3
votes
Do the canonical commutation relations have any connection to geometry?
Well, this is a fairly broad topic.
Here is one way CCRs arise from a rather large class of geometries: Given a Fedosov manifold $(M,\omega, \nabla)$ [i.e. a manifold $M$ endowed with a symplectic $...
- 213k
3
votes
Relationship between $\star$-products in phase-space QM and NC geometry
If the phase space is $\mathbb{R}^{2r+k}$, and $\theta^{\mu\nu}$ has rank $2r$ then there exists a bijective linear coordinate transformation that brings the anti-symmetric real matrix $\theta^{\mu\nu}...
- 213k
2
votes
Accepted
What is a fuzzy space?
Fuzzy spaces can refer to a few things, but mainly they refer to finite-dimensional spectral triples, specifically noncommutative ones.
So to understand the idea behind them it is useful to ...
- 56
2
votes
What is Kappa deformation in quantum gravity?
Firstly, $\kappa$-deformations is a really specific example of non-commutative geometry and non-commutative field theory.
As far as I'm aware kappa-deformations ($\kappa$-deformations) usually refers ...
- 56
1
vote
Effects of non-locality in the star-product of two fields
This is a basic feature of the star product, easiest to see in bland QM in phase space. (Just take D=2 dimensions x and p, and $\theta_{ij} =\hbar \epsilon_{ij}$; beyond the stringer tart-up, the ...
- 66.3k
1
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Relationship between $\star$-products in phase-space QM and NC geometry
I cannot pretend to understand the question (I would answer what I understand as a clumsier version of @Qmechanic 's answer), but, as a courtesy of the casual reader, I'd rewrite the first equation ...
- 66.3k
1
vote
Accepted
Implication of the Jacobian map for the structure of the Euclidean space-time
The Heisenberg-like commutation relation $$ \langle Y[D,Y]^{2m} \rangle= \gamma $$ (with $\gamma $ the chirality operator) appeared first in a preprint then a PRL article.
It was devised by A. ...
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