All Questions
Tagged with oa.operator-algebras mp.mathematical-physics
47 questions
47
votes
3
answers
10k
views
Quantum mechanics formalism and C*-algebras
Many authors (e.g Landsman, Gleason) have stated that in quantum mechanics, the observables of a system can be taken to be the self-adjoint elements of an appropriate C*-algebra. However, many ...
- 573
39
votes
6
answers
7k
views
A remark of Connes on non-standard analysis
In an interview (at http://www.alainconnes.org/docs/Inteng.pdf) Connes remarks that
I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point is ...
- 1,131
33
votes
1
answer
4k
views
What about nonassociative geometry?
At the conclusion of a conference delivered by Alain Connes in 2000 (video in French at 1:19:25), an audience member posed a question. Below is a polished translated transcription:
Audience:
You have ...
28
votes
6
answers
6k
views
Any real contribution of functional analysis to quantum theory as a branch of physics?
In the last paragraph of this last paper of Klaas Landsman, you can read:
Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of ...
23
votes
2
answers
3k
views
States in C*-algebras and their origin in physics?
in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$
Now, of course there is also in classical physics and quantum ...
- 243
17
votes
1
answer
2k
views
The cyclic subfactors theory: a quantum arithmetic?
Context: First recall some results:
Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980).
A Galois correspondence for depth 2 irreducible subfactors (Izumi-Longo-Popa ...
12
votes
3
answers
1k
views
What's algebraic approach to QM good for?
The algebraic formulation of quantum mechanics (and related stuff, like quantum thermodynamics & dynamical systems etc.) via C*-algebras provides a viewpoint based mostly on abstract functional ...
- 3,323
10
votes
5
answers
580
views
Observables and dimensional analysis
Here is a simple question about physical units that I hope has a simple satisfying answer. In mathematically sophisticated treatments of both quantum and classical physics one often speaks of an ...
- 101
10
votes
2
answers
793
views
What is the physical difference between states and unital completely positive maps?
Mathematically, completely positive maps on C*-algebras generalize positive linear functionals in that every positive linear functional on a C*-algebra $A$ is a completely positive map of $A$ into $\...
- 7,047
9
votes
1
answer
566
views
Non-perturbative Renormalization in the sense of Polchinski's equation. Do we have a mathematical formulation?
My question is about mathematical treatment of exact renormalization group in the sense of Polchinski's flow equation. In a heuristic form, Polchinski's equation looks like: $\partial_t S[\phi] = \...
- 91
9
votes
0
answers
290
views
A robust version of Schur's lemma?
Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this:
Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...
- 91
8
votes
1
answer
966
views
Murray-von Neumann classification of local algebras in Haag-Kastler QFT
The Haag-Kastler approach to quantum field theory (QFT) is one of the oldest approaches to rigorously define what a QFT is, it deals with nets of operator algebras: You start with a spacetime and ...
- 1,544
8
votes
1
answer
420
views
What is the general form of the duality transform for the Fock space?
I am interested in properties of the symmetric Fock space, looked at via the associated Wiener space. It is well known that for a Hilbert space $k$, the symmetric Fock space $$\mathcal{F}(L^2(\mathbb{...
- 81
7
votes
4
answers
1k
views
Quantum channels as categories: question 1.
A quantum channel is a mapping between Hilbert spaces, $\Phi : L(\mathcal{H}_{A}) \to L(\mathcal{H}_{B})$, where $L(\mathcal{H}_{i})$ is the family of operators on $\mathcal{H}_{i}$. In general, we ...
- 283
7
votes
2
answers
1k
views
C*-algebraic representation of observables vs self-adjoint operators one
I am trying to reconcile the "physicist" definition of an observable: self-adjoint operator on a Hilbert space, and the operational one as given by Strocchi in "An introduction to the mathematical ...
- 585
7
votes
1
answer
730
views
Formal series convergence in deformation quantization and $C^*$-condition
A link between formal series convergence in deformation quantization (strict deformation quantization) and producing $C^*$-algebras instead of mere $*$-algebras (which $(\mathcal{C}^{\infty}(M)[[t]],\...
