All Questions
Tagged with quantization mp.mathematical-physics
35 questions
71
votes
10
answers
11k
views
The Planck constant for mathematicians
The questions
Q1. What are simple ways to think mathematically about the physical meanings of the Planck constant?
Q2. How does the Planck constant appear in mathematics of quantum mechanics? In ...
- 24.7k
35
votes
5
answers
5k
views
Does quantum mechanics ever really quantize classical mechanics?
I was curious about a physics question which I thought might be suitable for mathoverflow. I looked at the answer to this question, but it's not what I'm looking for.
Basically, classical mechanics ...
- 7,972
15
votes
0
answers
472
views
Squeezing physics out of formal deformation quantizations
I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested ...
- 5,407
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
14
votes
0
answers
1k
views
The quantization problem: modern quantization procedures and current status
The quantization problem is one of the most well-known current problems of theoretical and mathematical physics. It is also part of Hilbert's sixth problem (on the axiomatization of physics - see here ...
13
votes
1
answer
678
views
Reconciling two notions of geometric quantization.
Let $(M,\omega)$ be a compact symplectic manifold and $(L,\nabla)$ a prequantum line bundle. There are two schemes to quantize this data:
Choose a polarization $P$ of $M$ and define the quantum ...
- 3,244
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
11
votes
2
answers
849
views
Physical intuition behind Kontsevich's deformation quantization formula
Kontsevich gives a construction that produces deformation quantization of $C^\infty(M)$ for general Poisson manifolds $M$. The resulting formula (on $\mathbb{R}^n$) is
$$
f\star g = \sum_{n=0}^\infty \...
- 181
11
votes
0
answers
1k
views
Yang-Mills theory with non-compact gauge groups G
Physicists are familiar working with Yang-Mills theory with compact and semi-simple gauge groups $G$ (Lie groups).
However, it is not entirely clear the formulation of Yang-Mills theory with non-...
- 10.5k
10
votes
1
answer
2k
views
basic questions on quantum integrable systems
I have been learning about (classical) integrable systems lately, e.g. in the examples of a Lax pair etc. I frequently run into the term 'quantum integrable system'. May I ask a few questions:
What ...
- 1,719
9
votes
1
answer
1k
views
Fedosov vs. Kontsevich deformation quantization : a beginner survey
I'm a condensed matter physicist who tries to understand the details of deformation quantization.
In my self-made training, I've found two huge pieces of work, namely
Fedosov, B. V. (1994). "A ...
- 245
9
votes
0
answers
544
views
From classical to quantum mechanics
Let ($X,\omega$) be a symplectic manifold (phase space of some physical system). Consider the algebra $\mathcal{C}^{\infty}(X,\mathbb{R})$ of smooth functions on $X$ and $[\omega]\in \textrm{H}^{2}_{\...
- 1,629
7
votes
7
answers
2k
views
Quantization of a classical system (e.g. the case of a billiard)
There have been already several questions asking for an introduction to quantum mechanics
for a mathematician, but this one is slightly different, and more restrictive.
I know (some)
quantum ...
- 26.1k
7
votes
1
answer
332
views
Question about a remark on quantization of Coulomb branches
I will follow the definition of Coulomb branches of $3d$ $\mathcal{N}=4$ gauge theories from the paper by Braverman, Finkelberg and Nakajima, Towards a mathematical definition of Coulomb branches of 3-...
- 3,318
7
votes
1
answer
3k
views
What is the relation between BRST quantization and gauge fixing quantization
To quantize gauge field, one usually use gauge-fixing procedure and then plus ghost field, my question is what the relation between BRST quantization and gauge fixing quantization is? Because it seems ...
- 781
7
votes
1
answer
731
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
7
votes
1
answer
413
views
Kontsevich Formality sign convention
Since my question is related to sign convention, I want to define everything from the very beginning. $T_{poly}^k(M)=\Gamma(\wedge^{k+1} TM)$ are the multi vector fields with shifted degree and with ...
7
votes
0
answers
221
views
Deformation quantization of infinite dimensional Poisson manifolds
In 1999, Karaali wrote a review of formal deformation quantization for a class she took with Weinstein.
She ends the paper with the following remark:
Another question that remains involves the ...
- 527
6
votes
4
answers
710
views
Higgs mechanism from a deformation quantization point of view
Is it possible to describe the Higgs mechanism from a deformation quantization point of view? How would one do it? Are there aspects of the Higgs mechanism and Higgs particle which one may see clearer ...
- 1,222
6
votes
3
answers
2k
views
Ambidexterity and Quantization
Here the nlab says about Hopkins-Lurie's ambidexterity paper:
"The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in ...
user62675
6
votes
2
answers
850
views
1-dimensional pure gauge theory
I am learning TQFT from compact Lie groups by Freed, Hopkins, Lurie,
and Teleman: https://arxiv.org/abs/0905.0731 , and got stuck very hard
even in the first section ($n = 1$), which was "trivial but ...
- 5,230
6
votes
3
answers
265
views
graded generalization of the Moyal–Weyl product
Has anyone written about the graded generalization of the Moyal–Weyl product/star product, that is, where the original algebra is already graded? Is it just a matter of signs?
- 3,880
5
votes
2
answers
536
views
Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization?
The question is in the title. The form of the condition looks like the Bohr-Sommerfeld quantization formula of angular momentum, is there a link between the two formulas?
- 585
5
votes
1
answer
295
views
Absent 2nd order terms in deformation quantization of Poisson manifolds
I am reading Kontsevich' famous paper on deformation quantization of Poisson manifolds. In section 1.4.2 on page 4 he gives the general formula for the star product associated to a Poisson structure ...
- 515
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
3
votes
1
answer
166
views
Reference request for quantum Teichmuller space
I would like to ask for some detailed reference for quantum Teichmuller theory, better in a mathematical taste. I read a little bit on Kashaev's or Chekhov and Fock's, but find that I need to fill ...
- 263
3
votes
1
answer
275
views
higher order Noether identities
Noether's second variational theorem gives a correspondence between symmetries of a Lagrangian and Noether identities, which are relations among the Euler–Lagrange equations.
How about relations ...
- 3,880
3
votes
1
answer
161
views
Reference of $\hbar$-differential operator from symplectic geometry perspective
I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian ...
- 587
3
votes
0
answers
413
views
From symplectic manifold to Hilbert spaces [closed]
What could be a mathematical model of such physical wish? I'm looking for something sending a symplectic manifold $(M,\omega)$ to a Hilbert space $H_{M}$ with the following properties:
1- We should ...
- 1,607
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
765
views
Problem of quantization: state of the art [closed]
As the title suggests, I'm interested in finding out the state-of-the-art in the problem of quantization.
Any suggestions and/or feedback would be greatly appreciated.
Regards.
- 279
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
2
votes
0
answers
90
views
Most general space for the Wigner-Weyl transform
The Wigner-Weyl transform $\mathfrak{W}$ is a bijective mapping between functions on a phase space and Hilbert space operators in order to map quantum mechanics into a phase-space formulation. Then ...
user478556
-1
votes
2
answers
684
views
Problem of quantization: state of the art
The "problem of quantization":
Find a vector space $Obs$ (as large as possible) of real-valued functions $f(p, q)$ on $R^{2n}$, containing the coordinate ...
- 279