# Noncommutative Geometries from first principles

Noncommutative Geometries from "first principles"

Hi everyone,

to give a motivation for studying specific models of noncommutative geometry, I would like to start this thread as a collecting tank of models of noncommutative geometry that are obtained as a limit of some kind from 'first principles', i.e. not an ad-hoc modification of the commutator [x_i,x_j] . It would be nice, if we could collect cases by briefly stating what theory in which limit they are obtained from, provide the specific form of the coordinate commutator, and cite a reference (if possible, respectively). So, as a start:

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From 2+1-dim. Spinfoams, by integrating out gravitational DOF, to arrive at the flat-space effective field theory:

$$[X_i, X_j]=i\hbar \kappa \epsilon_{ijk} X_k, \quad \kappa = 4\pi G$$

Reference: http://arxiv.org/abs/0705.2222
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As some knowledgeable people are around here, I hope some members share interest in this and contribute . For example I heard in string theory world-sheet coordinates do not commute, but I don't know much about it, i.e. which precise form in which case.

Related Beyond the Standard Model News on Phys.org

This thread is "above my pay grade" but for those who are more versed than I, Roger Penrose in THE ROAD TO REALITY has some relevant discussions, especially in 33.1, where on pages 961 and 962 he gives some non commutative examples.

He attributes much non commutative geometry insight to Alan Connes and goes on to say

...in quantum mechanics one frequently encounters algebras that are non commutative....Connes and his colleagues developed the idea of non commutative geometry with a view to producing a physical theory which includes the standard model of particle physics....the potential richness of the idea of non-commutative geometry does not seem to me to be at all strongly used, so far....twistor theory has some relations to spin network theory and to Ashtekar variables and possibly non-commutative geometry....

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Hi Naty,

so it is for me actually , which is why I hope that others pop in to contribute. Anyways, thanks for your reference. So from THE ROAD TO REALITY, p. 983, one at least has

$$[Z^{\alpha},\overline{Z}_{\beta}]=\hbar \delta^{\alpha}_{\beta},$$

and as the twistor components Z, if I understand correctly, are actually composed of spacetime coordinates, it would be interesting to see what this means for the coordinate commutator. However I couldn't find it worked out anywhere and I don't know if one link this framework to a coordinate commutator in a sensible way.

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Edit: As for Connes, I think the approach is a little different, as one considers a product of a commutative spin manifold M (accounting for spacetime) with a finite, noncommutative space accounting for matter content. While this as well seems very worthwile to study, we do have some threads about this, so that I would prefer to talk about scenarios in which the ordinary space/spacetime coordinates are noncommuting.

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