Noncommutative World Research Articles

Non-Commutative Worlds and Relativity

This paper shows how aspects of gauge theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a non-commutative framework for calculus and differential geometry.

Non-Commutative Worlds and Classical Constraints.

This paper reviews results about discrete physics and non-commutative worlds and explores further the structure and consequences of constraints linking classical calculus and discrete calculus formulated via commutators. In particular, we review how the formalism of generalized non-commutative electromagnetism follows from a first order constraint and how, via the Kilmister equation, relationships with general relativity follow from a second order constraint. It is remarkable that a second order constraint, based on interlacing the commutative and non-commutative worlds, leads to an equivalent tensor equation at the pole of geodesic coordinates for general relativity.

Quantum metric spaces of quantum maps

We show that any quantum family of quantum maps from a noncommutative space to a compact quantum metric space has a canonical quantum pseudo-metric structure. Here by a 'compact quantum metric space' we mean a unital C*-algebra together with a Lipschitz seminorm, in the sense of Rieffel, which induces the weak* topology on the state space of the C*-algebra. Our main result generalizes a classical result to noncommutative world.

Noncommutative Valuation of Options

The aim of this note is to show that the classical results in finance theory for pricing of derivatives, given by making use of the replication principle , can be extended to the noncommutative world. We believe that this could be of interest in quantum probability. The main result called the First fundamental theorem of asset pricing , states that a noncommutative stock market admits no-arbitrage if and only if it admits a noncommutative equivalent martingale probability.

$\mathbf A^1$-homotopy theory of noncommutative motives

In this article we continue the development of a theory of noncommutative motives, initiated in [30]. We construct categories of {\bf A}^1 -homotopy noncommutative motives, describe their universal properties, and compute their spectra of morphisms in terms of Karoubi–Villamayor's K -theory ( KV ) and Weibel's homotopy K -theory ( KH ). As an application, we obtain a complete classification of all the natural transformations defined on KV, KH . This leads to a streamlined construction of Weibel's homotopy Chern character from KV to periodic cyclic homology. Along the way we extend Dwyer–Friedlander's étale K -theory to the noncommutative world, and develop the universal procedure of forcing a functor to preserve filtered homotopy colimits.

K-theory and the bridge from motives to noncommutative motives

In this work we present a new approach to the theory of noncommutative motives and use it to explain the different flavors of algebraic K-theory of schemes and dg-categories. The work is divided into three main parts. In the first part we use the techniques of higher algebra developed in [63] to provide a universal characterization for the symmetric monoidal (∞,1)-category underlying the motivic stable A1- homotopy theory of Morel–Voevodsky [107,67]. More precisely, given a symmetric monoidal model category V together with an object X∈V, we characterize the underlying symmetric monoidal (∞,1)-category of the symmetric monoidal model category SpΣ(V,X) introduced by Hovey in [43], by means of a universal property amongst symmetric monoidal (∞,1)- categories. This characterization trivializes the problem of finding motivic monoidal realizations.In the second part we introduce a new approach to the theory of noncommutative motives by constructing a stable motivic homotopy theory for the noncommutative spaces of Kontsevich [56,55,54]. The key ingredient is a notion of Nisnevich topology in the noncommutative setting, compatible with the classical notion. This compatibility, together with the universal property proved in the first part, ensures the existence of a canonical monoidal map from the stable motivic theory of Morel–Voevodsky towards these new noncommutative motives that allow us to compare the two theories.In the last part of this paper we explain how this bridge can be used to explain the various flavors of algebraic K-theory of dg-categories. More precisely, we prove that the non-connective K-theory of dg-categories introduced by Schlichting [82] is the (non-commutative) Nisnevich sheafification of connective algebraic K-theory. Then we prove that its further (non-commutative) A1-localization is a tensor unit in our noncommutative motives. As a corollary we obtain a precise proof for an original conjecture of Kontsevich claiming that K-theory gives the correct mapping spaces in noncommutative motives. Our major application is the discovery of a canonical factorization of our motivic bridge through the (∞,1)-category of modules over the commutative algebra object representing homotopy invariant algebraic K-theory of schemes. The results in [77] imply that this bridge is fully faithful over a field k with resolutions of singularities, so that, at the motivic level, no information (below K-theory) is lost by passing to the noncommutative world.

Jacobians of Noncommutative Motives

In this article one extends the classical theory of (intermediate) Jacobians to the “noncommutative world”. Concretely, one constructs a ℚ-linear additive Jacobian functor N → J(N) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd periodic cyclic homology of N which is generated by algebraic curves; (ii) the abelian variety J(perf_(dg)(X)) (associated to the derived dg category perf_(dg)(X) of a smooth projective k-scheme X) identifies with the product of all the intermediate algebraic Jacobians of X. As an application, every semi-orthogonal decomposition of the derived category perf(X) gives rise to a decomposition of the intermediate algebraic Jacobians of X.

