a], f(0)=f(a)=0\right\}$$ i.e. the domain is the subspace of square-integrable functions ...
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How does Kirchhoff's voltage law relate to the spatial derivative of voltage?
I'm reading this libretexts article on the basics of transmission line theory. In it, they include this circuit diagram as a model of a uniform transmission line:
They then say that applying ...
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Spectral representation of a white stationary process
I am trying to better understand the spectral representation of stochastic processes. From the book "Spectral Analysis for physical applications" by Walden and Persival:
The spectral ...
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Existence of Schwinger Functions for QCD?
It seems to me the 'naive' approach to proving the existence of Yang-Mills in a rigorous context (via Osterwalder-Schrader $\to$ Wightman axioms), would be:
Study gauge invariant lattice QCD ...
- 6,278
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Haag-Kastler axioms
In the Haag-Kastler axioms, an algebra of observables $A(O)$ is associated to each open spacetime region $O$ of the Minkowski space. In several treatments, the algebra $A(O)$ is a $C^{*}$ algebra, and ...
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Different parts of spectrum appearing in the spectral theorem in terms of generalized eigenvectors
Question: How to exactly relate both expansions quoted below: Can one be "transformed" into the other? What is the interplay between the various parts of the spectrum appearing?
In Ref. 1 it ...
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How are Schwinger and Wightman functions used in practice?
In Reed & Simon's Methods of Mathematical Physics Volume II, they define a (Hermitian scalar) quantum field theory to be the quadruple $\langle \mathcal{H}, U, \varphi, D\rangle$ that satisfies ...
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In QM, does an algebra containing the Hamiltonian always evolve into itself?
Let $\mathcal{A}$ be an algebra of operators on a Hilbert space $\mathcal{H}$, and suppose it contains the Hamiltonian: $H\in\mathcal{H}$. The Heisenberg evolution for any $\hat{O}\in\mathcal{A}$ is
$$...
- 1,335
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How does Algebraic QFT deal with renormalization?
I'm reading David Wallace's essay on a critique of the algebraic approach to quantum field theory (AQFT). There he argued that AQFT failed to resolve the renormalization problem because it doesn't ...
- 1,535
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1
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Is rigged Hilbert space generally considered the correct structure for QM?
I am currently reading the quantum mechanics text by Ballentine and, over and over, arguments are made (e.g. Chapter 4.6 on constraining the wavevectors of free particles to be real) which rely on ...
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1
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Continuous spectrum defined in terms of bounded eigenfunctions
I believe this is a mathematical question, but everytime I expose my question to a mathematician the answer is 'because they do this in Quantum Mechanics'. So here I am: a very ignorant person in ...
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What is the importance of unitary (in-)equivalent representations?
Say we have two representations of the observables from an abstract $C^*$-algebra $\mathcal A$ on two Hilbert spaces $H_1$ and $H_2$, i.e. consider the maps $\pi_1,\pi_2: \mathcal A \longrightarrow \...
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