- 585
6
votes
1
answer
751
views
Time-Energy Uncertainty Relation in relativistic Quantum Mechanics
There is an old intriguing result in non-relativistic QM, stating (roughly) that there is an Heisenberg Time-Energy Uncertainty Relation.
Unfortunately, in QM time is not an operator like space, ...
- 7,853
6
votes
1
answer
244
views
What's the relation between spin model for subfactors theory and physics?
In the sense of subfactor theory, a spin model is a commuting square of the form
$$\begin{matrix}
\Delta &\subset & M_n(\mathbb{C})\cr
\cup &\ &\cup\cr
\mathbb{C} &\subset &w\...
6
votes
1
answer
476
views
$T\bar{T}$ deformation: Stress-energy momentum tensor deformed in CFT and in QFT for various $d$-dimensions
The $T\bar{T}$ deformation is based on the original work of Zamolochikov [1] explored deformations of two-dimensional conformal field
theories (CFT) by an operator that is quadratic in the stress-...
- 10.5k
6
votes
0
answers
378
views
What are some results that assume the Connes' embedding conjecture or any of its reformulations?
As you all (may) know, the Connes embedding conjecture was disproven last year. Also, as its Wikipedia page shows, there are multiple reformulations (but it is definitely not an exhaustive list):
...
- 265
6
votes
0
answers
197
views
A 'Fock'-type construction on a $C^*$-algebra
In very rough terms, let $A$ be a complex unital $C^*$-algebra. Assume it nuclear for convenience, but it doesn't matter much. Consider the 'Fock'-type $C^*$-algebra (don't know a better name for it)
$...
- 1,959
5
votes
2
answers
585
views
"Uncertainty principle" for self-adjoint operators in a finite von Neumann algebra
Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a ...
- 3,897
5
votes
1
answer
254
views
Well defined Tensoring of spectral triples
Hi,
I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it.
Question: In connes standard model he takes ...
- 133
5
votes
1
answer
897
views
Folium in GNS construction and von Neumann algebras
The GNS construction allows one to represent a $C^*$-algebra as the algebra of bounded operators on a Hilbert space when a state is fixed, this state being represented as a vector on the Hilbert space....
- 585
5
votes
0
answers
72
views
Braided monoidal category of (generalized) operator algebras
In the paper "Quantum Collections" Kornell (2016) proved that the category of $W^*$ (von Neumann) (and iirc also $C^*$) algebras, equipped with a suitable choice of tensor product, forms a ...
- 1,084
4
votes
1
answer
141
views
"Open systems" version of Stone's Theorem for one-parameter groups of quantum operations
Let $H$ be a Hilbert space, which we interpret as a space of quantum states.
If $U(t):H\to H$ is a unitary norm-continuous one-parameter group with $U(0)=I$, (essentially) Cauchy's functional ...
- 854
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
4
votes
0
answers
151
views
Is there a maximal finite depth infinite index irreducible subfactor?
A subfactor $N \subset M $ is irreducible if $N' \cap M = \mathbb{C} $.
It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
It's cyclic if its lattice of ...
3
votes
2
answers
1k
views
$C^*$ algebras and states
I have a question arising from von Neumann's C*-algebra formulation of quantum mechanics. In it, a state is a $\mathbb{C}$-linear functional on a C*-algebra $A$
$\rho:A\rightarrow\mathbb{C}$
...
- 103
3
votes
1
answer
125
views
Classification of $2k$-vectors modulo orthogonal transformations
Consider the following chain $\{A_1,A_2,A_3,\cdots,A_{n}\}$ of orbit spaces of even-rank anti-symmetric tensors, where
$$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k}}...
- 605
3
votes
1
answer
244
views
inclusion of von Neumann algebras implies reversing inequality of its modular operators
I'm working with von Neumann algebras and I stumbled with this statement in a work of Borchers (1999)
Since $\mathcal N \subseteq \mathcal M$, it follows by standard arguments that $\Delta_{\mathcal ...
- 424
3
votes
0
answers
219
views
Can any POVM be induced by a quantum instrument?
