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.
2,244
questions
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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 ...
- 381
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votes
0
answers
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+50
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 $|\...
- 111
6
votes
1
answer
106
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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 ...
- 1,902
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votes
2
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543
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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:...
- 381
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votes
2
answers
145
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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 ...
- 43
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answer
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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 ...
- 378
5
votes
1
answer
223
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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 ...
- 185
3
votes
0
answers
165
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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 ...
4
votes
0
answers
191
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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 ...
- 835
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answers
34
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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 ...
- 1,890
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35
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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}...
- 414
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votes
1
answer
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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 ...
- 225
1
vote
2
answers
120
views
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 ...
- 1,517
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2
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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 ...
- 10.4k
3
votes
3
answers
509
views
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 ...
1
vote
0
answers
34
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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 ...
- 111
5
votes
0
answers
84
views
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 ...
- 6,228
3
votes
1
answer
90
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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 ...
- 381
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votes
2
answers
128
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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 ...
- 6,117
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vote
1
answer
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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 ...
- 1,902
4
votes
1
answer
118
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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
$$...
- 1,203
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votes
1
answer
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How does Algebraic QFT deal with renormalization?
I'm reading David Wallace's essay on a critique of the algebraic approach to quantum field theory (AQFT). There he argued that AQFT failed to resolve the renormalization problem because it doesn't ...
- 1,305
17
votes
1
answer
926
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Is rigged Hilbert space generally considered the correct structure for QM?
I am currently reading the quantum mechanics text by Ballentine and, over and over, arguments are made (e.g. Chapter 4.6 on constraining the wavevectors of free particles to be real) which rely on ...
- 743
3
votes
1
answer
50
views
Continuous spectrum defined in terms of bounded eigenfunctions
I believe this is a mathematical question, but everytime I expose my question to a mathematician the answer is 'because they do this in Quantum Mechanics'. So here I am: a very ignorant person in ...
- 31
5
votes
1
answer
165
views
What is the importance of unitary (in-)equivalent representations?
Say we have two representations of the observables from an abstract $C^*$-algebra $\mathcal A$ on two Hilbert spaces $H_1$ and $H_2$, i.e. consider the maps $\pi_1,\pi_2: \mathcal A \longrightarrow \...
- 6,117
2
votes
1
answer
97
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Decay of the First Derivative of the Quantum Wave Function
I understand that the Hilbert space of all physical solutions of the Schrodinger equation have the property where
$$
\lim_{x\to\infty}\Psi=0
$$
For one of my assignments, I wanted to use
$$
\lim_{x\to\...
- 55
4
votes
2
answers
99
views
Gauge Symmetry of the Lagrangian
My teacher told the following statement to me during office hours. Is it correct and if so, how could one go about proving it?
Given a material system subject to holonomic and smooth constraints ...
- 356
14
votes
1
answer
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What does it mean for a field to be defined by a measure?
In Quantum Physics by Glimm and Jaffe they mention on p. 90 that
The Euclidean fields are defined by a probability measure $d\mu(\phi) = d\mu$ on the space of real distributions. Here $d\mu$ plays ...
- 1,902
2
votes
1
answer
60
views
Is the following map linear over the space of density matrices?
I have a map $\mathcal{N}$ from the space of two-qubit subnormalised density matrices $\mathcal{S}(\mathcal{H}_2 \otimes \mathcal{H}_2)$ to itself (positive operators with trace between 0 and 1). ...
- 25
1
vote
0
answers
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views
Prandtl boundary layer equations for two-dimensional steady laminar flow of incompressible fluid over a semi-infinite plate are given by [closed]
Prandtl boundary layer equations for two-dimensional steady laminar flow of incompressible fluid over a semi-infinite plate are given by
jpg
- 11
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0
answers
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The force distribution along curved surface of contact [closed]
Is it possible to calculate the stress or contact force distribution over a curved contact surface?
I will try to explain the general idea in the figure below. Object $O$ (could be assumed to be rigid ...
3
votes
1
answer
70
views
$\rm Tr[log( )]$ calculation to go from BCS to Ginzburg-Landau
It seems like calculating the effective action $|\Delta|^2 + Tr[ln(G^{-1})]$ give the Ginzburg Landau action.
\begin{equation}
G^{-1} =\begin{pmatrix} i\partial_t - H & \Delta \\
\Delta^* & i\...
- 31
1
vote
1
answer
90
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A theorem about functions of self-adjoint operators
It is very common (see e.g. page 18 of Ballentine's Quantum Mechanics: A Modern Development) for the following development to take place. We couch the discussion in Dirac's bra-ket notation noting ...
- 743
5
votes
0
answers
143
views
Does geometric quantization work for arbitrary "particle with constraint + potential" systems?
