Reviews of Articles on

Quantum Topodynamics

" Quantum Dynamics of the Space" Washington, DC 2005.

*Functional Integral Formalism of Quantum Topodynamics *The Quantum Set *The Quantum Topological Group *Quantum Toplogical Structures *Hilbert Ortholattice *The Quantum Processor

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 (F-transformation), and into the projection of rays of D and Q from the quantum space (F-representation). A 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). 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 NxN 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.

" Theory of the Functional Space" Washington, DC 2005.

*Functional Space *Inner Product and Global Properties of the Fundamental Set *Commutation Relations and Topological Structures

Functional space is different from Hilbert space; it has global properties that Hilbert space does not have.

" The Dirac Quantum Field" Amsterdam 2001.

*The Linear Dynamical Operator (Complex) *Hermiticity (im) *The Quantum Hamiltonian (Positive Definite) *The Dirac Hamiltonian (Dirac Matrices) *Bilinears (Modulus) *Dirac Spinors (Complex Numbers) *Charge Conjugation (+- Symmetry) *Propagators (2im) *Applications (Convergence).

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" Amsterdam 1999.

*Review of the Standard Formalism (States, Mappings, Operators) *The Quantum Set (Fourier Representation, Mappings, Open Sets) *Algebraic Structure (Quantum Topological Group, Group Manifold, Gauge Field) *Lattice Structure (Hilbert Orthologic, Properties, Complexity) *Quantum Topology (Quantum Space, Representation Space, Quantum Conditions) *Functional Integration (Phase Angels, Local Dynamics, Cosmology) *Quantum Computation (Coding, Quantum Interference, Holography).

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 integration. 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" Rotterdam 1997.

*Group Structure (Phase Angle, Compactness) *Yang-Mills-Einstein Field (Commutation Relations).

Quantum topological origin of gravitation and the compact nature of the quantum space are exposed.

Applications of quantum topoldynamical principles to derive Einstein's gravitational field along with Yang-Mills gauge field from a compact group structure.

" Differential Topology in Quantum Space" Rotterdam 1997.

*Functional Space (Fourier Representation, Transformations) *Differential Topology (Operators).

Analysis on the quantum set requires differential topology and functional analysis.

Fourier representation of the functional space and the operators require differential topological manifold.

" Quantum Topodynamics" Rotterdam 1997.

*The Set (Fourier Representation) *The Group Structure (Quantum Conditions, Commutation Relations, Graded Lie Algebra) *The Logic (Open Sets, Orthomodular Lattice, Information Capacity) *The Topology (Denumerability, Measurability) *The Functional Integral (Feynman Integral, Fourier Representation, Group Structure, Quantum Holography).

The algebraic structures on the quantum set that incorporate a quantum topological group with its group manifold and gauge field.

Elements of the quantum set are Fourier representation of the translation operators. Commutation relations define a quantum topological group built on the set with its graded Lie manifold and gauge field. Continuous mapping represented as a logical operation define the orthomodular structure built on the open sets. The exponential information capacity of the quantum space is the source of complexity. Functional space is denumerable and measurable in contrast to Hilbert space. In this representation we have a quantum topological group structure with compact phases and denumerable Fourier representation to perform the integration.

" Quantum Topology" Cairo 1989.

*Quantum Space (h Measure) *Representation Space (Reisz, <|>) *Projections ( Subspaces *Fourier Transformations *Translation Operators) *Functional Integrals (Functional Space, Convolution Integral) *Feynman Integral (Action) *Extended Symmetry (T->T, Extended Poincare).

Quantum theory is derived from the topology of the quantum space.

The measure h generates a functional quantum space. The complete Dirac bracket is abstract representation of the quantum space. Translation operators represent subspaces of the quantum space and they are connected via Fourier transformations. Dynamical operators have distributions in functional space via functional integral. This distribution is defined by the classical action. Topological transformations lead to extended Poincare group.

" A Model of the Quantum Computer" Amsterdam 1999.

*The Coding *The Input Matrix (Fundamental Group) *The Processor *The Quantum Computer.

Information is encoded in the Lie algebra of the topological group. Parallel unitary transformations are performed on the open sets and quantum interference generates quantum holographs which are the functional integrals.

" A Model of the Quantum Mind" Cairo 1986.

*Quantum Logic *Logic of the Sutras *The Group *Topological Solitons *The Model.

Quantum structure of the mind is based on the ortholattice structure of the quantum set and orthological structure of the sutras and formation of the topological solitons that are relevant to living systems.