Quantization Of Manifolds Research Articles

Quantum manifolds and integral (co)bordism groups in quantum partial differential equations

Quantum manifolds and integral (co)bordism groups in quantum partial differential equations

CONCEPTUAL ISSUES FOR NONCOMMUTATIVE GRAVITY ON ALGEBRAS AND FINITE SETS

We discuss some of the issues to be addressed in arriving at a definitive noncommutative Riemannian geometry that generalises conventional geometry both to the quantum domain and to the discrete domain. This also provides an introduction to our 1997 formulation based on quantum group frame bundles. We outline now the local formulae with general differential calculus both on the base "quantum manifold" and on the structure group Gauge transforms with nonuniversal calculi, Dirac operator, Levi–Civita condition, Ricci tensor and other topics are also covered. As an application we outline an intrinsic or relative theory of quantum measurement and propose it as a possible framework to explore the link between gravity in quantum systems and entropy.

Noncommutative geometry and fundamental physical interactions: The Lagrangian level—Historical sketch and description of the present situation

These notes comprise (i) a descriptive account of the history of the subject showing how physics and mathematics interwove to develop a mathematical concept of quantum manifold relevant to elementary particle theory; (ii) a detailed technical description, from scratch, of the spectral action formalism and computation.

A geometric approach to a noncommutative generalized d'Alembert equation

A new m-d'Alembert equation, m≥2, is introduced in the category of quantum manifolds 5,6,8, that extends the commutative generalized d'Alembert equation just considered in 9. For such a new equation we give theorems of existence of local and global solutions. Une équation nouvelle de m-d'Alembert (m≥2) est introduite dans la catégorie des variétés quantiques 5,6,8, qui étend l'équation generalisée de d'Alembert déjà considerée dans 9. Pour cette nouvelle équation nous donnons des théorèmes d'existence de solutions locales et globales.

Co)bordism Groups in Quantum PDEs

In this paper we formulate a theory of noncommutative manifolds (quantum manifolds) and for such manifolds we develop a geometric theory of quantum PDEs (QPDEs). In particular, a criterion of formal integrability is given that extends to QPDEs previously obtained by D. C. Spencer and H. Goldschmidt for PDEs for commutative manifolds, and by Prastaro for super PDEs. Quantum manifolds are seen as locally convex manifolds where the model has the structure A m 1 1×···×A m s s , with A≡A 1×···×A s a noncommutative algebra that satisfies some particular axioms (quantum algebra). A general theory of integral (co)bordism for QPDEs is developed that extends our previous for PDEs. Then, noncommutative Hopf algebras (full quantum p-Hopf algebras, 0≤p≤m−1) are canonically associated to any QPDE, Ek ⊂ Ĵk m(W) whose elements represent all the possible invariants that can be recognized for such a structure. Many examples of QPDEs are considered where we apply our theory. In particular, we carefully study QPDEs for quantum field theory and quantum supergravity. We show that the corresponding regular solutions, observed by means of quantum relativistic frames, give curvature, torsion, gravitino and electromagnetic fields as A-valued distributions on spacetime, where A is a quantum algebra. For such equations, canonical quantizations are obtained and the quantum and integral bordism groups and the full quantum p-Hopf algebras, 0≤p≤3, are explicitly calculated. Then, the existence of (quantum) tunnel effects for quantum superstrings in supergravity is proved.

(CO)Bordisms in PDEs and quantum PDEs

In this paper we announce some recent results on the quantum and integral (co)bordism in PDEs and quantum PDEs. We shall essentially prove that the technique of (co)bordism, introduced by Pontrjagin and Thom in algebraic topology can be generalized in the framework of partial differential equations, in order to obtain sufficient criteria that allow to decide when a p-dimensional compact closed integral manifold contained in a PDE E k ⊂ J n k ( W) is the boundary of a ( p + 1)-dimensional integral compact manifold contained also in E k (integral bordism) or eventually in the jet-space J n k ( W) containing E k (quantum bordism). Furthermore, we shall prove that such results can be extended to the category of quantum PDEs. Here, by the term “quantum manifold” (and as a consequence also “quantum PDE”) we mean a new structure that extends globally usual concepts of quantum spaces, and that is very useful for physical applications.

The quantum 2-sphere as a complex quantum manifold

We describe the quantum sphere of Podles for c = 0 by means of a stereographic projection which is analogous to that which exibits the classical sphere as a complex manifold. We show that the algebra of functions and the differential calculus on the sphere are covariant under the coaction of fractional transformations with SUq(2) coefficients as well as under the action of SUq(2) vector fields. Going to the classical limit we obtain the Poisson sphere. Finally, we study the invariant integration of functions on the sphere and find its relation with the translationally invariant integration on the complex quantum plane.

Rational Scalar Field Theories on Quantum Spheres

Quantum manifolds can be used to describe strong quantum fluctuations of geometry which eliminate the ultraviolet divergences of the fields at microscopic level. It is shown that the numbers of scalar wave functions on quantum spheres in 2, 3 and 4 dimensions are finite when the deformation parameter is a root of unity. The deformation parameter is determined by the radius of the universe and the scale of the ultraviolet cutoff. Moreover, it is pointed out that the non-commutative nature of the wave functions depends on their scales. The wave functions and their energies are almost classical below the intermediate scale. If we set the scale far beyond our scale, the effect is almost invisible for our scale. A possible scenario for the quantum evolution of the universe is also discussed

Algebras and manifolds:: Differential, difference, simplicial and quantum

Generalized manifolds and Clifford algebras depict the world at levels of resolution ranging from the classical macroscopic to the quantum microscopic. The coarsest picture is a differential manifold and algebra (dm), direct integral of familiar local Clifford algebras of spin operators in curved time-space. Next is a finite difference manifold (Δm) of Regge calculus. This is a subalgebra of the third, a Minkowskian simplicial manifold (Σm). The most detailed description is the quantum manifold (Qm), whose algebra is the free Clifford algebra S of quantum set theory. We surmise that each Σm is a classical “condensation” of a Qm. Quantum simplices have both integer and half-integer spins in their spectrum. A quantum set theory of nature requires a series of reductions leading from the Qm and a world descriptor W up through the intermediate Σm and Δm to a dm and an action principle. What may be a new algebraic language for topology, classical or quantum, is a by-product of the work.

The KS-transformation in hypercomplex form and the quantization of the negative-energy orbit manifold of the Kepler problem

In a previous note we have shown that the KS-transformation, introduced by Kustaanheimo and Stiefel into Celestial Mechanics for the regularization of the Kepler problem, may be formulated in terms of hypercomplex numbers as the product of a quaternion and its anti-involute, thus representing a particular morphism of the real algebra of quaternions-having for image the physical configuration space of the Kepler problem. In the present note we show, first, that this formulation allows a straight derivation of the Hopf fibering of the sphere S3 (characterized by unit quaternions) having the base space given by the sphere S2 (characterized by unit vectors), and secondly that the KS-transformation allows the ‘quantization of the symplectic manifold S2’ in the sense of Souriau, the associated quantum manifold S3 having a contact structure given by the bilinear relation characteristic of the KS-theory. Furthermore, after presenting a natural extension of the hypercomplex KS-transformation to the full phase space of the Kepler problem, we show that this extension allows the quantization of the manifold of Kepler orbits of fixed negative energy (manifold diffeomorphic to the symplectic product S2×S2). The energy levels satisfy a well known quantum integrality condition and the associated quantum manifold is diffeomorphic to the product manifold S3×S3 quotiented by a suitable equivalence relation.