Matrix geometries emergent from a point

D’Andrea, Francesco; Lizzi, Fedele; Martinetti, Pierre

doi:10.1142/s0129055x14500172

Cited by 3 publications

(4 citation statements)

References 24 publications

(54 reference statements)

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“…Let O θ be the order unit space spanned by π θ (f ), f ∈ A sa , and π θ (1) = P θ . The action of P θ on D, with a proper normalization factor 4 , yields the Dirac operator of the irreducible spectral triple of Moyal plane [40,17]. Proposition 6.5.…”

Section: Berezin Quantization Of the Planementioning

confidence: 99%

Spectral geometry with a cut-off: Topological and metric aspects

D’Andrea

Lizzi

Martinetti

2014

Journal of Geometry and Physics

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Inspired by regularization in quantum field theory, we study topological and metric properties of spaces in which a cut-off is introduced. We work in the framework of noncommutative geometry, and focus on the Connes distance associated to a spectral triple (A,H,D). A high momentum (short distance) cut-off is implemented by the action of a projection P on the Dirac operator D and/or on the algebra A. This action induces two new distances. We individuate conditions making them equivalent to the original distance. We also study the Gromov-Hausdorff limit of the set of truncated states, first for compact quantum metric spaces in the sense of Rieffel, then for arbitrary spectral triples. To this aim, we introduce a notion of "state with finite moment of order 1" for noncommutative algebras. We then focus on the commutative case, and show that the cut-off induces a minimal length between points, which is infinite if P has finite rank. When P is a spectral projection of D, we work out an approximation of points by non-pure states that are at finite distance from each other. On the circle, such approximations are given by Fejér probability distributions. Finally we apply the results to the Moyal plane and the fuzzy sphere, obtained as Berezin quantization of the plane and the sphere respectively. © 2014 Elsevier B.V

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Section: Berezin Quantization Of the Planementioning

confidence: 99%

Spectral geometry with a cut-off: Topological and metric aspects

D’Andrea

Lizzi

Martinetti

2014

Journal of Geometry and Physics

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“…Their renormalization has been studied in [21][22][23][24][25] and the phenomenological implications of the resulting effective actions have been carried out, e.g., in [26][27][28]. A more detailed discussion of the cutoff Λ may be found in [29], and the generalization to non-commutative spaces built from non-associative algebras has been pursued in [30].…”

Section: Introductionmentioning

confidence: 99%

Spectral dimensions from the spectral action

2015

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The generalised spectral dimension DS(T ) provides a powerful tool for comparing different approaches to quantum gravity. In this work, we apply this formalism to the classical spectral actions obtained within the framework of almost-commutative geometry. Analysing the propagation of spin-0, spin-1 and spin-2 fields, we show that a non-trivial spectral dimension arises already at the classical level. The effective field theory interpretation of the spectral action yields plateaustructures interpolating between a fixed spin-independent DS(T ) = dS for short and DS(T ) = 4 for long diffusion times T . Going beyond effective field theory the spectral dimension is completely dominated by the high-momentum properties of the spectral action, yielding DS(T ) = 0 for all spins. Our results support earlier claims that high-energy bosons do not propagate.

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“…is a strongly continuous one-parameter group of unitaries generated by the (unbounded, selfadjoint on a suitable domain) operator 8) and σ t (a) = T (z t ) * aT (z t ) is a strongly continuous one-parameter group of * -automorphisms generated by the derivation δ(a) = i[X, a]. In the notations above, if τ 0 := Ψ z0 , then τ t = Ψ zt for all t ∈ R; from [8, Prop.…”

Section: Pythagoras From Generalized Geodesicsmentioning

confidence: 99%

“…The proportionality constant can be computed by taking the trace of (4.17), thus proving that Q maps 1 to 1 A . Due to the above properties, 1 N Q sends probability measures into density matrices, and N σ sends density matrices into probability distributions 8 . The latter map is injective under the following assumptions: suppose G is a connected compact semisimple Lie group and P a rank-one projection (a pure state) which has the highest weight vector of the representation in its range.…”

Section: 5mentioning

confidence: 99%