All Questions
Tagged with hilbert-spaces mp.mathematical-physics
20 questions
2
votes
0
answers
92
views
Representations of the commutation relations and renormalization
I had a hard time deciding whether this question is more appropriate for physicsSE or MO, so I have cross-listed it for the time being. The physicsSE post can be found here.
In the lecture Is ...
- 721
6
votes
0
answers
159
views
Identification of Fock space and the $L^2$ space of tempered distributions
Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the ...
- 721
0
votes
0
answers
138
views
Question about a step in the proof of the min-max principle
I honestly do not think this is a hard question, maybe it is even obvious but I tried MSE and had no success so far, so I am reproducing the question Question about the proof of the min-max principle ...
- 1,305
1
vote
1
answer
354
views
Mach's principle, Newton's law and Hilbert sphere?
(This question has originally been posted on reddit, but I thought, that the question raised in the post above, might fit as well here on MO.)
I wanted to share with you something I stumbled upon ...
- 3,064
4
votes
1
answer
371
views
Motivation for Heisenberg's modeling of observables
What's the motivation for observables to be modeled by self-adjoint operators? I can't seem to find any place where this is laid out clearly. Maybe von Neumann's book is decent, but it's not ...
- 433
2
votes
0
answers
69
views
Mathematical reason for scatter states being special?
In infinite spectral theory, we have the discrete and continuous spectrum, which are called "bound" and "scatter states" in physics.
My understanding is, if $O \in B(H)$ is a self-...
- 569
4
votes
1
answer
1k
views
Is the "space of physical quantities" a field of transcendence degree $6$ or $7$ over the rationals?
Excuse my naive question and please let me explain it:
In everyday life we experience 3 spatial "dimensions" + time etc.
Usually the 3 dimensions are represented by a coordinate system and ...
- 3,064
1
vote
0
answers
127
views
Does convergence of tempered distributions implies convergence in $\mathcal{S}(\mathbb{R}^4,\mathbb{R})/\mathcal{S}_{0}$?
We can define the following symmetric semi-definite positive bi-linear form on
$\mathcal{S}(\mathbb{R}^{4},\mathbb{R})$ with values in $\mathbb{C}$,
\begin{equation}\label{prodintespaciales}
(h_{...
- 424
5
votes
1
answer
609
views
Initial conditions in the Klein-Gordon equation
I am interested in what are the largest families of initial conditions of the problem (in $\mathbb{R}^{4}$)
\begin{equation}\label{kg}
\left\lbrace
\begin{array}{ll}
(\square+m^2)F(x)=0\\
...
- 424
25
votes
4
answers
4k
views
Rigged Hilbert spaces and the spectral theory in quantum mechanics
I'm trying to learn some quantum mechanics by myself, and because of my mathematics background, I'm trying to understand it in a rigorous way. Since then, I've been intrigued by the use of rigged ...
- 1,305
3
votes
2
answers
411
views
Spin-statistic for free quantum fields
Short version of the question:
Can someone explains what physicist's 'Spin-statistic theorem' says rigorously in the context of Free Quantum fields when they are describe as the Fock space of some '1-...
- 42.4k
1
vote
0
answers
57
views
Continuity of a linear functional for sequences of projections
Let $T_+$ be the set of a positive trace-class operators over some separable Hilbert space and
$A: T_+ \to \mathbb{R}\cup \{\infty\}$ some linear functional.
In general, $A$ will not be continuous. ...
- 695
1
vote
1
answer
247
views
Linearity of the directional derivative of a convex functional at the minimum
Let $H$ be a Hilbert space, $T_+(H)$ the set of positive self-adjoint trace-class operators on $H$, and $f : T_+(H) \to [0,m]$ a non-negative, bounded, convex functional. I don't necessarily know that ...
- 695
6
votes
2
answers
539
views
Is there a reasonable notion of spectral theorem on a pre-Hilbert space?
I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...
- 269
8
votes
1
answer
359
views
Lax pairs in an abstract formalism
I am reading Integrals of Nonlinear Equations of
Evolution and Solitary Waves by Peter Lax and I'm having a hard time. The methods are pioneering, of course, but Lax does not bother much to provide ...
- 3,388
3
votes
1
answer
232
views
what is about the corresponding power series?
According to the papers The absolutely continuous spectrum of Jacobi matrices and these lecture notes:
periodicity ~ potential well or lattice (order)
lack of absolutely continued spectrum ~ Anderson ...
- 3,094
5
votes
2
answers
2k
views
On the domains and extensions of unbounded operators
I am not an expert in functional analysis but I was studying some, motivated from some mathematical physics considerations. I am not quite sure whether this is research-level, but let me state some ...
- 7,746
4
votes
1
answer
165
views
Scattering of relativistic particle by long-range potential
Let
$\mathcal{H}=L^2(\mathbb{R}^3)$,
$H_0=\sqrt{-\Delta+M^2}$, ($M$ is a positive constant, $\Delta$ is the laplacian)
and
$H=H_0+V(\vec{x})$
(where $V(\vec{x})$ is the operator of ...
- 552
2
votes
1
answer
178
views
Self-adjointness of a perturbed quantum mechanical Hamiltonian specified in an infinite matrix form
Consider an operator $H$ on the Hilbert space $\ell_2$ given as an infinite matrix with two pieces, one diagonal and one arbitrary:
$H_{ij}=E_i\delta_{ij}+V_{ij}$. This has a physical meaning in ...
- 679
0
votes
0
answers
155
views
General form of a symplectic map
A symplectic automorphism of a Hilbert space has the form $T=U(\cosh S+J\sinh S)$ for a unitary $U$, an antilinear involution $J$ and a positive operator $S$. In fact a version of this goes through in ...
- 1,411