Non-commutative Geometry Research Papers - Academia.edu (original) (raw)

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- by
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- Mathematical Physics, Quantum Physics, Non-commutative Geometry, Phase Space

Drawing inspiration from Dirac's work on functions of non commuting observables, we develop a fresh approach to phase space descriptions of operators and the Wigner distribution in quantum mechanics. The construction presented here is marked by its economy, naturalness and more importantly, by its potential for extensions and generalisations to situations where the underlying configuration space is non Cartesian.

- by Rajiah Simon
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- Quantum Physics, Quantum Mechanics, Non-commutative Geometry, Phase Space
- by George Georgescu
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- Pure Mathematics, Non-commutative Geometry, Total order, Fuzzy Sets and Systems
- by Andrea Posilicano
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- Lie Algebra, Pure Mathematics, Non-commutative Geometry, Time Dependent

The impacts of home-based telecommuting on travel behavior and personal vehicle emissions for participants in the State of California Telecommuting Pilot Project are analyzed using the most advanced emissions modeling tools currently available. A comparison of participants' telecommuting day travel behavior with their before-telecommuting behavior shows a 27% reduction in the number of personal vehicle trips, a 77% decrease in vehicle-miles traveled (VMT), and 39% (and 4%) decreases in the number of cold (and hot) engine starts. These decreases in travel translate into emissions reductions of: 48% for total organic gases (TOG), 64% for carbon monoxide (CO), 69% for nitrogen oxide (NOx), and 78% for particulate matter (PM). Although the authors developed the methodology to investigate the emissions impacts of telecommuting, the analysis technique can be applied to any demand management or other transportation strategy where all of the necessary model inputs are available. An analysis of the number of personal vehicle trips and VMT partitioned into commute-related and non-commute-related purposes revealed that non-commute personal vehicle trips increased by 0.5 trips per person-day on average, whereas the non-commute VMT decreased by 5.3 miles. This important finding supports (for one indicator, the number of trips) the hypothesis that non-commute travel generation is a potential negative impact of telecommuting. This finding demonstrates the need to monitor these changes as telecommuting moves into the mainstream. In this study, however, the small increase in non-commute trips has a negligible impact compared to the overall travel and emissions savings.

- by Patricia L Mokhtarian
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- Engineering, Behavior, Non-commutative Geometry, Emission reduction
- by Roberto La Scala and +2
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- Applied Mathematics, Symbolic Computation, Non-commutative Geometry, Maple Computer Algebra System
- by Samuel Kováčik
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- Quantum Mechanics, Non-commutative Geometry
- by Samuel Kováčik
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- Quantum Mechanics, Non-commutative Geometry

Emergent gravity is based on a novel form of the equivalence principle known as the Darboux theorem or the Moser lemma in symplectic geometry stating that the electromagnetic force can always be eliminated by a local coordinate transformation as far as spacetime admits a symplectic structure, in other words, a microscopic spacetime becomes noncommutative (NC). If gravity emerges from U(1) gauge theory on NC spacetime, this picture of emergent gravity suggests a completely new quantization scheme where quantum gravity is defined by quantizing spacetime itself, leading to a dynamical NC spacetime. Therefore the quantization of emergent gravity is radically different from the conventional approach trying to quantize a phase space of metric fields. This approach for quantum gravity allows a background independent formulation where spacetime as well as matter fields is equally emergent from a universal vacuum of quantum gravity.

In this paper we introduce an algebra embedding ι:K< X >→ S from the free associative algebra K< X > generated by a finite or countable set X into the skew monoid ring S = P * Σ defined by the commutative polynomial ring P = K[X× N^*] and by the monoid Σ = < σ > generated by a suitable endomorphism σ:P→ P. If P = K[X] is any ring of polynomials in a countable set of commuting variables, we present also a general Gröbner bases theory for graded two-sided ideals of the graded algebra S = ⊕_i S_i with S_i = P σ^i and σ:P → P an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of P. Moreover, using a suitable grading for the algebra P compatible with the action of Σ, we obtain a bijective correspondence, preserving Gröbner bases, between graded Σ-invariant ideals of P and a class of graded two-sided ideals of S. By means of the embedding ι this results in the unification, in the graded case, of the Gröbner bases ...

After an introduction to some basic issues in non-commutative geometry (Gel'fand duality, spectral triples), we present a "panoramic view" of the status of our current research program on the use of categorical methods in the setting of A.Connes' non-commutative geometry: morphisms/categories of spectral triples, categorification of Gel'fand duality. We conclude with a summary of the expected applications of "categorical non-commutative geometry" to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.