I was struck by the following line in Hall's Quantum Theory for Mathematicians (Ch. 23, p. 484):
In the case $N = T^*M$, for example, with the natural “vertical” polarization, geometric quantization ...
- 3,203
1
vote
0
answers
81
views
Epsilon dependence of the beta function
For some time now, I’ve been trying to prove that the beta function for a quantum field theory with coupling $g$ (in the typical case of a coupling with mass dimension $\varepsilon$ in dimensional ...
- 11
2
votes
1
answer
53
views
Is a continuum of bound states possible in a finite system Hamiltonian with a Coulomb potential?
I am wondering if it is possible to have a continuum of bound states in a finite system, for example a molecular system with a fixed number of nuclei and electrons. As a chemist, I'm used to ...
- 2,142
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vote
0
answers
51
views
Non-separability of Hilbert space [closed]
A Fock space constructed via a separable Hilbert space is separable, however the tensor product of a countable set of separable Hilbert spaces is not (infinite qbit chain). How is the previous ...
- 61
1
vote
1
answer
50
views
Proof of differentiate form of dynamical semigroups
I am studying some basics of the pure mathematical background for open quantum systems from Angel Rivas`s book which is "Open quantum systems, an introduction".
Here is a theorem (Page 6, ...
- 107
4
votes
1
answer
74
views
Infrared bound on Ising model
I'm currently trying to understand aspects of Hugo Duminil-Copin's Lectures on the Ising and Potts models on the hypercubic lattice. In section 4.3, he claims that for the Ising model in $\mathbb{Z}^d$...
- 1,890
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1
answer
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views
Integrating over a charge distribution without using Radon-Nikodym
Goodmorning,
I'm studying for a basic Electrostatic course and I have a doubt about how to justify in terms of measure theory the physicists' writing
$dq=\rho \ d\tau$ or $\rho = \frac{dq}{d\tau}$,
...
1
vote
0
answers
50
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Mathematical Physics Research [closed]
I'm a physics graduate coming out of a physics program that was so poorly structured and disorganized. In the last year and a half of undergraduate school, I developed a very keen interest in the ...
- 762
2
votes
0
answers
46
views
Is Santilli's hadronic mechanics sound and useful? [closed]
I'm a mathematician. Some math papers and books related to mutation algebras (a kind of nonassociative algebras which are Lie-admissible), and even an entry in the Encyclopedia of Mathematics (Lie-...
- 121
2
votes
0
answers
98
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Can quantum fields be smeared in space (rather than spacetime)?
I am interested in what is known about the possibility of smearing interacting quantum fields on a Cauchy slice. This is easy to do for free fields and their conjugate momentum, and indeed this is ...
1
vote
1
answer
54
views
Maupertuis' principle by variational method
On p.464 of Spivak's mechanics book, the author proves the equivalence of Maupertuis' principle and stationary action principle by considering variation of some path $c(t)$, such that other paths in ...
- 119
5
votes
1
answer
103
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Correlation Functions as Morphisms
In https://arxiv.org/abs/1911.07895, the authors consider a generalization of correlation functions to make sense of the $O(n)$ symmetry for $n \in \mathbb{R}$. As explained in Sec. 7, each field $\...
- 71
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votes
1
answer
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Domain issues for products of bosonic creation and annihilation operators
Consider the bosonic Fock space $F$ and let $a^\dagger$ and $a$ denote the "usual" creation operators. As far as I know, these are defined on the dense domain $\mathcal D(N^{1/2}):=\{ \psi \...
- 6,117
2
votes
1
answer
119
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Spectrum of $f(T)$, where $T$ is a self-adjoint operator
Consider on a Hilbert space $\mathcal{H}$ a self-adjoint operator $T$ with spectrum given by $\sigma(T)=\{\lambda_n\}_{n \in \mathbb{N}} \subseteq \mathbb{R}$ (let's suppose for simplicity that the ...
- 107
0
votes
1
answer
85
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On the domain of $P^2:=-\nabla^2$ for the infinite square well
A potential of the form
$$
V(x) = \left\{
\begin{split}
0&\quad \operatorname{in}\ [-a, a] \\
+\infty&\quad \operatorname{elsewhere}
\end{split}
\qquad a>0,
\right.
$$
compels us to solve
$...
- 1,386
2
votes
0
answers
105
views
On the definition of the bosonic one-particle reduced density matrix
Consider the bosonic/fermionic Fock space $F^\pm:=\bigoplus\limits_{N=0}^\infty H_N^{\pm}$, where $H^+_N:=\vee^N \mathfrak h$ and $H^-_N:=\wedge^N \mathfrak h$ for some (complex, separable) one-...
- 6,117
0
votes
0
answers
65
views
Can Quantum observables be modeled as real functions on the phase space?
In the context of justifying the failure of modeling quantum observables in the 'more natural' way as real functions on the phase space (i.e. similar to the mathematical image modeling classical ...