Fuzzy Galois connections categorically

This paper presents a systematic investigation of fuzzy (non-commutative) Galois connections in the sense of R. Bělohlávek [1], from the point of view of enriched category theory. The results obtained show that the theory of enriched categories makes it possible to present the theory of fuzzy Galois connections in a succinct way; and more importantly, it provides a useful method to express and to study the link and the difference between the commutative and the non-commutative worlds (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Non-commutative worlds

This paper presents a mathematical view of aspects of physics, showing how the forms of gauge theory, Hamiltonian mechanics and quantum mechanics arise from a non-commutative framework for calculus and differential geometry. This work is motivated by discrete calculus, as it is shown that classical discrete calculus embeds in a non-commutative context. It is shown how various processes are modeled by non-commutative discrete calculus, and how aspects of differential geometry, such as the Levi-Civita connection, arise naturally from commutator equations and the Jacobi identity. A new and generalized version of the Feynman–Dyson derivation of electromagnetic equations is given, with corresponding discrete models.

Field theory on an evolving fuzzy 2-sphere

I construct a field theory on an evolving fuzzy 2-sphere, which is based on the idea of the evolving non-commutative worlds of the previous paper [1]. The equations of motion are similar to the one that can be obtained by dropping the time-derivative term of the equation derived sometime ago by Banks, Peskin and Susskind for pure-into-mixed-state evolutions. The equations do not contain an explicit time, and therefore follow the spirit of the Wheeler–DeWitt equation. The basic properties of the field theory, such as action, gauge invariance and charge and momentum conservation, are studied. The continuum limit of the scalar field theory shows that the background geometry of the corresponding continuum theory is given by ds2 = −dt2 + t dΩ2, which saturates locally the cosmic holographic principle.

Non-unitary evolutions of noncommutative worlds with symmetry

Motivated by describing time-evolutions of noncommutative worlds, I discuss symmetry-preserving evolutions of noncommutative worlds of finite dimensional representation spaces. An interesting issue in such evolutions is that there can be transitions of representation spaces of symmetry, and therefore the evolutions are generally non-unitary or symmetry-violating. The central idea of this paper is that a main world evolves by emitting baby worlds which compensate the violations of the symmetry of the main world. Tracing out the states of the baby worlds, the symmetry-preserving evolutions of the density matrices of the main world are obtained. I give a simple example with SU(2) symmetry, which can be regarded as an evolving quantum two-sphere. This simple model has some attractive features resembling our universe: it gets born from a vacuum and its entropy of geometric origin grows. I also discuss the evolutions in Heisenberg picture.

Non-commutative fuzzy Galois connections

Fuzzy Galois connections were introduced by Bělohlavek in [4]. The structure considered there for the set of truth values is a complete residuated lattice, which places the discussion in a “commutative fuzzy world”. What we are doing in this paper is dropping down the commutativity, getting the corresponding notion of Galois connection and generalizing some results obtained by Bělohlavek in [4] and [7]. The lack of the commutative law in the structure of truth values makes it appropriate for dealing with a sentences conjunction where the order between the terms of the conjunction counts, gaining thus a temporal dimension for the statements. In this “non-commutative world”, we have not one, but two implications ([15]). As a consequence, a Galois connection will not be a pair, but a quadruple of functions, which is in fact two pairs of functions, each function being in a symmetric situation to his pair. Stating that these two pairs are compatible in some sense, we get the notion of strong L-Galois connection, a more operative and prolific notion, repairing the “damage” done by non-commutativity.

The non-commutative brane world

We propose a new higher-dimensional mechanism to localize scalar fields as well as fermionic and gauge fields. The underlying theory is a six-dimensional non-commutative field theory where non-commutativity is allowed along two extra infinite spatial dimensions and the four-dimensional brane is provided by a scalar soliton living in the non-commutative space. Making use of the powerful correspondence between non-commutative coordinates and operators on a single particle Hilbert space, we show that the non-commutative brane world admits localized chiral fermions and it ensures the localization of massless gauge fields. It may also give rise to a variety of different low-energy spectra since the localized zero mode may come along either with a discrete tower of degenerate heavy states or with a tower of Kaluza-Klein heavy states, or it may even be the only state in the low-energy spectrum.

CURVED D-BRANE GEOMETRY AND ZERO MODES

We present a path integral formalism to review the effective dynamics of an arbitrary and curved Dirichlet p-brane (Dp-brane) in open bosonic string and subsequently in type I superstring theory. We obtain the perturbative corrections to the Dirac–Born–Infeld dynamics up to [Formula: see text] in the presence of an antisymmetric two-form B field. A part of the corrections can be seen to be associated with an ultraviolet divergence. Renormalization of the Dp-brane collective coordinates is performed to show that the residual corrections lead to a noncommutative description on the world volume. The Dp-brane modes are analyzed and we show that its zero modes play a vital role in noncommutative geometry. Our analysis facilitates the noncommutative world volume description for an arbitrary B field.