- by Paolo Bertozzini
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- Quantum Gravity, Operator Algebras, Category Theory, Non-commutative Geometry

Here, in this paper we present two approaches outlining a conceptually and mathematically neat
frameworks within which we reproduce the mass generation mechanism essential for the Standard
Model. The rst framework is based on the inclusion of the gauge-group parameters into the
theory as scalar dynamical elds paralleling the standard Goldstone bosons. The advantage of
including the gauge-group parameters into the physical theory lies in that traditional gauge-
xing operations are nothing other than ordinary changes of variable which do not spoil gauge
invariance.
With a proper Lagrangian for these new elds of the -model type and appropriate rewriting
of the traditional Minimal Coupling Prescription we arrive at a general Massive Gauge The-
ory explicitly exhibiting gauge symmetry. When applied to the electroweak symmetry the new
prescription provides mass to the W() and Z vector bosons without the need for the Higgs
particle, leaving naturally the electromagnetic eld massless.
In the second part we present spontaneously broken NC theory, which is broken by a scalar
eld,providing a new symmetry breaking term to the gauge eld, a term not belonging to the
gauge eld itself. This theory was shown by explicit calculation to be one loop renormalizable,
and we therefore expect it to be unitary at tree-level.
Thus, we show how it is possible to obtain mass generation in the context of non Abelian gauge
eld theory, using a non commutative space-time, i.e. the space-time symmetry is further en-
larged. This is further con rmed by the modi ed dispersion relation that results from such a
geometry. Other e ects in the domain of Ultra High Energy Gamma Rays and Cosmic Rays are
also deduced and it is pointed out that we might already have observed such e ects.

- by Yan Soibelman
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- Algebraic Geometry, Category Theory, High Energy Physics, Non-commutative Geometry
- by Carlos Herdeiro
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- Field Theory, Quantum Physics, Early Universe, Non-commutative Geometry
- by Bilal Khan
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- Group Theory, Non-commutative Geometry, Public key cryptography, Decision Problem
- by Ronald Brown
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- Cognitive Science, Philosophy, Non-commutative Geometry, Quantum Logic

Starting with the quantum Liouville equation, we write the density operator as the product of elements respectively in the left and right ideals of an operator algebra and find that the Schrodinger picture may be expressed through two representation independent algebraic forms in terms of the density and phase operators. These forms are respectively the continuity equation, which involves the

- by Basil Hiley
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- Quantum Mechanics, Non-commutative Geometry, Berry phase, Continuity Equation
- by Viktor Levandovskyy
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- Non-commutative Geometry, Computer Algebra, Grobner bases, Grobner Basis

Using constructive methods in invariant theory, we define a map (with the minimal number of invariants) that distinguishes simultaneous similarity classes for non-commutative sequences over a field of characteristic neq2\neq2neq2. We also describe canonical forms for sequences of 2times22\times22times2 matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants.

- by Carlos Florentino
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- Engineering, Non-commutative Geometry, Mathematical Sciences, Invariant Theory
- by Vito Michele Abrusci
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- Non-commutative Geometry, Tableaux
- by Roberto Poli
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- Formal Ontology, Non-commutative Geometry

In this talk, motivated by the need to address foundational problems in relativistic quantum physics, we first historically introduce some basic elements of non-commutative geometry (Gel’fand-Naimark duality and Connes’ spectral triples); then (following arXiv:1007.4094) we speculatively suggest that, when coupled with Tomita-Takesaki modular theory and suitable notions of categorical covariance (higher C*-categories), non-commutative geometry provides a possible way for a formulation of a theory of quantum relativity, where non-commutative relational space-time might be spectrally reconstructed from an operational formalism of categories of observable algebras.
Part of this work is a joint collaboration with:
Dr.Roberto Conti (Sapienza Universita' di Roma),
Dr.Matti Raasakka (Paris 13 University).

An attempt is made to interpret the interactions of bosonic open strings as defining a non-cummulative, associative algebra, and to formulate the classical non-linear field theory of such strings in the language of non-commulative geometry. The point of departure is the BRST approach to string field theory. A setting is given in which there is a unique gauge invariant action, whose linearized approximation reproduces the conventional Veneziano spectrum. A derivation of conventional Veneziano model amplitudes from this gauge invariant action is sketched. Some brief comments are made about attempts to extend these results to open superstrings and to closed strings.

