Abstracts of Articles
Diaa A Ahmed
e-mail: diahmed@yahoo.com
Quantum theory demonstrated that the manifold and the dynamics are connected in such a mathematical manner that vectors and their duals are connected in the theory of functional spaces. "Quantum Topology" is an extension of the space-time manifold into a functional space that incorporates the quantum dynamics; the quantum space. The quantum manifold is expressed as an antilinear-bilinear form. Quantum space is the invariant arena where physical interactions are projected from.
Set theory offer a sound mathematical foundation to study topology, gauge group, and quantum logic structures of the quantum space. A coherent theory should find one mathematcal structure on the set that can be represented as topology, gauge group, and quantum logic to account for the dynamics. Fourier analysis gives us an insight into the connectedness of D and Q (Fourier Transformation), and into the projection of rays of D and Q from the quantum space (Fourier Representation). Quantum set is defined as the 2-fold infinite set of the dual coordinates of the quantum space D and Q provided by the Fourier representation. Topological structures; both quantum and classical are defined by commutation relations of D and Q. Continuous mathematical transformations on the quantum set generates a topological group that gives rise to a compact group manifold and a gauge field (Fibre Bundle Representation). To take account for the fibre bundle structure of the dynamics in quantum space, the group structure of the quantum set is introduced into the functional integral to formulate "Quantum Topodynamics".
The quantum logic approach to formulate the dynamics is to represent the continuous mathematical operation on the quantum set as logical operation, and to represent the algebraic structure as an orthomodular structure. This approach translates the group structure into the language of quantum logic (quantum numbers; properties) and gives us an insight into quantum computation and, a criterion for the finiteness of the functional integration in the theory on the basis of the global properties of the functional space. An immediate application to this is to use the fibre bundle structure for quantum computation. First, to represent the fundamental logical operation as quantum interference, and then to reflect the group structure in a matrix quantum interference devise, provided we devise an approperiate coding for the quantum numbers. This matrix processor allows N x N gauge potentials to act on the phases of N rays, and the quantum interference of these rays is equivelant to performing functional integration to generate continuous holographic output that represents mathematically the topology of the quantum state being computed.
Functional space is different from Hilbert space; it has global properties that Hilbert space does not have.
According to P A M Dirac (1966) the divergences arise from the running out of countability of the space of the states due to the violent fluctuations in the quantum field. The global topological property of the functional space does not depend on the non-countability of the space the states. I attempt here at a correct representation of the dynamical operator of the Dirac quantum field in quantum space. We will consider representation of the linear dynamical operator as a complex variable for the reasons stated here and see the result of this consideration on our understanding of the fluctuations and the divergences. Quantum space is in one to one correspondence with the representation space of quantum theory, and the quantum dynamical operator will be represented by a complex variable like the state vector of the representation space and therefore it will represent purely quantum dynamics in the quantum space. Hermitian operators oscillate in a complex plane around their real values. The mass term in the Hamiltonian represents an oscillatory component in the complex plane and contributes spin current. Spinors are anticommutative extension of complex numbers. Charge conjugation is a symmetry between the positive and negative solutions along the imaginary axes. The new field propagator we study has formal symmetry and more sensible behavior, then, we will apply this new field propagator to study a finite field theory (this article will be revised and completed in a later date).
Theory of the quantum space is founded on the theory of sets. The quantum of action is a measure on the quantum set that generates the quantum space. In quantum space there exist dual spaces p and q defined by the translation operators subject to the quantum conditions and their Fourier representation provide the physical elements of the quantum set. The algebraic structure (commutation relations) unfolds the quantum topological group structure with its graded Lie manifold and gauge field. Representing the continuous mappings as logical operations permit us to represent the algebraic structure as a Hilbert ortholattice with the Fourier components as its rays and the open sets translated as the properties of the Hilbert orthologic. Exponential information capacity of the functional space is the source of complexity. Topology of the complete Dirac bracket represent the space of distributions of the dynamical operators. The criterion for convergence of functional integration performed on a quantum topological group rests on the measure, compactness and denumerablity of the space. The role that the holographic representation and phase angels play in gravitation and cosmology through constructing the manifold from rays and functional integral. Translating the group structure into orthomodular structure and coding of the generators. Parallel unitary transformations act on the rays and quantum interference processors.
