Distances on a lattice from non-commutative geometry

Bimonte, Giuseppe; Lizzi, Fedele; Sparano, Giovanni

doi:10.1016/0370-2693(94)90302-6

Cited by 36 publications

(46 citation statements)

References 8 publications

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“…Other problems with discretizing non-commutative geometry, such as the construction of an appropriate lattice Dirac operator, are discussed in ref. [42].…”

Section: Jhep05(2000)023mentioning

confidence: 99%

Mode Transition of an Atmospheric Pressure Arc Plasma Jet Sustained by Pulsed DC Power

Hsu¹,

Wu²,

Chen

et al. 2009

jjap

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We present a lattice formulation of non-commutative Yang-Mills theory in arbitrary even dimensionality. The UV/IR mixing characteristic of noncommutative field theories is demonstrated at a completely non-perturbative level. We prove a discrete Morita equivalence between ordinary Yang-Mills theory with multi-valued gauge fields and non-commutative Yang-Mills theory with periodic gauge fields. Using this equivalence, we show that generic non-commutative gauge theories in the continuum can be regularized non perturbatively by means of ordinary lattice gauge theory with 't Hooft flux. In the case of irrational non-commutativity parameters, the rank of the gauge group of the commutative lattice theory must be sent to infinity in the continuum limit. As a special case, the construction includes the recent description of non-commutative Yang-Mills theories using twisted large-N reduced models. We study the coupling of non-commutative gauge fields to matter fields in the fundamental representation of the gauge group using the lattice formalism. The large mass expansion is used to describe the physical meaning of Wilson loops in non-commutative gauge theories. We also demonstrate Morita equivalence in the presence of fundamental matter fields and use this property to comment on the calculation of the beta-function in non-commutative quantum electrodynamics.correspondence has been used to suggest that non-commutative geometry provides a natural framework to describe non-perturbative aspects of string theory [2,5]. This belief is further supported by the fact that Matrix Theory [6] and the IIB-matrix model [7], which are conjectured to provide non-perturbative definitions of string theories, give rise to non-commutative Yang-Mills theory on toroidal compactifications [8]. The particular non-commutative toroidal compactification is interpreted as being the result of the presence of a background Neveu-Schwarz two-form field, and it can also be understood in the context of open-string quantization in D-brane backgrounds [9, 10]. Furthermore, in ref. [11] it has been shown that the IIB-matrix model with D-brane backgrounds is described by non-commutative Yang-Mills theory.The early motivation [12] for studying quantum field theory on non-commutative spacetimes was that, because of the spacetime uncertainty relation, the introduction of non commutativity would provide a natural ultraviolet regularization. However, more recent perturbative calculations [13]- [16] have shown that planar noncommutative Feynman diagrams contain exactly the same ultraviolet divergences that their commutative counterparts do, which implies that the non commutativity does not serve as an ultraviolet regulator. One therefore needs to introduce some other form of regularization to study the dynamics of non-commutative field theories. On the other hand, it has been found that the ultraviolet divergences in non-planar Feynman diagrams [16,17] exhibit an intriguing mixing of ultraviolet and infrared scales, which can also be described using string-theoreti...

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“…Other problems with discretizing non-commutative geometry, such as the construction of an appropriate lattice Dirac operator, are discussed in ref. [42].…”

Section: Jhep05(2000)023mentioning

confidence: 99%

Mode Transition of an Atmospheric Pressure Arc Plasma Jet Sustained by Pulsed DC Power

Hsu¹,

Wu²,

Chen

et al. 2009

jjap

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“…the quantity which carries the relevant physical unit). Finding the distances on a one-dimensional lattice was the goal of [1,2]. In these works [1,2], one uses the local discrete Dirac-Wilson operator, usually applied in lattice gauge theories 1 .…”

Section: Bi-graded Markovian Matrices As Non-local Dirac Operatorsmentioning

confidence: 99%

Finite 1D-Lattice Physics as Induced by Dirac–Markov Operators

Atzmon

Atzmon²

2012

Lett Math Phys

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Measuring distances on a lattice in noncommutative geometry involves square, symmetric and real "three-diagonal" matrices, with the sum of their elements obeying a supremum condition, together with a constraint forcing the absolute value of the maximal eigenvalue to be equal to 1. In even dimensions, these matrices are unipotent of order two, while in odd dimensions only their squares are Markovian. We suggest that these bi-graded Markovian matrices (i.e. consisting in the square roots of Markovian matrices) can be thought of as non-local Dirac operators. The eigenvectors of these matrices are spinors. Treating these matrices as determining the stochastic time evolution of states might explain why one observes only left handed neutrinos. Some other physical interpretations are suggested. We end by presenting a mathematical conjecture applying to q− graded Markovian matrices.

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“…In this way, one recovers the ordinary Riemann distance between points [15]. Indeed, one has for a spinor…”

Section: Noncommutative Geometry and Quantization Of Volumementioning

confidence: 99%

Noncommutative Geometry and Stochastic Processes

Frasca¹

2017

Lecture Notes in Computer Science

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The recent analysis on noncommutative geometry, showing quantization of the volume for the Riemannian manifold entering the geometry, can support a view of quantum mechanics as arising by a stochastic process on it. A class of stochastic processes can be devised, arising as fractional powers of an ordinary Wiener process, that reproduce in a proper way a stochastic process on a noncommutative geometry. These processes are characterized by producing complex values and so, the corresponding Fokker-Planck equation resembles the Schrödinger equation. Indeed, by a direct numerical check, one can recover the kernel of the Schrödinger equation starting by an ordinary Brownian motion. This class of stochastic processes needs a Clifford algebra to exist. In four dimensions, the full set of Dirac matrices is needed and the corresponding stochastic process in a noncommutative geometry is easily recovered as is the Dirac equation in the Klein-Gordon form being it the Fokker-Planck equation of the process. * marcofrasca@mclink.it

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Distances on a lattice from non-commutative geometry

Cited by 36 publications

References 8 publications

Mode Transition of an Atmospheric Pressure Arc Plasma Jet Sustained by Pulsed DC Power

Mode Transition of an Atmospheric Pressure Arc Plasma Jet Sustained by Pulsed DC Power

Finite 1D-Lattice Physics as Induced by Dirac–Markov Operators

Noncommutative Geometry and Stochastic Processes

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