I suspect this is the obvious result of something in operator algebras, but that's far outside my field.
Recall that a projection-valued measure is a map $E$ from a sigma-algebra $\mathcal{F}$ on ...
- 854
3
votes
0
answers
57
views
Integration of Weyl operators multiplied by quasifree state over a symplectic space
I am reading the book "An invitation to the Algebra of Canonical Commutation Relations" by Denes Petz. It is freely available for download here. In Chapter 9, he defines the Lebesgue measure on a ...
3
votes
0
answers
431
views
Bohr topos and quantization
Bohrification is a natural way to construct a quantum "phase space" (with some nice insights on foundational problems like non-contextuality through Kochen-Specker etc). I was wondering, since we get ...
- 585
2
votes
1
answer
179
views
Second quantization of partial isometry
If we have a unitary map from Hilbert space $H$ to $H$, we get a unitary map from $e^{H}$ to
$e^{H}$, where $e^{H}$ is the symmetric Fock space of $H$. But if we replace the unitary with partial ...
- 95
2
votes
1
answer
437
views
Reference request (or otherwise): Adjoint action
I am interested in clearing up a confusion of mine. I will try to make my question as clear as I can but I apologize in advance if this is not the case.
Given a unitary group of some unital ...
- 133
2
votes
0
answers
79
views
Algebraic machinery for boundary conditions: may spectral data be "lifted" via the Toeplitz extension?
Let $\tilde{\mathcal H}$ be a Hilbert space, and let $L(\tilde{\mathcal H})$ denote the corresponding space of linear operators. By fixing a basis, we can, via Fourier transform, identify an important ...
- 605
2
votes
0
answers
132
views
Extension of a bounded operator on manifold
I have a problem, which is quite urgent, as I have only today discovered an error in a proof i had in a thesis which is to be handed in tomorrow.
The problem, if stated in as full generality as ...
- 63
1
vote
2
answers
2k
views
Quantum channels, question 2: tensor products and composition of functions
Please be kind. I've been working on this for a long time and can't find an answer. Feel free to edit for clarity if you think the question can be better worded.
Background
It may help to see a ...
- 283
1
vote
1
answer
116
views
Is there a Bell inequality for each of $2\times 2$, $3\times 1$, $2\times1\times1$ and $1\times1\times1\times1$ configurations?
There was no answer in https://physics.stackexchange.com/questions/600494/is-there-a-bell-inequality-for-2-times-2-and-1-times1-times1-times1-configur. Hence posting in mathoverflow on the possibility ...
- 13.9k
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
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
0
answers
85
views
Characterization of the Subspace of Quasifree States of the CAR Algebra
Consider $\mathfrak U(\mathfrak H)$, the CAR algebra over a separable Hilbert space $\mathfrak H$. The states $E_{\mathfrak U}$ over this algebra are defined to be positive linear functionals of norm ...
- 605
1
vote
0
answers
455
views
How to correctly name "irreducible subrepresentation of an indecomposable representation"
I am studying the representation theory of some infinite dimensional algebras, for example infinite dimensional Lie-algebras, Kac-Moody algebras, W-algebras. These algebras arise as symmetry algebras ...
- 81
1
vote
0
answers
742
views
Tensor products as isomorphic functors in category theory
An earlier question that I posed sought to define a category with a set of quantum channels as arrows and the C$^{*}$-algebra that these channels map from and to as the object. So, for example, my ...
- 283
0
votes
0
answers
55
views
Status of generalization of timelike tube theorem to algebras of causal completions
The timelike tube theorem states that the additive algebra $A_{\text{add}}(U)$ of operators in a spacetime region $U$ is equal to the additive algebra $A_{\text{add}}(E(U))$ of the timelike envelope $...
user528837
-3
votes
1
answer
133
views
SU(2) and entangled particles [closed]
We have two particles $A$ and $B$ in a maximally entangled state $|\Psi\rangle \in \cal{H}_A \times \cal{H}_B$
$$
\left|\Psi\right\rangle = \frac{1}{\sqrt{2}} ( \left| 0
\right\rangle_A\otimes \left| ...
- 1