All Questions
Tagged with quantization qa.quantum-algebra
11 questions
5
votes
0
answers
198
views
Analogue of Kontsevich's formality theorem for quantization of Courant algebroids
In his 1997 preprint, M. Kontsevich proved the formality of the differential graded algebra controlling deformations of the associative and commutative algebra of functions on a manifold, seen as an ...
5
votes
0
answers
274
views
Deformation quantization of Poisson bracket without star-product
Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$,
$$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...
- 654
5
votes
0
answers
167
views
Distinguishing the Duflo star product
$\newcommand{\g}{\mathfrak g}\newcommand{\h}{\hbar}$
For a finite dimensional Lie algebra $\g$, he Duflo isomorphism is a complicated algebra isomorphism between the $\g$-invariant part $S(\g)^\g$ of ...
- 8,524
4
votes
0
answers
338
views
Quantization of $S^2$ as $C^*$-algebra?
The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695).
The particular question is about ...
- 24.9k
9
votes
0
answers
268
views
Existence/characterization/properties of $C^*$-algebras which "are" quantization of compact symplectic manifolds?
Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...
- 24.9k
2
votes
2
answers
327
views
Deformation quantization of a closed Riemann surface with genus >1
Quantization of of an elliptic curve can be done in different ways.
In C^*-algebraic version,
one can start with the C^*-algebra ...
- 309
4
votes
1
answer
147
views
Equivalence of star products on two differents Poisson algebras?
Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...
- 53
12
votes
2
answers
935
views
Is the quantum algebra unique (up to isomorphism) in deformation quantization ?
Consider a Poisson algebra A (i.e. commutative algebra with Poisson bracket).
Let $\hat A$ be a deformation quantization of the algebra A. We know that construction of deformation quantization and ...
- 24.9k
14
votes
1
answer
1k
views
Coherent states vs quantization of Lagrangian submanifold
Coherent states http://en.wikipedia.org/wiki/Coherent_states
are vectors in the Hilbert space which in certain sense are strongly localized
and "corresponds" to points in classical phase space (see ...
- 24.9k
10
votes
2
answers
1k
views
Quantization of conjugacy classes in a Lie group
Let $G$ be a Lie group (and to be safe, let's assume it is semisimple). Consider the action of $G$ on itself by conjugation, and form the GIT (algebro-geometric) quotient $G/\!/G$. Then let $\pi:G\...
- 18.7k
11
votes
1
answer
1k
views
Kontsevich's formality theorem from an explicit homotopy
Suppose that $X$ is a smooth manifold, whose $C^{\infty}$-functions we denote by $A$. Let $D_{poly}^*(A):=\bigoplus_{n\geq -1}Hom(A^{\otimes n+1},A)$ be the Lie algebra of polydifferential operators ...
- 386