- by mike johhh
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- Mathematical Physics, Algebra, Field Theory, Associative Algebra
- by Albert Schwarz
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- Non-commutative Geometry

Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic R 4 , and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called smooth quantum gravity by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard R 4 to the exotic R 4 is a quantization of geometry.

- by Marius Junge
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- Mathematics, Pure Mathematics, Non-commutative Geometry, von Neumann algebra
- by Yj Oh
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- Pure Mathematics, Non-commutative Geometry, Master Equation
- by Albert Schwarz and +1
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- High Energy Physics, Non-commutative Geometry, Matrix Theory, Noncommutative Geometry
- by Chong-sun Chu
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- Field Theory, High Energy Physics, Non-commutative Geometry, Mathematical Sciences

A satisfactory marriage between higher categories and operator algebras has never been achieved: although (monoidal) C*-categories have been systematically used since the development of the theory of superselection sectors, higher category theory has more recently evolved along lines closer to classical higher homotopy.
We present axioms for strict involutive n-categories (a vertical
categorification of dagger categories) and a definition for strict
higher C*-categories and Fell bundles (possibly equipped with
involutions of arbitrary depth), that were developed in
collaboration with Roberto Conti, Wicharn Lewkeeratiyutkul and Noppakhun Suthichitranont.

- by S. Hoedl
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- Particle Physics, String Theory, Non-commutative Geometry, Low Energy Buildngs
- by M. Bożejko
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- Mathematical Physics, Functional Analysis, Quantum Physics, High Energy Physics
- by Carlos Jimenez
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- Non-commutative Geometry, Quality Control, Humans, Calibration

We make use of Tomita-Takesaki modular theory in order to reconstruct non-commutative spectral geometries (formally similar to spectral triples) from suitable states over (categories of) operator algebras and further elaborate on the utility of such a formalism in an algebraic theory of quantum gravity, where space-time is spectrally reconstructed a posteriori from (partial) observables and states in a covariant quantum theory. Some relations with A.Carey-J.Phillips-A.Rennie modular spectral triples and with (loop) quantum gravity will be described.
This is an ongoing joint research with Dr. Roberto Conti
(Universit`a di Chieti-Pescara “G.D’Annunzio” - Italy)
Dr. Wicharn Lewkeeratiyutkul
(Chulalongkorn University - Bangkok - Thailand).

- by Vito Michele Abrusci
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- Cognitive Science, Mathematics, Computer Science, Philosophy

Relation between Bopp-Kubo formulation and Weyl-Wigner-Moyal symbol calculus, and non-commutative geometry interpretation of the phase space representation of quantum mechanics are studied. Harmonic oscillator in phase space via creation and annihilation operators, both the usual and qqq-deformed, is investigated. We found that the Bopp-Kubo formulation is just non-commuting coordinates representation of the symbol calculus. The Wigner operator for the qqq-deformed harmonic oscillator is shown to be proportional to the 3-axis spherical angular momentum operator of the algebra suq(2)su_{q}(2)suq(2). The relation of the Fock space for the harmonic oscillator and double Hilbert space of the Gelfand-Naimark-Segal construction is established. The quantum extension of the classical ergodiicity condition is proposed.

- by S. Baskoutas
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- Quantum Mechanics, High Energy Physics, Non-commutative Geometry, Phase Space
- by Marius Junge
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- Mathematics, Pure Mathematics, Non-commutative Geometry, von Neumann algebra

We outline the main features of the definitions and applications of crossed complexes and cubical omega\omegaomega-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the... more

- by Ronald Brown
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- Category Theory, Non-commutative Geometry, Homotopy Type Theory

Abstract. In these lectures 4 quantum physics in noncommutative spacetime is developed. It is based on the work of Doplicher et al. which allows for time-space noncommutativity. In the context of noncommutative quantum mechanics, some... more

- by Carlos Mendes
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- Engineering, Field Theory, Physics, Quantum Physics

We exhibit new examples of double quasi-Poisson brackets, based on some classification results and the method of fusion. This method was introduced by Van den Bergh for a large class of double quasi-Poisson brackets which are said differential, and our main result is that it can be extended to arbitrary double quasi-Poisson brackets. We also provide an alternative construction for the double quasi-Poisson brackets of Van den Bergh associated to quivers, and of Massuyeau-Turaev associated to the fundamental groups of surfaces.

- by Maxime Fairon
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- Non-commutative Geometry, Quantum Algebra, Non-Commutative Ring Theory