Quantum Geometry
&
New Concept of Space
QHtml Article
Quantum geometry is a generalization and extension of
classical geometry. It incorporates various ideas and concepts of
quantum physics, into the world of geometry. Quantum geometry deals with
quantum spaces. In classical geometry spaces are always understandable
as collections of points equipped with the appropriate additional structure. Quantum
spaces are not viewable in this way. In general, quantum spaces have no
points at all! They exhibit nontrivial quantum fluctuations of geometry
at all scales. At the formal level, quantum spaces are described by certain
noncommutative complex *algebras. The elements of these algebras are
interpretable as functions over the associated quantum spaces.
Classical geometry is the commutative sector of quantum geometry.
It is believed that quantum geometry could provide a consistent
description of spacetime at the level of ultrasmall distances where classical
concepts of the spacetime continuum are not applicable. In principle, this could give
the appropriate mathematical framework to formulate a coherent quantum
theory of fundamental interactions.



Subquantum Physics: World
Beyond the LimitsOf Uncertainty
Relations
Philosophical Article
Subquantum theories pretend to go deeper than quantum
mechanics. The idea is to provide a picture of physical reality which is based on
individual physical systems, completely causal, and statistically
compatible with quantum mechanics. Such a subquantum theory is logically
possible. A natural mathematical framework for developing the subquantum theory
is given by special extensions of C*algebras of quantum observables. These
structures naturally incorporate complementarity feature of
quantum mechanics and allow us to introduce consistently (thanks to a special
contextuality property of the subquantum world) the concept of a subquantum space. This
space consists of all possible subquantum states of a given physical
system. If a subquantum state of the system is known, then the values of
all quantum observables for this system are completely determined.
As a special topic, it is explained how to incorporate the principle of
locality into the subquantum theory, using the appropriate nonclassical
probability theory on the subquantum space and overcoming the obstacles
given by Bell's inequalities.


 

Quantum Principal Bundles
Fibered Article
This section is devoted to my theory of quantum
principal bundles. These objects constitute a very important class
of quantum spaces. The basic conceptual picture is taken from classical
geometrythe idea of a fibered space equipped with a free right action
of a Lie group, so that the fibers are the orbits of the action. In the
framework of quantum principal bundles, all the fibered space, the base manifold
and the structure
group are considered as quantum objects. The formalism of quantum principal
bundles provides powerful tools for the study of the internal structure
of quantum spaces, and gives a coherent mathematical framework for a formulation
of the theory of fields and particles over a quantum spacetime. Classical
spaces are viewable in the new light, too.



Diagrammatic Quantum Groups
Diagrammatic
Article
In this lecture, a diagrammatic formulation of generalized Hopf
algebras is outlined. The formalism covers various examples of quantum spaces
equipped with a grouplike structure, going far beyond the framework of standard braided
categories. In particular, the theory includes completely "pointless" objects,
overcoming an inherent geometrical inhomogeneity of standard structures. All
morphisms are built from the product and the coproduct maps.
Furthermore, instead of just one braiding, an infinite system of mutually related braid
operators emerges, expressing twisting properties of the product and coproduct. All braid type
identities are consequences of simple axioms. Another interesting feature of the theory is its
invariance under tensoring with matrix algebras.



 

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Quantum Activities
This section includes the information about my weekly Seminar
on Quantum Geometry at the Institute of Mathematics of UNAM.
It also features a
basic sinopsis of my periodic onesemestral course entitled "Numbers, Space,
Time and Light" presented at Mexican National Arts Center (CENART) and devoted
to exploring connections between Arts, Mathematics & Physics.