Gauge Theory of Gravitation" deals with gravitation as a quantum topological phenomenon on a large scale. Introducing the action of gravitation into functional space -through a phase angle- requires a change in our understanding of both gauge theory and general relativity. This displays its unity with the rest of gauge interactions and demonstrate that the manifold is compact.
Quantum Topodynamics" deals with the set that underlies the assumed oneness of the topology, group and logic structure of quantum space. The elements of the basic set of the quantum space are identified as the Fourier components of dual linear forms D, X. . Dynamics is identified as quantum mappings (commutation relations). The elements of the set 'rays' are the basis of topology and the quantum computer, it encodes the logic and perform functional integration, the physical picture based on holography permits for computability and the building up of the quantum computer.
Differential Topology in Quantum Space" deals with the method of analysis appropriate for the quantum space. To sum up; continuous transformations; Poincare-Lorentz, gauge, renormalization group, are needed to supplement functions and their function spaces, in fact it needed a continuous infinity of these and their duals. The concept of functional d -space; as a model is a central idea to understand the transformations and the physical nature of the space. Functions and geometry are no longer representing physical reality.
Each point is in fact a ray, a Fourier component of D, the metric itself should be represented in this way. This is a requirement of quantum mechanics, classical and quantum field theory and the Fourier representation of the space -an outcome of the transformations. The holographic principle recovers the Reimannian Einsteinian structure of the manifold. Charges act as solitons or holograms in the functional space; project a specific type of rays from the functional space through a certain phase angle, and here again the holographic principle recovers the geometric structure of the fields.
Quantum Topology" deals with the general quantum theory as the theory of quantum space. On the quantum level space-time and energy-momentum forms form a connected manifold; a functional quantum space. Many problems in quantum theory and field theory flow from not perceiving this symmetry and the functional nature of the quantum space.
Both topology, groups and logic are based on the concept of sets. If properties coincide with the open sets of topology, then logic and topology will have the same structure. If transformations are continuous in topology, then we will have topological groups; we can derive fields, therefore, quantum logic underlies the manifold and the fields and nature is based on the language of quantum logic.
Quantum theory and field theory based on sets and the derived topology, group and logic structures should address the question of computation and the mind; the quantum computer and the quantum mind.
The model of the quantum computer is developed on the fundamental set of the quantum space. The basic structure is the Lie algebra of the generators of the fundamental group where information is encoded. Parallel unitary transformations are performed on the elements of the set 'rays' and quantum interference perform the functional integration and the output are quantum holographs.
A Model of the Quantum Mind" is introduced in conjunction with the study of "Quantum Topology"; the theory of the quantum space on the basis of set theory. Some new developments are based on the logical structure of the "Yoga Sutras of Patanjali"; the ancient text on yoga psychology that describe the development of existential states of the mind; ones that have the nature of existence itself.
Quantum mind is the source of intuition; that discriminates the set into subsets, giving each a different flavor and realizes their morphisms. It sees all the histories in a functional space and when chooses a particular one, attention actualize and project it.
Computations are performed on the fabrics of topology, and stable structures -solitons- store the information and play the role of providing a macroscopic quantum computer -DNA- and macroscopic quantum effects.
All these morphisms are implicit in the set and the logic that is chosen is the logic of complexity; quantum logic.
Topology is for transformations. Global aspects derive from that.
Functional is the global aspect of functions. Action is global, integral, functional. Functional space amounts to infinite freedom. Each one of the solutions amounts to a choice. All exist in functional space.
Finiteness of h means that quantum space exists.
Quantum space is imaginary (complex). Dynamics of quantum space is conceptual (mappings). Quantum space animates the dynamics.
Quantum space is not space-time nor doe's it exists in space-time; it projects space-time; space-time is a construct in quantum space, construct of logic; mathematical logic.
This is a freedom that quantum space allows us; to think of existence as a construct of logic, and the logic exists in functional space.